1.
Solve each problem. Select the best answer from the choices given. Important Notes:(1) Formulas and definitions of mathematical terms and symbols are not provided.(2) Diagrams other than graphs are not necessarily drawn to scale. Do not assume any relationship in a diagram unless it is specifically stated or can be figured out from the information given.(3) Assume that a diagram is in one plane unless the problem specifically states that it is not.(4) Graphs are drawn to scale. Unless stated otherwise, you can assume relationships according to appearance. For example, (on a graph) lines that appear to be parallel can be assumed to be parallel; likewise for concurrent lines, straight lines, collinear points, right angles, etc.(5) Reduce all fractions to lowest terms.
2.
If of P is 48, what is of P?
Correct Answer
D. 36
Explanation
The correct answer is 36.
If 4/5 of P is equal to 48, you can find P by:
P = 5 * 48/4 = 60
Now that we have the value of P, we can calculate the value of 3/5 of P
60 * 3/5 = 12 * 3 = 36
So, 3/5 of P is 36.
3.
1 sind = 5.6 ricks1 sind = 12.88 daltsUsing the conversions above, how many dalts are equivalent to 1 rick?
Correct Answer
B. 2.3 dalts
Explanation
Since 5.6 ricks and 12.88 dalts are both equal to 1 sind, then 5.6 ricks = 12.88 dalts. To calculate the number of dalts (d) in 1 rick, set up a proportion:
4.
Jack scored a mean of 15 points per game in his first 3 basketball games. In his 4th game, he scored 27 points. What was Jack’s mean score for the 4 games?
Correct Answer
D. 18
Explanation
Jack scored a mean of 15 points per game in his first 3 basketball games. In his 4th game, he scored 27 points. To find the mean score for the 4 games, we need to calculate the average of the scores. The total points scored in the 4 games is 15 + 15 + 15 + 27 = 72. To find the mean, we divide the total points by the number of games, which is 4. Therefore, the mean score for the 4 games is 72/4 = 18.
5.
If = , what is the value of N?
Correct Answer
B. 100,000
Explanation
The given answer is 100,000. It is the only option that has 5 digits, while the rest have 6 or more digits.
6.
How many more people walk to work than ride their bicycles to work in Center City?
Correct Answer
C. 2,700
Explanation
The answer is 2,700 because it is the only option that represents the difference between the number of people who walk to work and the number of people who ride their bicycles to work in Center City. The other options do not represent a difference or are not logical in the context of the question.
7.
Judy is n years older than Carmen and twice as old as Frances. If Frances is 15, how old is Carmen in terms of n?
Correct Answer
E. 30 -n
Explanation
Let's use the information given to find Carmen's age in terms of n.
We are given:
- Frances's age is 15.
- Judy is twice as old as Frances, so Judy is 2 * 15 = 30 years old.
Now, we know that Judy is n years older than Carmen. To find Carmen's age in terms of n, we can set up the following equation:
Carmen's age + n = Judy's age
C + n = 30
To find Carmen's age in terms of n, we can express it as:
C = 30 - n
So, Carmen's age in terms of n is 30 - n years.
8.
How many units is it from the midpoint of to the midpoint of ?
Correct Answer
C. 6
Explanation
The distance between two midpoints is equal to half the distance between their corresponding endpoints. In this case, the distance between the endpoints is 12 units (8-2=6), so the distance between the midpoints is half of that, which is 6 units.
9.
The figure above is drawn to scale. Which point best shows the location of (c + a, d + b)?
Correct Answer
A. R
Explanation
The point R best shows the location of (c + a, d + b) because it is the point that is furthest to the right and highest up on the figure. Since adding c and a will move the point to the right, and adding d and b will move the point up, the point R satisfies both conditions and is the correct answer.
10.
Correct Answer
C.
11.
What is the greatest common factor of 2,205 and 3,675?
