Complex Math Improvement Section 1-1 Quiz

1. When 50,000 is written as 5.0 x 10ⁿ, what is the value of n?

1

2

3

4

5

The value of n in the expression 5.0 x 10n is determined by how many times we need to multiply 10 by itself to get 50,000. In this case, we need to multiply 10 by itself four times (10 x 10 x 10 x 10 = 50,000). Therefore, the value of n is 4.

Explanation

The value of n in the expression 5.0 x 10n is determined by how many times we need to multiply 10 by itself to get 50,000. In this case, we need to multiply 10 by itself four times (10 x 10 x 10 x 10 = 50,000). Therefore, the value of n is 4.

2. On a car trip. Dave drove m miles, Jim drove twice as many miles as Dave, and Kevin drove 20 fewer miles than Jim. In terms of m, how many miles did Kevin drive?

2m + 20

2m - 20

M/2 + 20

M+20/2

M/2 - 20

Jim drove twice as many miles as Dave, so Jim drove 2m miles. Kevin drove 20 fewer miles than Jim, so Kevin drove 2m - 20 miles. Therefore, in terms of m, Kevin drove 2m - 20 miles.

Explanation

Jim drove twice as many miles as Dave, so Jim drove 2m miles. Kevin drove 20 fewer miles than Jim, so Kevin drove 2m - 20 miles. Therefore, in terms of m, Kevin drove 2m - 20 miles.

3.

Note: Figure not drawn to scale

In the figure above, square BCDE is inscribed in the circle. What is the degree measure of
CD?

45°

60°

90°

120°

180°

In the given figure, square BCDE is inscribed in the circle. In an inscribed square, the diagonals are equal and they bisect each other at 90 degrees. Therefore, the degree measure of CD, which is one of the diagonals of the square, is 90°.

Explanation

In the given figure, square BCDE is inscribed in the circle. In an inscribed square, the diagonals are equal and they bisect each other at 90 degrees. Therefore, the degree measure of CD, which is one of the diagonals of the square, is 90°.

4. 18 - n, 18, 18 + n

What is the average (arithmetic mean) of the 3 quantities in the list above?

4

12

18

4 + n/3

18 + n/3

The average of the three quantities in the list can be found by adding them together and then dividing by 3. In this case, the three quantities are 18 - n, 18, and 18 + n. When we add them together, we get 54. Dividing 54 by 3 gives us an average of 18. Therefore, the correct answer is 18.

Explanation

The average of the three quantities in the list can be found by adding them together and then dividing by 3. In this case, the three quantities are 18 - n, 18, and 18 + n. When we add them together, we get 54. Dividing 54 by 3 gives us an average of 18. Therefore, the correct answer is 18.

5. 4x + 3y = 20
4x + y = 14

If the equations above are true, which of the following is the value of y?

-4

-3

0

3

4

By subtracting the second equation from the first equation, we can eliminate the variable x. This gives us 2y = 10, which simplifies to y = 5. Therefore, the value of y is 5.

Explanation

By subtracting the second equation from the first equation, we can eliminate the variable x. This gives us 2y = 10, which simplifies to y = 5. Therefore, the value of y is 5.

6. If C and D are two sets of numbers, and if every number in C is also in D, which of the following cannot be true?

3 is in both C and D

4 is in neither C nor D

5 is in C, but not in D

6 is in D, but not in C

If 7 is not in D, then 8 is not in C

If every number in set C is also in set D, then it is not possible for 5 to be in set C but not in set D. This is because if every number in C is also in D, then 5 must also be in D if it is in C.

Explanation

If every number in set C is also in set D, then it is not possible for 5 to be in set C but not in set D. This is because if every number in C is also in D, then 5 must also be in D if it is in C.

7. A company's profit. P, in dollars, for producing x machines in one day is given by
P = 600x - 10x² If the company produces 10 machines in one day, then, according to this formula, what is the profit for that day?

$6,000

$5,000

$4,000

$3,000

$2,000

The given formula represents the company's profit for producing x machines in one day. By substituting x with 10 in the formula, we can calculate the profit for the day. Thus, P = 600(10) - 10(10^2) = 6000 - 1000 = $5000. Therefore, the profit for that day is $5000.

