Mini GMAT Quant. Test - 10 Questions

1. A student's average (arithmetic mean) test score on 4 tests is 78. What must be the student's score on a 5th test for the student's average score on the 5 tests to be 80?

80

82

84

86

88

To find the student's score on the 5th test, we can use the formula for average: (sum of all scores) / (number of tests). The sum of the scores on the first 4 tests is 78 * 4 = 312. To have an average score of 80 on 5 tests, the sum of all 5 scores must be 80 * 5 = 400. Therefore, the student's score on the 5th test must be 400 - 312 = 88.

Explanation

To find the student's score on the 5th test, we can use the formula for average: (sum of all scores) / (number of tests). The sum of the scores on the first 4 tests is 78 * 4 = 312. To have an average score of 80 on 5 tests, the sum of all 5 scores must be 80 * 5 = 400. Therefore, the student's score on the 5th test must be 400 - 312 = 88.

2.

Lines m and n intersect, as shown in the figure above. What is the value of u - w? (1) u = 90 (2) Lines m and n are perpendicular.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

From statement (1), we know that u = 90. Since u is one of the angles formed by the intersection of lines m and n, we can conclude that u is a right angle. Therefore, w must be the complementary angle to u, which means w = 90. Therefore, u - w = 90 - 90 = 0.

From statement (2), we know that lines m and n are perpendicular. This means that the angles formed by their intersection must be right angles. Therefore, u is a right angle and w is a right angle. Again, u - w = 90 - 90 = 0.

In both cases, we can determine that u - w = 0, so each statement alone is sufficient to answer the question.

Explanation

From statement (1), we know that u = 90. Since u is one of the angles formed by the intersection of lines m and n, we can conclude that u is a right angle. Therefore, w must be the complementary angle to u, which means w = 90. Therefore, u - w = 90 - 90 = 0.

From statement (2), we know that lines m and n are perpendicular. This means that the angles formed by their intersection must be right angles. Therefore, u is a right angle and w is a right angle. Again, u - w = 90 - 90 = 0.

In both cases, we can determine that u - w = 0, so each statement alone is sufficient to answer the question.

3. On a map Town G is 10 centimeters due east of Town H and 8 centimeters due south of Town J. Which of the following is closest to the straight-line distance, in centimeters, between Town H and Town J on the map?

6

13

18

20

24

Town G is located 10 centimeters due east of Town H and 8 centimeters due south of Town J. To find the straight-line distance between Town H and Town J, we can draw a right-angled triangle with Town G as the right angle. The distance between Town H and Town J can be found using the Pythagorean theorem, which states that the square of the hypotenuse (the straight-line distance) is equal to the sum of the squares of the other two sides. In this case, the distance between Town H and Town J is the hypotenuse, and the distances due east and due south from Town G are the other two sides. Using the Pythagorean theorem, we can calculate that the straight-line distance between Town H and Town J is closest to 13 centimeters.

Explanation

Town G is located 10 centimeters due east of Town H and 8 centimeters due south of Town J. To find the straight-line distance between Town H and Town J, we can draw a right-angled triangle with Town G as the right angle. The distance between Town H and Town J can be found using the Pythagorean theorem, which states that the square of the hypotenuse (the straight-line distance) is equal to the sum of the squares of the other two sides. In this case, the distance between Town H and Town J is the hypotenuse, and the distances due east and due south from Town G are the other two sides. Using the Pythagorean theorem, we can calculate that the straight-line distance between Town H and Town J is closest to 13 centimeters.

4. What is the total value of Company H&rsquos stock? (1) Investor P owns of the shares of Company H&rsquos total stock. (2) The total value of Investor Q&rsquos shares of Company H&rsquos stock is $16000.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The total value of Company H's stock cannot be determined based on the given statements alone. Statement (1) only provides information about the ownership of the shares by Investor P, but it does not give any information about the total value of the stock. Statement (2) only provides information about the total value of Investor Q's shares, but it does not give any information about the total value of the stock or the ownership of other investors. Therefore, both statements together are not sufficient to determine the total value of Company H's stock.

Explanation

The total value of Company H's stock cannot be determined based on the given statements alone. Statement (1) only provides information about the ownership of the shares by Investor P, but it does not give any information about the total value of the stock. Statement (2) only provides information about the total value of Investor Q's shares, but it does not give any information about the total value of the stock or the ownership of other investors. Therefore, both statements together are not sufficient to determine the total value of Company H's stock.

5. If the length of a certain rectangle is 2 greater than the width of the rectangle, what is the perimeter of the rectangle? (1) The length of each diagonal of the rectangle is 10. (2) The area of the rectangular region is 48.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) alone is sufficient to find the perimeter of the rectangle. If the length of each diagonal is 10, we can use the Pythagorean theorem to find the relationship between the length and width of the rectangle. Since the length is 2 greater than the width, we can set up the equation x^2 + (x+2)^2 = 10^2, where x is the width. Solving this equation will give us the width, and we can then calculate the perimeter using the formula P = 2(length + width). Therefore, statement (1) alone is sufficient to find the perimeter.

Explanation

Statement (1) alone is sufficient to find the perimeter of the rectangle. If the length of each diagonal is 10, we can use the Pythagorean theorem to find the relationship between the length and width of the rectangle. Since the length is 2 greater than the width, we can set up the equation x^2 + (x+2)^2 = 10^2, where x is the width. Solving this equation will give us the width, and we can then calculate the perimeter using the formula P = 2(length + width). Therefore, statement (1) alone is sufficient to find the perimeter.

6. If x and y are integers, is x + y greater than 0? (1) x is greater than 0. (2) y is less than 1.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

From statement (1), we know that x is greater than 0. However, this does not provide any information about the value of y. Therefore, statement (1) alone is not sufficient to determine whether x + y is greater than 0.

From statement (2), we know that y is less than 1. However, this does not provide any information about the value of x. Therefore, statement (2) alone is not sufficient to determine whether x + y is greater than 0.

Combining both statements, we still do not have enough information to determine whether x + y is greater than 0. Therefore, both statements together are not sufficient.

Explanation

From statement (1), we know that x is greater than 0. However, this does not provide any information about the value of y. Therefore, statement (1) alone is not sufficient to determine whether x + y is greater than 0.

From statement (2), we know that y is less than 1. However, this does not provide any information about the value of x. Therefore, statement (2) alone is not sufficient to determine whether x + y is greater than 0.

Combining both statements, we still do not have enough information to determine whether x + y is greater than 0. Therefore, both statements together are not sufficient.

7. A glass was filled with 10 ounces of water, and 0.01 ounce of the water evaporated each day during a twenty-day period. What percent of the original amount of water evaporated during this period?

0.002%

0.02%

0.2%

2%

20%

During the twenty-day period, 0.01 ounce of water evaporated each day. To find the total amount of water evaporated, we multiply 0.01 ounce by 20 days, which equals 0.2 ounces. To find the percentage, we divide the amount evaporated (0.2 ounces) by the original amount of water (10 ounces) and multiply by 100. This gives us a percentage of 2%.

Explanation

During the twenty-day period, 0.01 ounce of water evaporated each day. To find the total amount of water evaporated, we multiply 0.01 ounce by 20 days, which equals 0.2 ounces. To find the percentage, we divide the amount evaporated (0.2 ounces) by the original amount of water (10 ounces) and multiply by 100. This gives us a percentage of 2%.

8. At a garage sale, all of the prices of the items sold were different. If the price of a radio sold at the garage sale was both the 15th highest price and the 20th lowest price among the prices of the items sold, how many items were sold at the garage sale?

33

34

35

36

37

not-available-via-ai

Explanation

not-available-via-ai

9. If and which of the following is NOT a possible value of ?

-2

-1

0

1

2

The possible values of x are -2, -1, 0, and 2. Therefore, the value 1 is not a possible value for x.

Explanation

The possible values of x are -2, -1, 0, and 2. Therefore, the value 1 is not a possible value for x.

10. A farm used two harvesting machines, H and K, to harvest 100 acres of wheat. Harvesting machine H, working alone at its constant rate, harvested 40 acres of wheat in 8 hours. Then harvesting machine K was brought in, and harvesting machines H and K, working together at their respective contant rates, harvested the remaining acres of wheat in 5 hours. Harvesting machine K harvested how many acres of wheat per hour? (1) Working alone at its constant rate, pump X would have taken 80 minutes to fill the tank with water. (2) Working alone at its constant rate, pump Y would have taken 120 minutes to fill the tank with water.

7

8

12

13

15

not-available-via-ai

Explanation

not-available-via-ai

11. If S is a set of ten consecutive integers, is the integer 5 in S? (1) The integer –3 is in S. (2) The integer 4 is in S.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

From statement (1), we know that the integer -3 is in S. Since S is a set of ten consecutive integers, we can determine that the lowest integer in S is -3 and the highest integer in S is 6. However, we still don't have enough information to determine if the integer 5 is in S.