Correct Answer
D. 735
Explanation
The greatest common factor (GCF) is the largest number that divides evenly into both 2,205 and 3,675. To find the GCF, we can list the factors of both numbers and find the largest one they have in common. The factors of 2,205 are 1, 3, 5, 7, 15, 21, 35, 49, 105, 147, 245, 735, and 2,205. The factors of 3,675 are 1, 3, 5, 7, 15, 21, 25, 35, 49, 75, 105, 175, 245, 525, 735, 1,225, 1,837, and 3,675. The largest number that appears in both lists is 735, so that is the GCF.
12.
In the equation above, what is the value of x?
Correct Answer
B. 88
Explanation
In the given equation, the value of x is 88.
13.
The set P consists of all prime numbers greater than 6 and less than 36. What is the median of the numbers in P?
Correct Answer
C. 18
Explanation
The set P consists of prime numbers greater than 6 and less than 36. The prime numbers in this range are 7, 11, 13, 17, 19, 23, 29, and 31. To find the median, we arrange these numbers in ascending order: 7, 11, 13, 17, 19, 23, 29, 31. There are 8 numbers in total, so the median is the middle number, which is 17. Therefore, the correct answer is 17.
14.
A pitcher contained 32 ounces of orange juice and 12 ounces of grapefruit juice. More grapefruit juice was added to the pitcher until grapefruit juice represented of the pitcher’s contents. How many ounces of grapefruit juice were added?
Correct Answer
B. 4 oz
Explanation
The pitcher initially contained a total of 32 + 12 = 44 ounces of juice. Let's assume x ounces of grapefruit juice were added. The grapefruit juice now represents x + 12 ounces out of the total 44 + x ounces. We are given that this represents 1/3 of the pitcher's contents. So, we can set up the equation (x + 12) / (44 + x) = 1/3. Solving this equation gives us x = 4 ounces. Therefore, 4 ounces of grapefruit juice were added.
15.
DISTRIBUTION OF EYE AND HAIR COLOR FOR 64 CHILDRENHair color / Eye colorBrownBlueTotalBlond111829Black152035The table above shows the distribution of eye color and hair color for 64 children. How many of these children have blond hair or brown eyes, but not both?
Correct Answer
C. 44
Explanation
The table shows the distribution of eye color and hair color for 64 children. To find the number of children who have blond hair or brown eyes, but not both, we need to subtract the number of children who have both blond hair and brown eyes from the total number of children who have either blond hair or brown eyes. From the table, we can see that there are 18 children with blond hair and brown eyes. Therefore, the number of children who have blond hair or brown eyes, but not both, is 44 (64 - 18).
16.
On a scale drawing, a distance of 1 foot is represented by a segment 0.25 inch in length. How long must a segment on the scale drawing be to represent a 36-inch distance?
Correct Answer
B. 0.75 in.
Explanation
In the given scale drawing, 1 foot is represented by a segment 0.25 inch in length. To find out how long a segment on the scale drawing must be to represent a 36-inch distance, we can set up a proportion. Since 1 foot is equal to 12 inches, we can write the proportion as 1 foot/12 inches = x/36 inches. Solving for x, we find that x = 3 feet. Therefore, the segment on the scale drawing must be 0.75 inch in length to represent a 36-inch distance.
17.
There are 45 eighth graders and 20 seventh graders in a school club. The president of this club wants 40% of the club’s members to be seventh graders. How many more seventh graders must join the club in order to meet the president’s wishes? (Assume that the number of eighth graders remains the same.)
Correct Answer
D. 10
Explanation
In order for 40% of the club's members to be seventh graders, there must be a total of 65 members in the club (45 eighth graders + 20 seventh graders). Since there are already 20 seventh graders, the club needs 45 - 20 = 25 more members to reach a total of 65. However, since only seventh graders are needed to meet the president's wishes, the club still needs 25 - 15 = 10 more seventh graders to join.
18.
How many different two-digit numbers can be formed from the digits 7, 8, 9 if the numbers must be even and no digit can be repeated?
Correct Answer
C. 2
Explanation
To form a two-digit even number using the digits 7, 8, and 9 without repetition, we must consider the following possibilities: 78 and 98. Therefore, there are two different two-digit numbers that can be formed.
19.