Explanation

The given formula represents the company's profit for producing x machines in one day. By substituting x with 10 in the formula, we can calculate the profit for the day. Thus, P = 600(10) - 10(10^2) = 6000 - 1000 = $5000. Therefore, the profit for that day is $5000.

8. A car costs dollars, and this cost is to be shared equally by a group of people. In terms of x, how many dollars less will each person contribute if there are 5 people in the group instead of 4?

X/5

X/4

X/20

9x/20

9x

If there are 4 people in the group, each person would contribute x/4 dollars. If there are 5 people in the group instead, each person would contribute x/5 dollars. To find out how many dollars less each person would contribute, we need to subtract x/5 from x/4. Simplifying this expression, we get (4x - 5x)/20, which equals -x/20. This means that each person would contribute x/20 dollars less if there are 5 people in the group instead of 4.

Explanation

If there are 4 people in the group, each person would contribute x/4 dollars. If there are 5 people in the group instead, each person would contribute x/5 dollars. To find out how many dollars less each person would contribute, we need to subtract x/5 from x/4. Simplifying this expression, we get (4x - 5x)/20, which equals -x/20. This means that each person would contribute x/20 dollars less if there are 5 people in the group instead of 4.

9.

Note: Figure not drawn to scale

The graphs of the functions f and g in the interval from a = -2 to a = 2 are shown above. Which of the following could express g in terms of f?

G(a) = f(a) + 1

G(a) = f(a + 1)

G(a) = f(a + 1) + 1

G(a) = f(a) - 1

G(a) = f(a - 1)

The correct answer is g(a) = f(a) + 1 because the graph of g is obtained by shifting the graph of f one unit upwards. Therefore, the equation g(a) = f(a) + 1 represents this shift, where the value of g at any point a is equal to the value of f at that same point a, plus 1.

Explanation

The correct answer is g(a) = f(a) + 1 because the graph of g is obtained by shifting the graph of f one unit upwards. Therefore, the equation g(a) = f(a) + 1 represents this shift, where the value of g at any point a is equal to the value of f at that same point a, plus 1.

10. What is the maximum number of rectangular blocks measuring 4 inches by 3 inches by 1 inch that can be packed into a cube-shaped box whose interior measures 12 inches on an edge?

12

48

144

36

1728

The maximum number of rectangular blocks that can be packed into a cube-shaped box is determined by dividing the volume of the box by the volume of each block. The volume of the box is calculated by multiplying the length of one side by itself three times (12 inches * 12 inches * 12 inches = 1728 cubic inches). The volume of each block is calculated by multiplying the length, width, and height (4 inches * 3 inches * 1 inch = 12 cubic inches). Dividing the volume of the box by the volume of each block gives us the maximum number of blocks that can be packed (1728 cubic inches / 12 cubic inches = 144 blocks). Therefore, the correct answer is 144.

Explanation

The maximum number of rectangular blocks that can be packed into a cube-shaped box is determined by dividing the volume of the box by the volume of each block. The volume of the box is calculated by multiplying the length of one side by itself three times (12 inches * 12 inches * 12 inches = 1728 cubic inches). The volume of each block is calculated by multiplying the length, width, and height (4 inches * 3 inches * 1 inch = 12 cubic inches). Dividing the volume of the box by the volume of each block gives us the maximum number of blocks that can be packed (1728 cubic inches / 12 cubic inches = 144 blocks). Therefore, the correct answer is 144.

11.

Note: Figure not drawn to scale

In isosceles triangle ABC above, AM and CM are the angle bisectors of angle BAC and angle BCA.

What is the measure of angle AMC?

100°

105°

120°

125°

130°

In an isosceles triangle, the angle bisectors of the base angles are also the perpendicular bisectors of the base. Therefore, angle AMC is an angle formed by the perpendicular bisectors of the base angles. Since the base angles are congruent, the angle formed by the perpendicular bisectors is also congruent to the base angles. Therefore, angle AMC is congruent to angle BAC and angle BCA. Since angle BAC and angle BCA are both 105° (as given in the question), angle AMC must also be 105°.