From statement (2), we know that the integer 4 is in S. This gives us some information about the range of S, but it still doesn't tell us if the integer 5 is in S.

By combining both statements, we know that the lowest integer in S is -3 and the highest integer in S is 6, and that the integer 4 is in S. However, we still don't have enough information to determine if the integer 5 is in S. Therefore, both statements together are not sufficient to answer the question.

Explanation

From statement (1), we know that the integer -3 is in S. Since S is a set of ten consecutive integers, we can determine that the lowest integer in S is -3 and the highest integer in S is 6. However, we still don't have enough information to determine if the integer 5 is in S.

From statement (2), we know that the integer 4 is in S. This gives us some information about the range of S, but it still doesn't tell us if the integer 5 is in S.

By combining both statements, we know that the lowest integer in S is -3 and the highest integer in S is 6, and that the integer 4 is in S. However, we still don't have enough information to determine if the integer 5 is in S. Therefore, both statements together are not sufficient to answer the question.

12. If x is equal to the sum of the even integers from 40 to 60, inclusive, and y is the number of even integers from 40 to 60, inclusive, what is the value of x+y ?

550

551

560

561

572

The value of x is equal to the sum of the even integers from 40 to 60, inclusive. The even integers in this range are 40, 42, 44, 46, 48, 50, 52, 54, 56, and 60. Adding these numbers together gives us a total of 502.

The value of y is the number of even integers from 40 to 60, inclusive. Counting these numbers gives us a total of 11 even integers.

To find the value of x+y, we simply add 502 and 11 together, which equals 513. Therefore, the correct answer is 513.

Explanation

The value of x is equal to the sum of the even integers from 40 to 60, inclusive. The even integers in this range are 40, 42, 44, 46, 48, 50, 52, 54, 56, and 60. Adding these numbers together gives us a total of 502.

The value of y is the number of even integers from 40 to 60, inclusive. Counting these numbers gives us a total of 11 even integers.

To find the value of x+y, we simply add 502 and 11 together, which equals 513. Therefore, the correct answer is 513.

13. In how many different ways can a chairperson and a secretary be selected from a committee of 9 people?

24

36

72

80

120

In order to select a chairperson and a secretary from a committee of 9 people, we need to consider the order in which they are selected. The chairperson can be selected in 9 different ways (as there are 9 people to choose from), and once the chairperson is selected, the secretary can be selected in 8 different ways (as there are now only 8 people left to choose from). Therefore, the total number of ways to select a chairperson and a secretary is 9 * 8 = 72.

Explanation

In order to select a chairperson and a secretary from a committee of 9 people, we need to consider the order in which they are selected. The chairperson can be selected in 9 different ways (as there are 9 people to choose from), and once the chairperson is selected, the secretary can be selected in 8 different ways (as there are now only 8 people left to choose from). Therefore, the total number of ways to select a chairperson and a secretary is 9 * 8 = 72.

14. Three printing presses, R, S, and T, working together at their respective constant rates, can do a certain printing job in 4 hours. S and T, working together at their respective constant rates, can do the same job in 5 hours. How many hours would it take R, working alone at its constant rate, to do the same job?

8

10

12

15

20

If R, S, and T can complete the job together in 4 hours, it means that their combined rate is 1/4 of the job per hour. If S and T can complete the job together in 5 hours, their combined rate is 1/5 of the job per hour. Subtracting the combined rate of S and T from the combined rate of R, S, and T gives the rate of R alone, which is 1/4 - 1/5 = 1/20 of the job per hour. Therefore, it would take R 20 hours to complete the job alone.

Explanation

If R, S, and T can complete the job together in 4 hours, it means that their combined rate is 1/4 of the job per hour. If S and T can complete the job together in 5 hours, their combined rate is 1/5 of the job per hour. Subtracting the combined rate of S and T from the combined rate of R, S, and T gives the rate of R alone, which is 1/4 - 1/5 = 1/20 of the job per hour. Therefore, it would take R 20 hours to complete the job alone.

15. Lines k and m are parallel to each other. Is the slope of line k positive? (1) Line k passes through the point (3, 2). (2) Line m passes through the point (-3, 2).

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The slope of a line can be determined by comparing the y-coordinates of two points on the line and the x-coordinates of the same two points. In this case, neither statement alone provides enough information to determine the slope of line k. Statement (1) only provides the coordinates of one point on line k, while statement (2) only provides the coordinates of one point on line m. Therefore, both statements together are still not sufficient to determine the slope of line k.

Explanation

The slope of a line can be determined by comparing the y-coordinates of two points on the line and the x-coordinates of the same two points. In this case, neither statement alone provides enough information to determine the slope of line k. Statement (1) only provides the coordinates of one point on line k, while statement (2) only provides the coordinates of one point on line m. Therefore, both statements together are still not sufficient to determine the slope of line k.

16. Three thousand families live in a certain town. How many families who live in the town own neither a car nor a television set? (1) Of the families who live in the town, 2,980 own a car. (2) Of the families who live in the town, 2,970 own both a car and a television set.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

not-available-via-ai

Explanation

not-available-via-ai

17. Working independently at their respective constant rates, pumps X and Y took 48 minutes to fill an empty tank with water. What fraction of the water in the full tank came from pump X? (1) Working alone at its constant rate, pump X would have taken 80 minutes to fill the tank with water. (2) Working alone at its constant rate, pump Y would have taken 120 minutes to fill the tank with water.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) alone is sufficient to answer the question. Since pump X would have taken 80 minutes to fill the tank alone, and it took 48 minutes for both pumps to fill the tank, it means that pump Y took 48 - 80 = -32 minutes to fill the tank alone. This is not possible, so pump Y did not contribute any water to the tank. Therefore, all the water in the tank came from pump X.

Explanation

Statement (1) alone is sufficient to answer the question. Since pump X would have taken 80 minutes to fill the tank alone, and it took 48 minutes for both pumps to fill the tank, it means that pump Y took 48 - 80 = -32 minutes to fill the tank alone. This is not possible, so pump Y did not contribute any water to the tank. Therefore, all the water in the tank came from pump X.

18. In a poll involving two yes- or no-questions, all the people who were asked the two questions answered "yes" or "no". If 75 percent of these people answered "yes" to the first question, 55 percent of the people answered "yes" to the second question, and 20 percent of the people answered "no" to both questions, what percentage of the people answered "yes" to both questions?

70%

50%

25%

20%

10%

not-available-via-ai

Explanation

not-available-via-ai

19. Two different pipes fill the same pool. Pipe A fills of the capacity of the pool in 3 hours while pipe B fills of it in 6 hours. If the two pipes fill the pool simultaneously with their respective rates how long will it take for the pool to be filled to capacity?

3 hours

3 hours 20 minutes

3 hours 36 minutes

4 hours 20 minutes

4 hours 30 minutes

Pipe A fills the pool in 3 hours, which means it fills 1/3 of the pool's capacity per hour. Pipe B fills the pool in 6 hours, which means it fills 1/6 of the pool's capacity per hour. When both pipes are filling the pool simultaneously, their combined rate is 1/3 + 1/6 = 1/2 of the pool's capacity per hour. Therefore, it will take 2 hours for the pool to be filled to half of its capacity. Since the question asks for the pool to be filled to full capacity, it will take twice as long, which is 2 x 2 = 4 hours. Additionally, 1/2 hour is equal to 30 minutes, so the total time is 4 hours + 30 minutes = 4 hours 30 minutes.

Explanation

Pipe A fills the pool in 3 hours, which means it fills 1/3 of the pool's capacity per hour. Pipe B fills the pool in 6 hours, which means it fills 1/6 of the pool's capacity per hour. When both pipes are filling the pool simultaneously, their combined rate is 1/3 + 1/6 = 1/2 of the pool's capacity per hour. Therefore, it will take 2 hours for the pool to be filled to half of its capacity. Since the question asks for the pool to be filled to full capacity, it will take twice as long, which is 2 x 2 = 4 hours. Additionally, 1/2 hour is equal to 30 minutes, so the total time is 4 hours + 30 minutes = 4 hours 30 minutes.

20. If the symbol ▼ represents either addition, subtraction, multiplication, or division, what is the value of 6 ▼ 2? (1) 10 ▼ 5 = 2 (2) 4 ▼ 2 = 2

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

From statement (1), we know that 10 ▼ 5 = 2. However, we don't know the specific operation represented by ▼. Therefore, we cannot determine the value of 6 ▼ 2 based on statement (1) alone.

From statement (2), we know that 4 ▼ 2 = 2. Again, we don't know the specific operation represented by ▼. Therefore, we cannot determine the value of 6 ▼ 2 based on statement (2) alone.

When we consider both statements together, we still don't have enough information to determine the specific operation represented by ▼. Therefore, the value of 6 ▼ 2 cannot be determined.

Hence, statement (1) alone is sufficient, but statement (2) alone is not sufficient.