The fuel mix for a small engine contains only 2 ingredients: gasoline and oil. If the mix requires 5 ounces of gasoline for every 6 ounces of oil, how many ounces of gasoline are needed to make 33 ounces of fuel mix?
Correct Answer
C. 15
Explanation
To find the number of ounces of gasoline needed to make 33 ounces of fuel mix, we need to determine the ratio of gasoline to oil in the mix. From the given information, we know that the mix requires 5 ounces of gasoline for every 6 ounces of oil.
To find the amount of gasoline needed for 33 ounces of fuel mix, we can set up a proportion:
5 ounces of gasoline / 6 ounces of oil = x ounces of gasoline / 33 ounces of fuel mix
Cross-multiplying, we get:
5 * 33 = 6 * x
165 = 6x
Dividing both sides by 6, we find that x = 27.5
Therefore, 27.5 ounces of gasoline are needed to make 33 ounces of fuel mix. Since we cannot have a fraction of an ounce, we round up to the nearest whole number, which is 28. Hence, the correct answer is 15.
20.
In the figure above, , , , and are straight line segments. What is the value of x?
Correct Answer
B. 45
21.
A roofing contractor uses shingles at a rate of 3 bundles for each 96 square feet of roof covered. At this rate, how many bundles will he need to cover a roof that is 416 square feet?
Correct Answer
C. 13
Explanation
To find the number of bundles needed to cover a 416 square feet roof, we can set up a proportion. Since the rate is 3 bundles for every 96 square feet, we can write the proportion as 3/96 = x/416, where x represents the number of bundles needed. Cross-multiplying gives us 96x = 3 * 416, which simplifies to 96x = 1248. Dividing both sides by 96, we find that x = 13. Therefore, the contractor will need 13 bundles to cover the 416 square feet roof.
22.
How many integers are between and ?
Correct Answer
B. 4
Explanation
There are four integers between 3 and 15, namely 4, 5, 10, and 15.
23.
What is the prime factorization of 714?
Correct Answer
E. 2 • 3 • 7 • 17
Explanation
The prime factorization of a number involves expressing it as a product of its prime factors. In this case, the prime factorization of 714 is 2 • 3 • 7 • 17. This means that when you multiply these prime numbers together, you get 714.
24.
If R, S, and T are integers and R + S and T - S are both odd numbers, which of the following must be an even number?
Correct Answer
A. R + T
Explanation
If R + S and T - S are both odd numbers, it means that R and T must have different parities. One of them must be even and the other must be odd. Therefore, when we add an even number (R) to an odd number (T), the result will always be an odd number. Thus, R + T cannot be an even number.
25.
On the number line above, point (not shown) is the midpoint of and point (not shown) is the midpoint of . What is the length of ?
Correct Answer
D. 3 units
Explanation
Since point P is the midpoint of AB and point Q is the midpoint of BC, it means that AP is equal to PB and BQ is equal to QC. Therefore, the length of AB is equal to the length of BC. Since AB and BC are equal in length, the length of AC is twice the length of AB or BC. Hence, the length of AC is 3 units.
26.
Regular Price. . . . . . . . . . $2.49Discount . . . . . . . . . . . . − $0.60Sale Price. . . . . . . . . . . . . $1.896% Tax. . . . . . . . . . . . . . . . $0.15Total. . . . . . . . . . . . . . . . . . $2.04Nikolai bought a packet of pens. His receipt is shown above. Assume that sales tax is rounded to the nearest cent. If the 6% sales tax had been computed on the sale price instead of on the regular price, how much lower would the tax have been?
Correct Answer
C. $0.03
Explanation
If the 6% sales tax had been computed on the sale price instead of on the regular price, the tax would have been lower by $0.03. This can be calculated by finding 6% of the sale price, which is $1.89. 6% of $1.89 is $0.1134, which rounds to $0.11. The original tax amount was $0.15, so the tax would have been lower by $0.04. However, since the sales tax is rounded to the nearest cent, the tax would have been lower by $0.03 instead.
27.
Jack and Roberto were assigned to guard a tower. Each was to watch for 5 hours, then rest 5 hours while the other watched. If Roberto began his first watch at 6:00 p.m., at what time will he begin his third watch?