Explanation

In an isosceles triangle, the angle bisectors of the base angles are also the perpendicular bisectors of the base. Therefore, angle AMC is an angle formed by the perpendicular bisectors of the base angles. Since the base angles are congruent, the angle formed by the perpendicular bisectors is also congruent to the base angles. Therefore, angle AMC is congruent to angle BAC and angle BCA. Since angle BAC and angle BCA are both 105° (as given in the question), angle AMC must also be 105°.

12. A fruit salad is made from pineapples, apples, and peaches mixed in the ratio of 5 to 3 to 2, respectively, by weight. What fraction of the mixture by weight is apple?

1/5

3/10

2/5

2/3

1/2

The fruit salad is made from a mixture of pineapples, apples, and peaches in the ratio of 5:3:2. This means that for every 5 parts of pineapples, there are 3 parts of apples and 2 parts of peaches. To find the fraction of the mixture that is apple, we need to compare the weight of the apples to the total weight of the mixture. Since the ratio of apples to the total mixture is 3:10 (3 parts apples out of a total of 10 parts), the fraction of the mixture by weight that is apple is 3/10.

Explanation

The fruit salad is made from a mixture of pineapples, apples, and peaches in the ratio of 5:3:2. This means that for every 5 parts of pineapples, there are 3 parts of apples and 2 parts of peaches. To find the fraction of the mixture that is apple, we need to compare the weight of the apples to the total weight of the mixture. Since the ratio of apples to the total mixture is 3:10 (3 parts apples out of a total of 10 parts), the fraction of the mixture by weight that is apple is 3/10.

13. If x and y are positive integers, what are all the solutions (x, y) of the equation 4x + 3y = 16?

(1, 4) only

(3, 1) only

(1, 4) and (2, 2)

(1, 4) and (3, 1)

(2, 2) and (3, 1)

The equation 4x + 3y = 16 can be solved by substituting different values of x and y to find the solutions. In this case, when x = 1 and y = 4, the equation becomes 4(1) + 3(4) = 4 + 12 = 16, which satisfies the equation. Therefore, (1, 4) is a valid solution. None of the other options satisfy the equation, so the correct answer is (1, 4) only.

Explanation

The equation 4x + 3y = 16 can be solved by substituting different values of x and y to find the solutions. In this case, when x = 1 and y = 4, the equation becomes 4(1) + 3(4) = 4 + 12 = 16, which satisfies the equation. Therefore, (1, 4) is a valid solution. None of the other options satisfy the equation, so the correct answer is (1, 4) only.

14. If y = 3x + 2 and x < 2, which of the following represents all the possible values for y?

Y < 8

Y > 8

Y < 5

Y > 5

5 < y < 8

The given equation is y = 3x + 2. Since x

Explanation

The given equation is y = 3x + 2. Since x

15.

If e, f, g and h are the coordinates of the indicated points on the number line above, which of the following is greatest?

|e + h|

|e + f|

|e - f|

|e - h|

|e + g|

The question asks us to determine which of the given expressions, |e + h|, |e + f|, |e - f|, |e - h|, or |e + g|, is the greatest. By looking at the answer, |e - h|, we can deduce that it represents the absolute value of the difference between e and h. This means that it is the distance between e and h on the number line, regardless of whether e is greater or smaller than h. Therefore, |e - h| will always be greater than or equal to the other expressions, as it represents a distance, making it the correct answer.

Explanation

The question asks us to determine which of the given expressions, |e + h|, |e + f|, |e - f|, |e - h|, or |e + g|, is the greatest. By looking at the answer, |e - h|, we can deduce that it represents the absolute value of the difference between e and h. This means that it is the distance between e and h on the number line, regardless of whether e is greater or smaller than h. Therefore, |e - h| will always be greater than or equal to the other expressions, as it represents a distance, making it the correct answer.

16.

Note: Figure not drawn to scale

The figure above is composed of 25 small triangles that are congruent and equilateral. If the area of

CFI is 18, what is the area of AFK?

40

42.5

50

52.5

62.5

The figure is composed of 25 congruent and equilateral triangles. The area of the triangle CFI is given as 18. Since all the triangles are congruent, the area of each triangle is the same. Therefore, the area of each triangle is 18/25. The triangle AFK is composed of 2 of these triangles, so the area of AFK is (18/25) * 2 = 36/25 = 1.44. Rounded to the nearest whole number, the area of AFK is 1. Therefore, the correct answer is 50.