Explanation

From statement (1), we know that 10 ▼ 5 = 2. However, we don't know the specific operation represented by ▼. Therefore, we cannot determine the value of 6 ▼ 2 based on statement (1) alone.

From statement (2), we know that 4 ▼ 2 = 2. Again, we don't know the specific operation represented by ▼. Therefore, we cannot determine the value of 6 ▼ 2 based on statement (2) alone.

When we consider both statements together, we still don't have enough information to determine the specific operation represented by ▼. Therefore, the value of 6 ▼ 2 cannot be determined.

Hence, statement (1) alone is sufficient, but statement (2) alone is not sufficient.

21. A certain candy manufacturer reduced the weight of candy bar M by 20 percent but left the price unchanged. What was the resulting percent increase in the price per ounce of the candy bar M?

5%

10%

15%

20%

25%

When the weight of candy bar M is reduced by 20 percent but the price remains unchanged, the price per ounce of the candy bar increases. To find the resulting percent increase in price per ounce, we can use the formula: (New price per ounce - Original price per ounce) / Original price per ounce * 100. Since the price remains unchanged, the new price per ounce is the same as the original price per ounce. Therefore, the resulting percent increase in the price per ounce is 25%.

Explanation

When the weight of candy bar M is reduced by 20 percent but the price remains unchanged, the price per ounce of the candy bar increases. To find the resulting percent increase in price per ounce, we can use the formula: (New price per ounce - Original price per ounce) / Original price per ounce * 100. Since the price remains unchanged, the new price per ounce is the same as the original price per ounce. Therefore, the resulting percent increase in the price per ounce is 25%.

22. If 5x – 5x - 3 = (124)(5y), what is y in terms of x?

X

X - 6

- 3/620 of x

2x + 3

2x + 6

Explanation

23.

In the figure above, O is the center of the circle. If the area of the shaded region is 2π, what is the value of x?

To find the value of x, we need to determine the length of the arc of the shaded region. Since the area of the shaded region is given as 2π, we can use the formula for the area of a sector of a circle (A = (θ/360) * π * r^2) to find the angle of the sector. The area of the sector is equal to the area of the shaded region plus the area of the triangle formed by the radii of the sector. By subtracting the area of the triangle from the total area of the sector, we can find the area of the shaded region. Setting this equal to 2π, we can solve for the angle of the sector, which is equal to x.

Explanation

To find the value of x, we need to determine the length of the arc of the shaded region. Since the area of the shaded region is given as 2π, we can use the formula for the area of a sector of a circle (A = (θ/360) * π * r^2) to find the angle of the sector. The area of the sector is equal to the area of the shaded region plus the area of the triangle formed by the radii of the sector. By subtracting the area of the triangle from the total area of the sector, we can find the area of the shaded region. Setting this equal to 2π, we can solve for the angle of the sector, which is equal to x.

24. Each of the numbers w x y and z is equal to either 0 or 1. What is the value of (1) (2)

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Since each of the numbers w, x, y, and z can only be either 0 or 1, the value of w & x & y & z will be 0 if any of the variables is 0, and it will be 1 only if all the variables are 1. Therefore, knowing the value of any one variable (either from statement 1 or statement 2) is sufficient to determine the value of w & x & y & z. Hence, each statement alone is sufficient.

Explanation

Since each of the numbers w, x, y, and z can only be either 0 or 1, the value of w & x & y & z will be 0 if any of the variables is 0, and it will be 1 only if all the variables are 1. Therefore, knowing the value of any one variable (either from statement 1 or statement 2) is sufficient to determine the value of w & x & y & z. Hence, each statement alone is sufficient.

25. 70, 75, 80, 85, 90, 105, 105, 130,130,130 The list shown consists of the times, in seconds, that it took each of 10 students to run a distance of 400 meters. If the standard deviation of the 10 running times is 22.4 sec, rounded to nearest tenth of a second , how many of the 10 running times are more than 1 standard deviation below the mean of the 10 running times?

One

Two

Three

Four

Five

The mean of the 10 running times can be calculated by summing all the times and dividing by 10. The standard deviation is a measure of how spread out the data is from the mean. If a value is more than 1 standard deviation below the mean, it means that it is significantly lower than the average. In this case, since the standard deviation is 22.4 sec, any running time that is more than 22.4 sec below the mean would be considered more than 1 standard deviation below. Looking at the list, there are two running times that meet this criteria, which are 70 sec and 75 sec. Therefore, the answer is Two.

Explanation

The mean of the 10 running times can be calculated by summing all the times and dividing by 10. The standard deviation is a measure of how spread out the data is from the mean. If a value is more than 1 standard deviation below the mean, it means that it is significantly lower than the average. In this case, since the standard deviation is 22.4 sec, any running time that is more than 22.4 sec below the mean would be considered more than 1 standard deviation below. Looking at the list, there are two running times that meet this criteria, which are 70 sec and 75 sec. Therefore, the answer is Two.

26. On Saturday morning, Malachi will begin a camping vacation and he will return home at the end of the first day on which it rains. If on the first three days of the vacation the probability of rain on each day is 0.2, what is the probability that Malachi will return home at the end of the day on the following Monday?

0.008

0.128

0.488

0.512

0.640

The probability of Malachi returning home at the end of the first day it rains is 0.2. Since it is given that Malachi will return home at the end of the first day it rains, the probability of it raining on the first day is 0.2. Therefore, the probability of it not raining on the first day is 1 - 0.2 = 0.8. Since it is not specified when the camping vacation starts, we assume it starts on Saturday. Therefore, the probability of it not raining on Saturday, Sunday, and Monday is (0.8)^3 = 0.512. Therefore, the probability of Malachi returning home at the end of the day on Monday is 0.512 * 0.2 = 0.1024, which is approximately 0.128.

Explanation

The probability of Malachi returning home at the end of the first day it rains is 0.2. Since it is given that Malachi will return home at the end of the first day it rains, the probability of it raining on the first day is 0.2. Therefore, the probability of it not raining on the first day is 1 - 0.2 = 0.8. Since it is not specified when the camping vacation starts, we assume it starts on Saturday. Therefore, the probability of it not raining on Saturday, Sunday, and Monday is (0.8)^3 = 0.512. Therefore, the probability of Malachi returning home at the end of the day on Monday is 0.512 * 0.2 = 0.1024, which is approximately 0.128.

27. Of the families in City X in 1994, 40 percent owned a personal computer. The number of families in City X owning a computer in 1998 was 30 percent greater than it was in 1994, and the total number of families in City X was 4 percent greater in 1998 than it was in 1994. What percent of the families in City X owned a personal computer in 1998?

50%

52%

56%

70%

74%

In 1994, 40% of the families in City X owned a personal computer. In 1998, the number of families owning a computer increased by 30% compared to 1994. Additionally, the total number of families in City X increased by 4% in 1998 compared to 1994. Therefore, the percentage of families owning a personal computer in 1998 can be calculated by adding the 30% increase to the initial 40% ownership rate. This gives us a total of 70% ownership. However, since the total number of families also increased by 4%, the percentage of families owning a personal computer in 1998 will be slightly less than 70%. The closest option is 50%, which is the correct answer.

Explanation

In 1994, 40% of the families in City X owned a personal computer. In 1998, the number of families owning a computer increased by 30% compared to 1994. Additionally, the total number of families in City X increased by 4% in 1998 compared to 1994. Therefore, the percentage of families owning a personal computer in 1998 can be calculated by adding the 30% increase to the initial 40% ownership rate. This gives us a total of 70% ownership. However, since the total number of families also increased by 4%, the percentage of families owning a personal computer in 1998 will be slightly less than 70%. The closest option is 50%, which is the correct answer.

28. Joshua and Jose work at an auto repair center with 4 other workers. For a survey on healthcare insurance, 2 of the 6 workers will be randomly chosen to be interviewd. What is the probability that Joshua and Jose both will be chosen?

The probability that Joshua will be chosen is 1/6, since there are 2 workers out of 6 who will be chosen. The probability that Jose will be chosen is 1/5, since there will be 1 worker out of 5 left to be chosen after Joshua is chosen. Therefore, the probability that both Joshua and Jose will be chosen is (1/6) * (1/5) = 1/30.

Explanation

The probability that Joshua will be chosen is 1/6, since there are 2 workers out of 6 who will be chosen. The probability that Jose will be chosen is 1/5, since there will be 1 worker out of 5 left to be chosen after Joshua is chosen. Therefore, the probability that both Joshua and Jose will be chosen is (1/6) * (1/5) = 1/30.

29. The operation ○ is defined by the equation = where y &ne -x. If then =

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Explanation

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30. In the xy-plane line l and line k intersect at the point . What is the slope of line l? (1) The product of the slopes of line l and line k is &ndash1. (2) Line k passes through the origin.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The slope of line l can be determined by using both statements together. Statement (1) tells us that the product of the slopes of line l and line k is -1. This means that the slopes of the two lines are negative reciprocals of each other. Statement (2) tells us that line k passes through the origin. Since line k passes through the origin, its y-intercept is 0, which means its slope is also 0. Therefore, the slope of line l can be determined by finding the negative reciprocal of 0, which is undefined. Thus, both statements together are sufficient to determine the slope of line l.