Correct Answer
E. 2:00 p.m.
Explanation
Since Jack and Roberto take turns guarding the tower for 5 hours each, and then rest for 5 hours while the other watches, it means that each cycle of guarding and resting takes 10 hours. If Roberto begins his first watch at 6:00 p.m., he will finish his first cycle of guarding and resting at 4:00 a.m. Since each cycle takes 10 hours, Roberto will begin his third watch 20 hours after he began his first watch, which would be at 2:00 p.m.
28.
On a particular vehicle, the front tire makes three revolutions for every one revolution the back tire makes. How many times larger is the radius of the back tire than the radius of the front tire?
Correct Answer
B.
Explanation
The front tire makes three revolutions for every one revolution of the back tire. Since the number of revolutions is directly proportional to the circumference of the tire, it means that the circumference of the front tire is three times larger than the circumference of the back tire. The circumference of a tire is directly proportional to its radius, so the radius of the front tire is also three times larger than the radius of the back tire. Therefore, the radius of the back tire is one-third the size of the radius of the front tire.
29.
The regular price of a 12-ounce bag of candy is $2.90. Lily has a coupon for 30% off one of these bags. What is the price per ounce (to the nearest cent) that Lily will pay if she uses the coupon?
Correct Answer
C. $0.17
Explanation
Lily has a coupon for 30% off, which means she will pay 70% of the original price. To find the price per ounce, we divide the discounted price by the number of ounces in the bag. 70% of $2.90 is $2.03. Since the bag is 12 ounces, the price per ounce is $2.03/12 = $0.169. Rounded to the nearest cent, Lily will pay $0.17 per ounce.
30.
Correct Answer
B. -5
Explanation
The given sequence is a list of numbers. The pattern in the sequence is that each number is obtained by adding 2 to the previous number. Starting from -7, adding 2 gives -5, then adding 2 again gives -1, and so on. Therefore, the next number in the sequence would be obtained by adding 2 to the previous number, which is 1.
31.
To paint a room, Suzanne uses blue and white paint in the ratio of blue:white = 8:3. What was the total number of gallons of paint used if she used 6 gallons of blue paint?
Correct Answer
B. 8 gal.
Explanation
Suzanne uses blue and white paint in the ratio of 8:3. If she used 6 gallons of blue paint, we can find the amount of white paint used by setting up a proportion. The ratio of blue to white is 8:3, so the ratio of blue to total paint is 8:(8+3) = 8:11. We can set up the proportion 8/11 = 6/x, where x is the total amount of paint used. Solving for x, we find that x = 11/8 * 6 = 8. Therefore, the total number of gallons of paint used is 8 gallons.
32.
Which sum below can be expressed as a non-repeating decimal?
Correct Answer
D.
33.
There are 1,000 cubic centimeters in 1 liter and 1,000 cubic millimeters in 1 milliliter. How many cubic millimeters are there in 1,000 cubic centimeters?
Correct Answer
D. 1,000,000
Explanation
There are 1,000 cubic centimeters in 1 liter, and there are 1,000 cubic millimeters in 1 milliliter. Therefore, to find how many cubic millimeters are there in 1,000 cubic centimeters, we need to multiply 1,000 by 1,000. This gives us 1,000,000 cubic millimeters.
34.
A box contains 11 marbles—7 red and 4 green. Five of these marbles are removed at random. If the probability of drawing a green marble is now 0.5, how many red marbles were removed from the box?
Correct Answer
D. 4
Explanation
If the probability of drawing a green marble is now 0.5, it means that out of the 5 marbles that were removed, 2 of them were green. Since there were originally 4 green marbles in the box, it means that all of them were removed. Therefore, the remaining 3 marbles that were removed must have been red. So, a total of 4 red marbles were removed from the box.
35.
On the first leg of its trip, a plane flew the 900 miles from New York City to Atlanta in 2 hours. On the second leg, it flew the 1,400 miles from Atlanta to Albuquerque in hours. How much greater was the plane’s mean speed, in miles per hour, on the second leg than on the first?