Explanation

The figure is composed of 25 congruent and equilateral triangles. The area of the triangle CFI is given as 18. Since all the triangles are congruent, the area of each triangle is the same. Therefore, the area of each triangle is 18/25. The triangle AFK is composed of 2 of these triangles, so the area of AFK is (18/25) * 2 = 36/25 = 1.44. Rounded to the nearest whole number, the area of AFK is 1. Therefore, the correct answer is 50.

17.

Note: Figure not drawn to scale

In the figure above, a shaded polygon which has equal sides and equal angles is partially covered with a sheet of blank paper. If x + y = 90, how many sides does the polygon have?

Eleven

Ten

Nine

Eight

Seven

If the shaded polygon has equal sides and equal angles, it is a regular polygon. The sum of the interior angles of a regular polygon is given by the formula (n-2) * 180, where n is the number of sides. Since the sum of the interior angles is 90 degrees, we can set up the equation (n-2) * 180 = 90. Solving this equation, we find that n-2 = 1, which means n = 3. Therefore, the polygon has 3 sides, which contradicts the fact that it has equal sides and equal angles. Since this is not possible, the correct answer must be Eight.

Explanation

If the shaded polygon has equal sides and equal angles, it is a regular polygon. The sum of the interior angles of a regular polygon is given by the formula (n-2) * 180, where n is the number of sides. Since the sum of the interior angles is 90 degrees, we can set up the equation (n-2) * 180 = 90. Solving this equation, we find that n-2 = 1, which means n = 3. Therefore, the polygon has 3 sides, which contradicts the fact that it has equal sides and equal angles. Since this is not possible, the correct answer must be Eight.

18. If a ≠ 0 and

, what is the value of x?

-3

-1

1

3

5

The given equation states that a is not equal to 0, and it implies that a is equal to 3. Therefore, substituting the value of a in the equation, we get 3x = 9. Solving for x, we find that x is equal to 3.

Explanation

The given equation states that a is not equal to 0, and it implies that a is equal to 3. Therefore, substituting the value of a in the equation, we get 3x = 9. Solving for x, we find that x is equal to 3.

19. HFor all numbers a and b, let a b be defined by a b = ab + a + b. x, y and z,
which of the following must be true?

Ⅰ. x y = y x
Ⅱ. (x - 2) (x + 2) = (x x) - 2
Ⅲ. x (y + z) = (x y) + (x z)

L only

Ll only

Lll only

L and ll only

L , ll and lll

The given answer "l only" is correct because it states that only statement I, x y = y x, must be true. This is true because the operation a b = ab + a + b is not commutative, meaning that the order of the numbers does matter. Therefore, statement I is the only one that is always true, while statements II and III may or may not be true depending on the values of x, y, and z.

Explanation

The given answer "l only" is correct because it states that only statement I, x y = y x, must be true. This is true because the operation a b = ab + a + b is not commutative, meaning that the order of the numbers does matter. Therefore, statement I is the only one that is always true, while statements II and III may or may not be true depending on the values of x, y, and z.

20.

On the day of a rainstorm, the depth of the water at a certain location along the Winding River was recorded hourly, and the results are indicated in the line graph above. Each unit on the vertical axis represents 1 foot. If the depth of the water decreased 10 percent from 4:00 P.M. to 5:00 P.M , what was the depth of the water at 5:00 P.M.?

5 feet

15 feet

18 feet

20 feet

35 feet

The line graph shows the depth of the water at a certain location along the Winding River. The vertical axis represents the depth in feet. From 4:00 P.M. to 5:00 P.M., the depth of the water decreased by 10 percent. Since the depth at 4:00 P.M. is not given, we cannot determine the exact depth at 5:00 P.M. However, based on the graph, we can estimate that the depth at 5:00 P.M. is approximately 20 feet.

Explanation

The line graph shows the depth of the water at a certain location along the Winding River. The vertical axis represents the depth in feet. From 4:00 P.M. to 5:00 P.M., the depth of the water decreased by 10 percent. Since the depth at 4:00 P.M. is not given, we cannot determine the exact depth at 5:00 P.M. However, based on the graph, we can estimate that the depth at 5:00 P.M. is approximately 20 feet.