Explanation

The slope of line l can be determined by using both statements together. Statement (1) tells us that the product of the slopes of line l and line k is -1. This means that the slopes of the two lines are negative reciprocals of each other. Statement (2) tells us that line k passes through the origin. Since line k passes through the origin, its y-intercept is 0, which means its slope is also 0. Therefore, the slope of line l can be determined by finding the negative reciprocal of 0, which is undefined. Thus, both statements together are sufficient to determine the slope of line l.

31.

The figure above shows two entries, indicated by m and n, in an addition table. What is the value of n + m? (1) d + y = -3 (2) e + z = 12

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

From statement (1), we know that d + y = -3. However, this does not provide any information about the values of m and n.

From statement (2), we know that e + z = 12. Similarly, this does not provide any information about the values of m and n.

However, when we combine both statements, we can solve for the values of m and n. Since both equations involve different variables, we cannot directly solve for m and n. Therefore, BOTH statements TOGETHER are sufficient to determine the value of n + m.

Explanation

From statement (1), we know that d + y = -3. However, this does not provide any information about the values of m and n.

From statement (2), we know that e + z = 12. Similarly, this does not provide any information about the values of m and n.

However, when we combine both statements, we can solve for the values of m and n. Since both equations involve different variables, we cannot directly solve for m and n. Therefore, BOTH statements TOGETHER are sufficient to determine the value of n + m.

32. Which of the following is equal to ?

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Explanation

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33. If and are integers is ? (1) (2)

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The answer is both statements together are sufficient but neither statement alone is sufficient. This means that both statements provide enough information to answer the question, but each statement on its own does not provide enough information.

Explanation

The answer is both statements together are sufficient but neither statement alone is sufficient. This means that both statements provide enough information to answer the question, but each statement on its own does not provide enough information.

34. In the figure shown what is the area of the circular region with center O and diameter BC? (1) (2)

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

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Explanation

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35.

In the parallelogram shown, what is the value of x? (1) y = 2x (2) x + z = 120

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

From statement (1), we know that y = 2x. Since opposite angles in a parallelogram are equal, we can conclude that y = x. Therefore, statement (1) alone is sufficient to determine the value of x. We do not need statement (2) to find the value of x. Hence, the correct answer is that EACH statement ALONE is sufficient.

Explanation

From statement (1), we know that y = 2x. Since opposite angles in a parallelogram are equal, we can conclude that y = x. Therefore, statement (1) alone is sufficient to determine the value of x. We do not need statement (2) to find the value of x. Hence, the correct answer is that EACH statement ALONE is sufficient.

36. In the xy-plane, line k passes through the point (1,1) and line m passes through the point (1,-1).Are the lines k and m perpendicular to each other? (1) Lines k and m intersect at the point (1,-1). (2) Line k intersects the x axis at the point (1,0).

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The given statements are not sufficient to determine whether lines k and m are perpendicular to each other. Statement (1) tells us that the lines intersect at a certain point, but it does not provide any information about their slopes or angles. Statement (2) tells us that line k intersects the x-axis at a certain point, but it does not provide any information about the slope or angle of line m. Therefore, without additional information, we cannot determine whether the lines are perpendicular.

Explanation

The given statements are not sufficient to determine whether lines k and m are perpendicular to each other. Statement (1) tells us that the lines intersect at a certain point, but it does not provide any information about their slopes or angles. Statement (2) tells us that line k intersects the x-axis at a certain point, but it does not provide any information about the slope or angle of line m. Therefore, without additional information, we cannot determine whether the lines are perpendicular.

37. In the figure shown the length of line segment QS is . What is the perimeter of equilateral triangle PQR ?

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Explanation

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38. If r, s, and t are positive integers, is r + s + t even? (1) r + s is even. (2) s + t is even.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The given question asks whether the sum of three positive integers, r + s + t, is even.

Statement (1) tells us that r + s is even. However, this does not provide any information about the parity of t. It is possible that t is odd, which would make the sum r + s + t odd. Therefore, statement (1) alone is not sufficient to determine whether r + s + t is even.

Statement (2) tells us that s + t is even. However, this does not provide any information about the parity of r. It is possible that r is odd, which would make the sum r + s + t odd. Therefore, statement (2) alone is not sufficient to determine whether r + s + t is even.

When we consider both statements together, we still do not have enough information to determine the parity of r + s + t. It is possible that r is odd and t is even, or vice versa, which would make the sum r + s + t odd. Therefore, statements (1) and (2) together are not sufficient to determine whether r + s + t is even.

Hence, the correct answer is that statements (1) and (2) together are not sufficient.

Explanation

The given question asks whether the sum of three positive integers, r + s + t, is even.

Statement (1) tells us that r + s is even. However, this does not provide any information about the parity of t. It is possible that t is odd, which would make the sum r + s + t odd. Therefore, statement (1) alone is not sufficient to determine whether r + s + t is even.

Statement (2) tells us that s + t is even. However, this does not provide any information about the parity of r. It is possible that r is odd, which would make the sum r + s + t odd. Therefore, statement (2) alone is not sufficient to determine whether r + s + t is even.

When we consider both statements together, we still do not have enough information to determine the parity of r + s + t. It is possible that r is odd and t is even, or vice versa, which would make the sum r + s + t odd. Therefore, statements (1) and (2) together are not sufficient to determine whether r + s + t is even.

Hence, the correct answer is that statements (1) and (2) together are not sufficient.

39. If x is the product of the positive integers from 1 to 8, inclusive, and if i, k, m, and p are positive integers such that x = 2i3k5m7p, then i + k + m + p =

4

7

8

11

12

To find i, k, m, and p in the equation x = 2^i * 3^k * 5^m * 7^p, we can break down the prime factorization of x, which is the product of the positive integers from 1 to 8, inclusive.

x = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8

Now, let's find the prime factorization of x:

x = 2^3 * 3^2 * 5^1 * 7^1

So, we can see that i = 3, k = 2, m = 1, and p = 1.

To find i + k + m + p:

i + k + m + p = 3 + 2 + 1 + 1 = 7

So, i + k + m + p is equal to 7.

Explanation

To find i, k, m, and p in the equation x = 2^i * 3^k * 5^m * 7^p, we can break down the prime factorization of x, which is the product of the positive integers from 1 to 8, inclusive.

x = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8

Now, let's find the prime factorization of x:

x = 2^3 * 3^2 * 5^1 * 7^1

So, we can see that i = 3, k = 2, m = 1, and p = 1.

To find i + k + m + p:

i + k + m + p = 3 + 2 + 1 + 1 = 7

So, i + k + m + p is equal to 7.

40. One day Hotel Merinda rented 75 percent of its rooms, including $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/math»$ of its luxury rooms. If 60 percent of Hotel Merinda's rooms are luxury rooms, what percent of the rooms that were NOT rented are luxury rooms?

80%

40%

30%

33.33%

10%

Since 60% of the hotel's rooms are luxury rooms, it means that 40% of the rooms are not luxury rooms.
If 75% of the hotel's rooms were rented, it means that 25% of the rooms were not rented.
Out of those 25% of rooms that were not rented, 40% of them are luxury rooms.
Therefore, the percentage of the rooms that were not rented and are luxury rooms is 40% of 25%, which is equal to 10%.
However, the question is asking for the percentage of the rooms that were not rented and are luxury rooms, so we need to find the complement of 10%, which is 100% - 10% = 90%.
Therefore, the correct answer is 90%.

Explanation

Since 60% of the hotel's rooms are luxury rooms, it means that 40% of the rooms are not luxury rooms.
If 75% of the hotel's rooms were rented, it means that 25% of the rooms were not rented.
Out of those 25% of rooms that were not rented, 40% of them are luxury rooms.
Therefore, the percentage of the rooms that were not rented and are luxury rooms is 40% of 25%, which is equal to 10%.
However, the question is asking for the percentage of the rooms that were not rented and are luxury rooms, so we need to find the complement of 10%, which is 100% - 10% = 90%.
Therefore, the correct answer is 90%.

41. The mass of 1 cubic meter of a substance is 800 kilograms under certain conditions. What is the volume, in cubic centimeters of 1 gram of this substance under these condition (1 kilogram= 1000 grams and 1 cubic meter = 1,000,000 cubic centimeters.

0.80

1.25

8.00

12.50

80.00

The mass of 1 cubic meter of the substance is 800 kilograms. To find the volume of 1 gram of the substance, we need to convert the mass from kilograms to grams and the volume from cubic meters to cubic centimeters. Since 1 kilogram is equal to 1000 grams and 1 cubic meter is equal to 1,000,000 cubic centimeters, we can calculate the volume as follows:
Volume = (1 gram / 800 kilograms) * (1,000,000 cubic centimeters / 1 cubic meter) = 1,250 cubic centimeters. Therefore, the volume of 1 gram of the substance under these conditions is 1.25 cubic centimeters.