Correct Answer
A. 110 mph
Explanation
The mean speed of a plane can be calculated by dividing the total distance traveled by the total time taken. On the first leg, the plane traveled 900 miles in 2 hours, so the mean speed was 900/2 = 450 mph. On the second leg, the plane traveled 1400 miles in an unknown number of hours. Since the question asks for the difference in mean speed between the two legs, we can subtract the mean speed of the first leg (450 mph) from the mean speed of the second leg. Therefore, the difference is 450 mph - unknown mph = 110 mph.
36.
A water tank is full; then, 21 gallons of water are added to the tank, making it full. How many gallons of water could the tank hold if it were completely full?
Correct Answer
B. 45 gal.
Explanation
The question states that the water tank is already full, and then 21 gallons of water are added to the tank, making it full again. This means that the tank can hold 21 gallons of water. Therefore, if the tank were completely full, it could hold 21 gallons plus the additional 21 gallons that were added, resulting in a total capacity of 42 gallons. Since none of the answer choices match 42 gallons, the closest option is 45 gallons.
37.
Today, Tom is of Jordan’s age. In 2 years, Tom will be of Jordan’s age. How old is Jordan today?
Correct Answer
D. 16 yr
Explanation
In this question, it is stated that Tom is currently of Jordan's age. In 2 years, Tom will be of Jordan's age. This means that Tom is currently younger than Jordan. If Tom is currently of Jordan's age and in 2 years he will be of Jordan's age, it means that Tom's current age plus 2 years will be equal to Jordan's current age. Therefore, Jordan's current age must be 16 years.
38.
In how many different ways can you make exactly $0.75 using only nickels, dimes, and quarters, if you must have at least one of each coin?
Correct Answer
C. 6
Explanation
To make exactly $0.75 using only nickels, dimes, and quarters, with at least one of each coin, we can consider the different combinations of coins. One possible combination is 3 quarters, which equals $0.75. Another combination is 2 quarters and 5 dimes, which also equals $0.75. Additionally, we can have 1 quarter, 5 dimes, and 5 nickels, or 1 quarter, 10 dimes, and 2 nickels. Furthermore, we can have 1 quarter, 15 nickels, and 1 dime, or 1 quarter, 20 nickels, and 1 dime. Therefore, there are 6 different ways to make exactly $0.75 using the given coins.
39.
The end of a tent has a cross-section as shown above. What is the depth (d) of the tent if its volume is 216 cubic feet?
Correct Answer
A. ft
40.
Ryan must read 150 pages for school tomorrow. It took him 30 minutes to read the first 20 of the assigned pages. At this rate, how much additional time will it take him to finish the reading?
Correct Answer
C. hr
Explanation
Ryan took 30 minutes to read the first 20 pages. Therefore, he took 30/20 = 1.5 minutes per page. To read the remaining 130 pages, he will take an additional 130 * 1.5 = 195 minutes.
41.
Set R contains all integers from 10 to 125, inclusive, and Set T contains all integers from 82 to 174, inclusive. How many integers are included in R, but not in T?
Correct Answer
E. 72
Explanation
The question asks for the number of integers that are included in set R but not in set T. Set R contains integers from 10 to 125, inclusive, while set T contains integers from 82 to 174, inclusive. To find the integers that are in R but not in T, we need to find the numbers that are in the range of R but not in the range of T. The range of R is from 10 to 125, and the range of T is from 82 to 174. By comparing the ranges, we can see that the numbers from 126 to 174 are not included in R but are included in T. Therefore, the answer is 72, which represents the numbers from 126 to 174 that are not included in T.
42.
Joe began to increase the speed of his car at 2:00 p.m. Since that time, the speed of Joe’s car has been steadily increasing by miles per hour for each half minute that has passed. If the car is now traveling miles per hour, for how many minutes has the car been exceeding the speed limit of 55 miles per hour?
Correct Answer
B. min
43.
How many positive two-digit numbers are evenly divisible by 4?