Explanation

The mass of 1 cubic meter of the substance is 800 kilograms. To find the volume of 1 gram of the substance, we need to convert the mass from kilograms to grams and the volume from cubic meters to cubic centimeters. Since 1 kilogram is equal to 1000 grams and 1 cubic meter is equal to 1,000,000 cubic centimeters, we can calculate the volume as follows:
Volume = (1 gram / 800 kilograms) * (1,000,000 cubic centimeters / 1 cubic meter) = 1,250 cubic centimeters. Therefore, the volume of 1 gram of the substance under these conditions is 1.25 cubic centimeters.

42. What is the remainder when the positive integer n is divided by 3? (1) The remainder when n is divided by 2 is 1. (2) The remainder when n + 1 is divided by 3 is 2.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

From statement (1), we know that the remainder when n is divided by 2 is 1. This means that n is an odd number. However, we don't have any information about the remainder when n is divided by 3 from this statement.

From statement (2), we know that the remainder when n + 1 is divided by 3 is 2. This means that (n + 1) is 2 more than a multiple of 3. Simplifying this, we get n is 1 less than a multiple of 3. In other words, n is 2 more than a multiple of 3. This means that the remainder when n is divided by 3 is 2.

Therefore, statement (2) alone is sufficient to determine the remainder when n is divided by 3, but statement (1) alone is not sufficient.

Explanation

From statement (1), we know that the remainder when n is divided by 2 is 1. This means that n is an odd number. However, we don't have any information about the remainder when n is divided by 3 from this statement.

From statement (2), we know that the remainder when n + 1 is divided by 3 is 2. This means that (n + 1) is 2 more than a multiple of 3. Simplifying this, we get n is 1 less than a multiple of 3. In other words, n is 2 more than a multiple of 3. This means that the remainder when n is divided by 3 is 2.

Therefore, statement (2) alone is sufficient to determine the remainder when n is divided by 3, but statement (1) alone is not sufficient.

43. In a survey of 200 college graduates, 30 percent said they had received student loans during their college careers, and 40 percent said they had received scholarships. What percent of those surveyed said that they had received neither student loans nor scholarships during their college careers? (1) 25 percent of those surveyed said that they had received scholarships but no loans. (2) 50 percent of those surveyed who said that they had received loans also said that they had received scholarships.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) alone is sufficient to determine the percent of those surveyed who said they had received neither student loans nor scholarships. It states that 25 percent of those surveyed said they had received scholarships but no loans. Since 30 percent said they had received student loans and 40 percent said they had received scholarships, we can subtract these percentages from 100 to find the percentage of those who received neither. Therefore, the percent of those surveyed who said they received neither student loans nor scholarships is 100 - 30 - 40 - 25 = 5 percent.

Similarly, statement (2) alone is also sufficient to determine the percent of those surveyed who said they had received neither student loans nor scholarships. It states that 50 percent of those surveyed who said they had received loans also said they had received scholarships. Since 30 percent said they had received student loans, we can subtract this percentage from 100 to find the percentage of those who received neither. Therefore, the percent of those surveyed who said they received neither student loans nor scholarships is 100 - 30 = 70 percent.

Hence, each statement alone is sufficient to determine the percent of those surveyed who said they had received neither student loans nor scholarships.

Explanation

Statement (1) alone is sufficient to determine the percent of those surveyed who said they had received neither student loans nor scholarships. It states that 25 percent of those surveyed said they had received scholarships but no loans. Since 30 percent said they had received student loans and 40 percent said they had received scholarships, we can subtract these percentages from 100 to find the percentage of those who received neither. Therefore, the percent of those surveyed who said they received neither student loans nor scholarships is 100 - 30 - 40 - 25 = 5 percent.

Similarly, statement (2) alone is also sufficient to determine the percent of those surveyed who said they had received neither student loans nor scholarships. It states that 50 percent of those surveyed who said they had received loans also said they had received scholarships. Since 30 percent said they had received student loans, we can subtract this percentage from 100 to find the percentage of those who received neither. Therefore, the percent of those surveyed who said they received neither student loans nor scholarships is 100 - 30 = 70 percent.

Hence, each statement alone is sufficient to determine the percent of those surveyed who said they had received neither student loans nor scholarships.

44. The infinite sequence a1, a2,…, an,… is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an = an-4 for n > 4. What is the sum of the first 97 terms of the sequence?

72

74

75

78

80

The given sequence follows a pattern where each term is equal to the term four places before it. By examining the first few terms of the sequence, we can see that it repeats every four terms: 2, -3, 5, -1, 2, -3, 5, -1, ... Therefore, we can group the terms into sets of four: (2, -3, 5, -1), (2, -3, 5, -1), ... The sum of each set of four terms is 3, so we can calculate the number of sets in 97 terms by dividing 97 by 4, which gives us 24 with a remainder of 1. Therefore, the sum of the first 97 terms is 24 * 3 + 2 = 74.

Explanation

The given sequence follows a pattern where each term is equal to the term four places before it. By examining the first few terms of the sequence, we can see that it repeats every four terms: 2, -3, 5, -1, 2, -3, 5, -1, ... Therefore, we can group the terms into sets of four: (2, -3, 5, -1), (2, -3, 5, -1), ... The sum of each set of four terms is 3, so we can calculate the number of sets in 97 terms by dividing 97 by 4, which gives us 24 with a remainder of 1. Therefore, the sum of the first 97 terms is 24 * 3 + 2 = 74.

45.

In parallelogram PQRS shown, if PQ = 4 and QR = 6, what is the area of PQRS?

To find the area of a parallelogram, we need to multiply the base by the height. In this case, the base is PQ and the height is QR. Therefore, the area of PQRS is 4 * 6 = 24.

Explanation

To find the area of a parallelogram, we need to multiply the base by the height. In this case, the base is PQ and the height is QR. Therefore, the area of PQRS is 4 * 6 = 24.

46. What is the value of │x + 7│? (1) │x + 3│= 14 (2) (x + 2)2 = 169

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) gives the absolute value of (x + 3) as 14. This means that (x + 3) can either be 14 or -14. Therefore, x can be either 11 or -17. However, this does not give us the value of │x + 7│ directly.

Statement (2) gives us (x + 2)² = 169. Taking the square root of both sides, we get x + 2 = ±13. This means that x can be either 11 or -15. However, this also does not give us the value of │x + 7│ directly.

By combining both statements, we can narrow down the possible values of x to 11. Therefore, we can determine that the value of │x + 7│ is 18.

Explanation

Statement (1) gives the absolute value of (x + 3) as 14. This means that (x + 3) can either be 14 or -14. Therefore, x can be either 11 or -17. However, this does not give us the value of │x + 7│ directly.

Statement (2) gives us (x + 2)² = 169. Taking the square root of both sides, we get x + 2 = ±13. This means that x can be either 11 or -15. However, this also does not give us the value of │x + 7│ directly.

By combining both statements, we can narrow down the possible values of x to 11. Therefore, we can determine that the value of │x + 7│ is 18.

47. What is the sum of a certain pair of consecutive odd integers? (1) At least one of the integers is negative. (2) At least one of the integers is positive.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) tells us that at least one of the integers is negative. This means that the sum of the consecutive odd integers could be negative, depending on the specific values of the integers. Therefore, statement (1) alone is not sufficient to determine the sum of the integers.

Statement (2) tells us that at least one of the integers is positive. This means that the sum of the consecutive odd integers could be positive, depending on the specific values of the integers. Therefore, statement (2) alone is not sufficient to determine the sum of the integers.

However, when we consider both statements together, we know that one integer is negative and one integer is positive, which means their sum will always be an odd number. Therefore, both statements together are sufficient to determine the sum of the integers.

Explanation

Statement (1) tells us that at least one of the integers is negative. This means that the sum of the consecutive odd integers could be negative, depending on the specific values of the integers. Therefore, statement (1) alone is not sufficient to determine the sum of the integers.

Statement (2) tells us that at least one of the integers is positive. This means that the sum of the consecutive odd integers could be positive, depending on the specific values of the integers. Therefore, statement (2) alone is not sufficient to determine the sum of the integers.

However, when we consider both statements together, we know that one integer is negative and one integer is positive, which means their sum will always be an odd number. Therefore, both statements together are sufficient to determine the sum of the integers.

48. If x and y are integers and x > 0, is y > 0? (1) 7x – 2y > 0 (2) -y < x

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

To determine if y is greater than 0, we need to analyze both statements together.

From statement (1), we have 7x - 2y > 0. This means that 7x > 2y. However, we do not have any information about the relation between x and y, so we cannot determine if y is greater than 0 based on this statement alone.

From statement (2), we have -y -x. Again, we do not have enough information to determine if y is greater than 0 based on this statement alone.

Since neither statement alone provides enough information, and we need both statements together to determine if y is greater than 0, the correct answer is that statements (1) and (2) together are not sufficient.

Explanation

To determine if y is greater than 0, we need to analyze both statements together.

From statement (1), we have 7x - 2y > 0. This means that 7x > 2y. However, we do not have any information about the relation between x and y, so we cannot determine if y is greater than 0 based on this statement alone.

From statement (2), we have -y -x. Again, we do not have enough information to determine if y is greater than 0 based on this statement alone.

Since neither statement alone provides enough information, and we need both statements together to determine if y is greater than 0, the correct answer is that statements (1) and (2) together are not sufficient.

49. If what is the value of ? (1) (2)

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

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Explanation

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50. Is an odd integer? (1) is an even integer. (2) is an odd integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The question asks whether the given integer is odd or not.

Statement (1) alone tells us that the integer is even. Since an even integer cannot be odd, we can conclude that the given integer is not odd.

Statement (2) alone tells us that the integer is odd. This directly answers the question and confirms that the given integer is indeed odd.

Therefore, each statement alone is sufficient to determine whether the integer is odd or not.

Explanation

The question asks whether the given integer is odd or not.

Statement (1) alone tells us that the integer is even. Since an even integer cannot be odd, we can conclude that the given integer is not odd.

Statement (2) alone tells us that the integer is odd. This directly answers the question and confirms that the given integer is indeed odd.

Therefore, each statement alone is sufficient to determine whether the integer is odd or not.

51. If s4v3x7 < 0, is svx < 0? (1) v < 0 (2) x > 0

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

From statement (1), we know that v
From statement (2), we know that x > 0. However, this does not provide any information about the value of v. Therefore, we cannot determine if svx
Since neither statement alone provides enough information to determine if svx

Explanation

From statement (1), we know that v
From statement (2), we know that x > 0. However, this does not provide any information about the value of v. Therefore, we cannot determine if svx
Since neither statement alone provides enough information to determine if svx

52. What is the ratio of the average (arithmetic mean) height of students in class X to the average height of students in class Y? (1) The average height of the students in class X is 120 centimeters. (2) The average height of the students in class X and class Y combined is 126 centimeters.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The ratio of the average height of students in class X to the average height of students in class Y cannot be determined with the given information. Statement (1) tells us the average height of students in class X, but it does not provide any information about the average height of students in class Y. Statement (2) tells us the average height of students in both classes combined, but it does not provide any information about the individual averages of each class. Therefore, both statements together are not sufficient to determine the ratio.

Explanation

The ratio of the average height of students in class X to the average height of students in class Y cannot be determined with the given information. Statement (1) tells us the average height of students in class X, but it does not provide any information about the average height of students in class Y. Statement (2) tells us the average height of students in both classes combined, but it does not provide any information about the individual averages of each class. Therefore, both statements together are not sufficient to determine the ratio.

53. For a recent play performance, the ticket prices were $25 per adult and $15 per child. A total of 500 tickets were sold for the performance. How many of the tickets sold were for adults? (1) Revenue from ticket sales for this performance totaled $10,500. (2) The average (arithmetic mean) price per ticket sold was $21.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) alone is sufficient to determine the number of adult tickets sold. We can calculate the number of adult tickets by dividing the total revenue from ticket sales ($10,500) by the price per adult ticket ($25). This will give us the total number of tickets sold, and we can subtract the number of child tickets (which we don't know) from this total to find the number of adult tickets. Therefore, statement (1) alone is sufficient to answer the question.

Explanation

Statement (1) alone is sufficient to determine the number of adult tickets sold. We can calculate the number of adult tickets by dividing the total revenue from ticket sales ($10,500) by the price per adult ticket ($25). This will give us the total number of tickets sold, and we can subtract the number of child tickets (which we don't know) from this total to find the number of adult tickets. Therefore, statement (1) alone is sufficient to answer the question.

54. Which of the following fractions has a decimal equivalent that is a terminating decimal?

A fraction has a decimal equivalent that is a terminating decimal if its denominator is a power of 10. This is because a terminating decimal can be written as a fraction with a denominator of 10, 100, 1000, and so on. Therefore, the fraction that has a decimal equivalent that is a terminating decimal is the one with a denominator of 10.

Explanation

A fraction has a decimal equivalent that is a terminating decimal if its denominator is a power of 10. This is because a terminating decimal can be written as a fraction with a denominator of 10, 100, 1000, and so on. Therefore, the fraction that has a decimal equivalent that is a terminating decimal is the one with a denominator of 10.

55. A certain list consist of several different integers. Is the product of all integers in the list positive? (1) The product of the greatest and smallest of the integers in the list is positive. (2) There is an even number of integers in the list.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The product of the greatest and smallest integers in the list being positive does not guarantee that all integers in the list are positive. However, the fact that there is an even number of integers in the list implies that there must be an equal number of positive and negative integers. Therefore, the product of all integers in the list cannot be positive. Thus, both statements together are sufficient to determine that the product of all integers in the list is not positive.

Explanation

The product of the greatest and smallest integers in the list being positive does not guarantee that all integers in the list are positive. However, the fact that there is an even number of integers in the list implies that there must be an equal number of positive and negative integers. Therefore, the product of all integers in the list cannot be positive. Thus, both statements together are sufficient to determine that the product of all integers in the list is not positive.

56. Is += 1 ? (1) x 0 (2) x 1

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

This answer suggests that both statement (1) and statement (2) together are sufficient to determine whether += 1 is true or not. However, neither statement alone is sufficient to make this determination.

Explanation

This answer suggests that both statement (1) and statement (2) together are sufficient to determine whether += 1 is true or not. However, neither statement alone is sufficient to make this determination.

57. Over a certain time period, did the number of shares of stock in Ruth's portfolio increase? (1) Over the time period, the ratio of the number of shares of stock to the total number of shares of stocks and bonds in Ruth's portfolio increased. (2) Over the time period, the total number of shares of stock and bonds in Ruth's portfolio increased.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) alone tells us that the ratio of shares of stock to the total number of shares of stocks and bonds in Ruth's portfolio increased. However, it does not provide any information about the actual number of shares or whether the total number of shares increased or not.

Statement (2) alone tells us that the total number of shares of stock and bonds in Ruth's portfolio increased. However, it does not provide any information about the ratio of shares or whether the number of shares of stock increased or not.

By combining both statements, we can conclude that both the ratio of shares and the total number of shares increased, which implies that the number of shares of stock in Ruth's portfolio also increased. Therefore, both statements together are sufficient to answer the question.

Explanation

Statement (1) alone tells us that the ratio of shares of stock to the total number of shares of stocks and bonds in Ruth's portfolio increased. However, it does not provide any information about the actual number of shares or whether the total number of shares increased or not.

Statement (2) alone tells us that the total number of shares of stock and bonds in Ruth's portfolio increased. However, it does not provide any information about the ratio of shares or whether the number of shares of stock increased or not.

By combining both statements, we can conclude that both the ratio of shares and the total number of shares increased, which implies that the number of shares of stock in Ruth's portfolio also increased. Therefore, both statements together are sufficient to answer the question.

58. There are 6 pairs of balls with each pair in a different color. If we select 2 from the 12 balls without replacement, what's the probability of selecting 2 balls with the same color?

The probability of selecting 2 balls with the same color can be calculated by dividing the number of favorable outcomes (selecting 2 balls of the same color) by the total number of possible outcomes. Since there are 6 pairs of balls in different colors, there are 6 different colors to choose from. The first ball can be any of the 12, and the second ball must be of the same color as the first. Therefore, there are 12 possible choices for the first ball and only 1 choice for the second ball. The total number of possible outcomes is 12 * 11. Thus, the probability is 1/66.

Explanation

The probability of selecting 2 balls with the same color can be calculated by dividing the number of favorable outcomes (selecting 2 balls of the same color) by the total number of possible outcomes. Since there are 6 pairs of balls in different colors, there are 6 different colors to choose from. The first ball can be any of the 12, and the second ball must be of the same color as the first. Therefore, there are 12 possible choices for the first ball and only 1 choice for the second ball. The total number of possible outcomes is 12 * 11. Thus, the probability is 1/66.

59. If a code word is defined to be a sequence of different letters chosen from the 10 letters A, B, C, D, E, F, G, H, I, and J, what is the ratio of the number of 5-letter code words to the number of 4-letter code words?

5 to 4

3 to 2

2 to 1

5 to 1

6 to 1

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Explanation

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60. When tossed, a certain coin has an equal chance of landing on either side. If the coin is tossed 4 times, what is the probability that it will land on the same side each time?

Explanation

61. Is x2 + y2 > 6? (1) (x + y)2 > 6 (2) xy = 2

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

From statement (1), we know that (x + y)^2 > 6. However, we cannot determine if x^2 + y^2 > 6 or x^2 + y^2
From statement (2), we know that xy = 2. However, we cannot determine the values of x and y individually or their sum. Therefore, statement (2) alone is not sufficient to answer the question.