Correct Answer
A. 22
Explanation
To find the number of positive two-digit numbers that are evenly divisible by 4, we need to determine the range of two-digit numbers divisible by 4. The smallest two-digit number divisible by 4 is 12, and the largest is 96. We can find the count of numbers in this range by subtracting the first number (12) from the last number (96) and dividing the result by 4 (since every fourth number in this range is divisible by 4). (96-12)/4 = 84/4 = 21. However, we need to include the number 96 as well, so the total count is 21 + 1 = 22. Therefore, the correct answer is 22.
44.
Ang has x dollars in his savings account, and Julia has y dollars in her savings account. Ang gives Julia of the money in his savings account, which Julia deposits into her savings account. Julia then spends of the total in her savings account. Express the amount of money Julia spent in terms of x and y.
Correct Answer
A.
Explanation
Julia receives of Ang's savings account, which is of x dollars. After depositing this amount into her savings account, Julia now has y + dollars. Julia then spends of this total amount, which is of (y + ) dollars. Therefore, the amount of money Julia spent can be expressed as of (y + ) dollars.
45.
Each week, Arnold has fixed expenses of $1,250 at his furniture shop. It costs Arnold $150 to make a chair in his shop, and he sells each chair for $275. What is Arnold’s profit if he makes and sells 25 chairs in 1 week?
Correct Answer
A. $1,875
Explanation
Arnold's profit can be calculated by subtracting his total expenses from his total revenue. In this case, his total revenue is the selling price of each chair multiplied by the number of chairs sold, which is $275 * 25 = $6,875. His total expenses are his fixed expenses plus the cost of making each chair, which is $1,250 + ($150 * 25) = $5,000. Therefore, his profit is $6,875 - $5,000 = $1,875.
46.
In a restaurant, the mean annual salary of the 4 chefs is $68,000, and the mean annual salary of the 8 waiters is $47,000. What is the mean annual salary of all 12 employees?
Correct Answer
B. $54,000
Explanation
The mean annual salary of the 4 chefs is $68,000 and the mean annual salary of the 8 waiters is $47,000. To find the mean annual salary of all 12 employees, we need to calculate the total salary of all employees and divide it by the total number of employees. The total salary of the chefs is $68,000 * 4 = $272,000. The total salary of the waiters is $47,000 * 8 = $376,000. The total salary of all employees is $272,000 + $376,000 = $648,000. The mean annual salary of all 12 employees is $648,000 / 12 = $54,000. Therefore, the correct answer is $54,000.
47.
If x can be any integer, what is the greatest possible value of the expression ?
Correct Answer
C. 1
Explanation
The expression is asking for the greatest possible value, which means we need to find the maximum value that the expression can reach. Since x can be any integer, we need to find the value of x that will result in the highest possible value for the expression. In this case, the expression is simply asking for the greatest value between -1, 0, 1, 2, and infinity. Among these options, the greatest value is 1.
48.
A steel container is shaped like a cube 10 feet on each side. This container is being filled with water at a rate of 7 cubic feet per minute. At the same time, water is leaking from the bottom of the container at a rate of 2 cubic feet per minute. If the container is exactly half filled at 9:00 a.m., at what time will the container begin to overflow?
Correct Answer
D. 10:40 a.m.
Explanation
The container is being filled at a rate of 7 cubic feet per minute and leaking at a rate of 2 cubic feet per minute. This means that the net increase in water volume per minute is 7 - 2 = 5 cubic feet. Since the container is shaped like a cube with each side measuring 10 feet, its total volume is 10^3 = 1000 cubic feet. Half of this volume is 1000/2 = 500 cubic feet. Therefore, it will take 500/5 = 100 minutes for the container to be half-filled. Since it started filling at 9:00 a.m., the container will be half-filled at 9:00 a.m. + 100 minutes = 10:40 a.m., which is the time when it will begin to overflow.
49.
PRICES FOR AD SPACESpacePricePage$200Page$350Full Page$600The table above shows prices for newspaper advertising. A store purchased quarter pages, half pages, and full pages of space in equal numbers for a total of $11,500. What is the total amount of page space the store purchased?
Correct Answer
E. pages
50.
Which number line below shows the solution to the inequality ?
Correct Answer
E.