By combining both statements, we still cannot determine if x^2 + y^2 > 6 or x^2 + y^2

Explanation

From statement (1), we know that (x + y)^2 > 6. However, we cannot determine if x^2 + y^2 > 6 or x^2 + y^2
From statement (2), we know that xy = 2. However, we cannot determine the values of x and y individually or their sum. Therefore, statement (2) alone is not sufficient to answer the question.

By combining both statements, we still cannot determine if x^2 + y^2 > 6 or x^2 + y^2

62. If n and k are positive integers is an even integer? (1) n is divisible by 8. (2) k is divisible by 4.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) alone tells us that n is divisible by 8. However, this does not guarantee that n is even. For example, n could be 16, which is even, or it could be 24, which is not even. Therefore, statement (1) alone is not sufficient to determine if n is an even integer.

Statement (2) alone tells us that k is divisible by 4. This does not provide any information about n, so it is not sufficient to determine if n is an even integer.

When we consider both statements together, we still do not have enough information to determine if n is an even integer. The statements provide information about n and k separately, but do not provide any relationship between them. Therefore, statements (1) and (2) together are not sufficient to determine if n is an even integer.

Explanation

Statement (1) alone tells us that n is divisible by 8. However, this does not guarantee that n is even. For example, n could be 16, which is even, or it could be 24, which is not even. Therefore, statement (1) alone is not sufficient to determine if n is an even integer.

Statement (2) alone tells us that k is divisible by 4. This does not provide any information about n, so it is not sufficient to determine if n is an even integer.

When we consider both statements together, we still do not have enough information to determine if n is an even integer. The statements provide information about n and k separately, but do not provide any relationship between them. Therefore, statements (1) and (2) together are not sufficient to determine if n is an even integer.

63. Pierre is driving 50 miles per hour for the first 20 miles of a 40-mile trip. What must be his average speed in the remaining 20 miles in order for his total average speed to be 60 miles per hour?

60

65

70

75

80

To find the average speed for the entire trip, we can use the formula: average speed = total distance / total time. We know that the total distance is 40 miles and the total average speed is 60 miles per hour. Let's assume that Pierre drives at an average speed of x miles per hour for the remaining 20 miles. We can set up the equation: 60 = 40 / (20/50 + 20/x). Simplifying this equation, we get: 60 = 40 / (2/5 + 20/x). Cross-multiplying, we get: 60 * (2/5 + 20/x) = 40. Simplifying further, we get: 2/5 + 20/x = 2/3. Solving for x, we find that x = 75. Therefore, Pierre must drive at an average speed of 75 miles per hour for the remaining 20 miles in order for his total average speed to be 60 miles per hour.

Explanation

To find the average speed for the entire trip, we can use the formula: average speed = total distance / total time. We know that the total distance is 40 miles and the total average speed is 60 miles per hour. Let's assume that Pierre drives at an average speed of x miles per hour for the remaining 20 miles. We can set up the equation: 60 = 40 / (20/50 + 20/x). Simplifying this equation, we get: 60 = 40 / (2/5 + 20/x). Cross-multiplying, we get: 60 * (2/5 + 20/x) = 40. Simplifying further, we get: 2/5 + 20/x = 2/3. Solving for x, we find that x = 75. Therefore, Pierre must drive at an average speed of 75 miles per hour for the remaining 20 miles in order for his total average speed to be 60 miles per hour.

64. The membership of a committee consists of 3 English teachers, 4 Mathematics teachers, and 2 Social Studies teachers. If 2 committee members are to be selected at random to write the committee's report, what is the probability that the two members selected will be an English teacher and a Mathematics teacher?

To find the probability of selecting an English teacher and a Mathematics teacher, we need to calculate the probability of selecting one English teacher and one Mathematics teacher, and then multiply these probabilities together.

The probability of selecting an English teacher as the first member is 3/9, since there are 3 English teachers out of a total of 9 committee members.

After selecting an English teacher, there are now 8 committee members left, including 4 Mathematics teachers. So the probability of selecting a Mathematics teacher as the second member is 4/8.

Multiplying these probabilities together, we get (3/9) * (4/8) = 12/72 = 1/6.

Therefore, the probability that the two members selected will be an English teacher and a Mathematics teacher is 1/6.

Explanation

To find the probability of selecting an English teacher and a Mathematics teacher, we need to calculate the probability of selecting one English teacher and one Mathematics teacher, and then multiply these probabilities together.

The probability of selecting an English teacher as the first member is 3/9, since there are 3 English teachers out of a total of 9 committee members.

After selecting an English teacher, there are now 8 committee members left, including 4 Mathematics teachers. So the probability of selecting a Mathematics teacher as the second member is 4/8.

Multiplying these probabilities together, we get (3/9) * (4/8) = 12/72 = 1/6.

Therefore, the probability that the two members selected will be an English teacher and a Mathematics teacher is 1/6.

65. Mary walked from home to work in the morning at an average speed of 3 miles per hour. She returned in the evening taking the same route running an average speed of 6 miles per hour. What was her average speed, in miles per hour, for the whole journey to and from work?

3.5

4

4.5

5

5.5

Mary's average speed for the whole journey can be calculated by taking the total distance traveled and dividing it by the total time taken. Since Mary traveled the same route both ways, the total distance is the same for the morning and evening trips. Let's assume the distance is D miles.

In the morning, Mary traveled at a speed of 3 miles per hour, so the time taken for the morning trip is D/3 hours.

In the evening, Mary traveled at a speed of 6 miles per hour, so the time taken for the evening trip is D/6 hours.

The total time taken for the whole journey is D/3 + D/6 = (2D + D)/6 = 3D/6 = D/2 hours.

Therefore, the average speed for the whole journey is D miles / (D/2 hours) = 2 miles per hour.

Hence, the correct answer is 2, which is not listed as an option.

Explanation

Mary's average speed for the whole journey can be calculated by taking the total distance traveled and dividing it by the total time taken. Since Mary traveled the same route both ways, the total distance is the same for the morning and evening trips. Let's assume the distance is D miles.

In the morning, Mary traveled at a speed of 3 miles per hour, so the time taken for the morning trip is D/3 hours.

In the evening, Mary traveled at a speed of 6 miles per hour, so the time taken for the evening trip is D/6 hours.

The total time taken for the whole journey is D/3 + D/6 = (2D + D)/6 = 3D/6 = D/2 hours.

Therefore, the average speed for the whole journey is D miles / (D/2 hours) = 2 miles per hour.

Hence, the correct answer is 2, which is not listed as an option.

66.

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Explanation

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67. What is the value of ? (1) (2)

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

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Explanation

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68. A certain company assigns employees to offices in such a way that some of the offices can be empty and more than one employee can be assigned to an office. In how many ways can the company assign 3 employees to 2 different offices?

5

6

7

8

9

The company can assign the employees in the following ways:
1. All 3 employees in one office and the other office empty.
2. 2 employees in one office and 1 in the other office.
3. 1 employee in one office and 2 in the other office.
4. All 3 employees in one office and the other office empty.
5. 2 employees in one office and 1 in the other office.
6. 1 employee in one office and 2 in the other office.
7. All 3 employees in one office and the other office empty.
8. 2 employees in one office and 1 in the other office.
Therefore, there are 8 ways in which the company can assign 3 employees to 2 different offices.

Explanation

The company can assign the employees in the following ways:
1. All 3 employees in one office and the other office empty.
2. 2 employees in one office and 1 in the other office.
3. 1 employee in one office and 2 in the other office.
4. All 3 employees in one office and the other office empty.
5. 2 employees in one office and 1 in the other office.
6. 1 employee in one office and 2 in the other office.
7. All 3 employees in one office and the other office empty.
8. 2 employees in one office and 1 in the other office.
Therefore, there are 8 ways in which the company can assign 3 employees to 2 different offices.

69. For the cube shown above what is the degree measure of PQR?

30

45

60

75

90

The degree measure of PQR is 60. This can be determined by observing that PQR is one of the faces of the cube. In a cube, each face is a square, and the sum of the interior angles of a square is always 360 degrees. Since PQR is a right angle, it is one of the interior angles of the square face. Therefore, the degree measure of PQR is 360/4 = 90 degrees. However, since the question is asking for the degree measure of PQR, and not the entire square face, the correct answer is 60 degrees.

Explanation

The degree measure of PQR is 60. This can be determined by observing that PQR is one of the faces of the cube. In a cube, each face is a square, and the sum of the interior angles of a square is always 360 degrees. Since PQR is a right angle, it is one of the interior angles of the square face. Therefore, the degree measure of PQR is 360/4 = 90 degrees. However, since the question is asking for the degree measure of PQR, and not the entire square face, the correct answer is 60 degrees.

70. Each week a certain salesman is made a fixed amount equal to $300 plus a commission equal to 5 percent of the amount of these sales that week over $1,000. What is the total amount the salesman was paid last week? (1) The total amount the salesman was paid last week is equal to 10 percent of the amount of these sales last week. (2) The salesman's sales last week total $5,000.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The correct answer is EACH statement ALONE is sufficient.

Statement (1) provides the information that the total amount the salesman was paid last week is equal to 10 percent of the amount of these sales last week. This allows us to calculate the total amount the salesman was paid last week based on the sales amount.

Statement (2) provides the information that the salesman's sales last week total $5,000. This gives us the sales amount needed to calculate the total amount the salesman was paid last week.

Therefore, each statement alone provides enough information to calculate the total amount the salesman was paid last week.

Explanation

The correct answer is EACH statement ALONE is sufficient.

Statement (1) provides the information that the total amount the salesman was paid last week is equal to 10 percent of the amount of these sales last week. This allows us to calculate the total amount the salesman was paid last week based on the sales amount.

Statement (2) provides the information that the salesman's sales last week total $5,000. This gives us the sales amount needed to calculate the total amount the salesman was paid last week.

Therefore, each statement alone provides enough information to calculate the total amount the salesman was paid last week.

71. If M is the least common multiple of 90, 196, and 300, which of the following is NOT a factor of M?

600

700

900

2,100

4,900

To find the least common multiple (LCM) of 90, 196, and 300, we need to find the smallest number that is divisible by all three numbers. The LCM of these numbers is 2,100. Therefore, all the given options are factors of the LCM except for 600. Hence, 600 is NOT a factor of the LCM.

Explanation

To find the least common multiple (LCM) of 90, 196, and 300, we need to find the smallest number that is divisible by all three numbers. The LCM of these numbers is 2,100. Therefore, all the given options are factors of the LCM except for 600. Hence, 600 is NOT a factor of the LCM.

72. Is xy > x2y2? (1) 14x2 = 3 (2) y2 = 1

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The question asks whether xy > x^2y^2.

Statement (1) tells us that 14x^2 = 3. However, this does not provide any information about the value of y, so we cannot determine the relationship between xy and x^2y^2 based on this statement alone.

Statement (2) tells us that y^2 = 1. This means that y could be either 1 or -1. Again, we do not have enough information to determine the relationship between xy and x^2y^2 based on this statement alone.

When we consider both statements together, we still cannot determine the relationship between xy and x^2y^2. Even though we know the value of x^2y^2 from statement (1) and the value of y from statement (2), we do not have enough information to determine the relationship between xy and x^2y^2.

Therefore, the answer is that statements (1) and (2) together are not sufficient to answer the question.

Explanation

The question asks whether xy > x^2y^2.

Statement (1) tells us that 14x^2 = 3. However, this does not provide any information about the value of y, so we cannot determine the relationship between xy and x^2y^2 based on this statement alone.

Statement (2) tells us that y^2 = 1. This means that y could be either 1 or -1. Again, we do not have enough information to determine the relationship between xy and x^2y^2 based on this statement alone.

When we consider both statements together, we still cannot determine the relationship between xy and x^2y^2. Even though we know the value of x^2y^2 from statement (1) and the value of y from statement (2), we do not have enough information to determine the relationship between xy and x^2y^2.

Therefore, the answer is that statements (1) and (2) together are not sufficient to answer the question.

73. Is n divisible by 12? (1) is an integer. (2) is an integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

The question asks whether n is divisible by 12. Statement (1) alone does not provide any information about n, so we cannot determine whether n is divisible by 12. Similarly, statement (2) alone does not provide any information about n. Therefore, neither statement alone is sufficient to answer the question.

Explanation

The question asks whether n is divisible by 12. Statement (1) alone does not provide any information about n, so we cannot determine whether n is divisible by 12. Similarly, statement (2) alone does not provide any information about n. Therefore, neither statement alone is sufficient to answer the question.

74. Each of the 45 boxes on shelf J weighs less than each of the 44 boxes on shelf K. What is the median weight of the 89 boxes on these shelves? (1) The heaviest box on shelf J weighs 15 pounds. (2) The lightest box on shelf K weighs 20 pounds.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) alone is sufficient to determine the median weight of the boxes. Since the heaviest box on shelf J weighs 15 pounds and each of the 45 boxes on shelf J weighs less than each of the 44 boxes on shelf K, it can be concluded that the median weight of the boxes on shelf J is less than 15 pounds.

Statement (2) alone does not provide any information about the weights of the boxes on shelf J or the relationship between the weights of the boxes on shelf J and shelf K. Therefore, it is not sufficient to determine the median weight of the boxes.

Hence, statement (1) alone is sufficient to determine the median weight of the 89 boxes on these shelves.

Explanation

Statement (1) alone is sufficient to determine the median weight of the boxes. Since the heaviest box on shelf J weighs 15 pounds and each of the 45 boxes on shelf J weighs less than each of the 44 boxes on shelf K, it can be concluded that the median weight of the boxes on shelf J is less than 15 pounds.

Statement (2) alone does not provide any information about the weights of the boxes on shelf J or the relationship between the weights of the boxes on shelf J and shelf K. Therefore, it is not sufficient to determine the median weight of the boxes.

Hence, statement (1) alone is sufficient to determine the median weight of the 89 boxes on these shelves.

75. 2x + y = 12 &le 12 For how many ordered pairs (x y) that are solutions of the system above are x and y both integers?

7

10

12

13

14

The system of equations is 2x + y = 12 and x + y ≤ 12. To find the number of ordered pairs (x, y) that are solutions to this system where both x and y are integers, we can analyze the possible values of x and y. Since x and y are integers, we can start by finding the range of values for x and y that satisfy the given conditions. From the equation 2x + y = 12, we know that y must be an even number for 2x + y to equal 12. Therefore, the possible values for y are 2, 4, 6, 8, 10, and 12. For each value of y, we can solve the equation 2x + y = 12 to find the corresponding value of x. By doing this, we find that there are 13 ordered pairs (x, y) that are solutions to the system where both x and y are integers.

Explanation

The system of equations is 2x + y = 12 and x + y ≤ 12. To find the number of ordered pairs (x, y) that are solutions to this system where both x and y are integers, we can analyze the possible values of x and y. Since x and y are integers, we can start by finding the range of values for x and y that satisfy the given conditions. From the equation 2x + y = 12, we know that y must be an even number for 2x + y to equal 12. Therefore, the possible values for y are 2, 4, 6, 8, 10, and 12. For each value of y, we can solve the equation 2x + y = 12 to find the corresponding value of x. By doing this, we find that there are 13 ordered pairs (x, y) that are solutions to the system where both x and y are integers.

76. Pat invested x dollars in a fund that paid 8 percent annual interest, compounded annually. Which of the following represents the value, in dollars, of Pat's investment plus interest at the end of 5 years?

5(0.08x

5(1.08x)

[1 + 5(0.08)]x

(1.08)5x

(1.08x)5

The given expression, (1.08)5x, represents the value of Pat's investment plus interest at the end of 5 years. The expression is derived from the formula for compound interest, which is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount (initial investment), r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years. In this case, the principal amount is x, the annual interest rate is 8% (or 0.08), and the interest is compounded annually (n = 1). Therefore, the correct answer is (1.08)5x.

Explanation

The given expression, (1.08)5x, represents the value of Pat's investment plus interest at the end of 5 years. The expression is derived from the formula for compound interest, which is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount (initial investment), r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years. In this case, the principal amount is x, the annual interest rate is 8% (or 0.08), and the interest is compounded annually (n = 1). Therefore, the correct answer is (1.08)5x.

77. If is a positive integer and is the remainder when is divided by 8 what is the value of ? (1) is odd. (2) is not divisible by 8.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) alone is sufficient because if is odd, then the possible remainders when divided by 8 are 1, 3, 5, or 7. Therefore, we can determine the value of based on this information alone. Statement (2) alone is not sufficient because it does not provide any information about the remainder when divided by 8.

Explanation

Statement (1) alone is sufficient because if is odd, then the possible remainders when divided by 8 are 1, 3, 5, or 7. Therefore, we can determine the value of based on this information alone. Statement (2) alone is not sufficient because it does not provide any information about the remainder when divided by 8.

78. Is is > 0 ? (1) (2)

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Since there is no information provided in either statement (1) or statement (2), we cannot determine the value of "is". Therefore, statement (1) alone is not sufficient to answer the question. However, since statement (2) also does not provide any information, it is also not sufficient to answer the question. Therefore, the correct answer is that neither statement alone is sufficient, but statement (1) alone is sufficient.

Explanation

Since there is no information provided in either statement (1) or statement (2), we cannot determine the value of "is". Therefore, statement (1) alone is not sufficient to answer the question. However, since statement (2) also does not provide any information, it is also not sufficient to answer the question. Therefore, the correct answer is that neither statement alone is sufficient, but statement (1) alone is sufficient.

79. If where and are positive integers then

16

24

48

96

144

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Explanation

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