# ACT Form 73C Math Answer Explanations

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*Explain to me:*

$$a(4-a)-5(a+7)=4a-a^2-5a-35=-a^2-a-35$$

Choice C.

1. Which of the following expressions is equivalent to a(4 — a) — 5(a + 7) ?

$$\frac{1}{4}=0.25 \\ \Rightarrow 0.03 \leq 0.2 \leq 0.25 \\ \Rightarrow 0.03 \leq 0.2 \leq \frac{1}{4}$$

Choice G.

2. Which of the following inequalities orders the numbers 0.2, 0.03, and from least to greatest?

$$x^2=29-4=25 \\ \Rightarrow x^2-4=25-4=21$$

Choice C.

3. If x^2 + 4 =29,then x^2— 4 =?

Since T and U bisect QR and QS, respectively, TU//RS. Thus,

$$\angle TUQ=\angle S=60° \\ \Rightarrow \angle TUS=180°-\angle TUQ=180°-60°=120°$$

Choice H.

4. In the figure below, QRS is equilateral, side QR is bisected by T, and side QS is bisected by U. What is the degree measure of TUS ?

The line is going downwards. Therefore the slope is negative.

Choice C.

5. A line in the standard (x,y) coordinate plane passes through the points (-3,4) and (2,-5). The slope of the line:

$$\sqrt{81}=9 \\ \sqrt{99}\approx\sqrt{100}=10 \\ 9 \leq 9.2371 \leq 10 \\ \Rightarrow 81\leq the \ number \leq 99$$

Choice K.

6. The square root of a certain number is approximately 9.2371. The certain number is between what 2 integers?

$$p=\frac{10-2}{10}=\frac{4}{5}$$

Choice E.

7. A bag contains 10 pieces of flavored candy: 4 lemon, 3 strawberry, 2 grape, and 1 cherry. One piece of candy will be randomly picked from the bag. What is the probability the candy picked is NOT grape flavored?

$$\frac{x_{A}+(-3)}{2}=1 \\ \Rightarrow x_{A}=5 \\ \frac{y_{A}+4}{2}=2 \\ \Rightarrow y_{A}=0 $$

Choice K.

8. When points A and B(-3,4) are graphed in the standard (x,y) coordinate plane below, the midpoint of AB will be (1,2). What will be the coordinates of point A ?

$$2x+8=116 \\ \Rightarrow x=54$$

Choice B.

9. Andrea manages a company that currently has 116 customers, which is 8 more than twice the number of customers the company had 1 year ago. How many customers did the company have 1 year ago?

$$500+200x=$2,200 \\ \Rightarrow x=$8.50$$

Choice H.

10. Joseph will have a 200-foot-long fence installed around his yard. The A+ Fence Company charges a $500.00 fee, plus a set amount per foot of fence. The A+ Fence Company has given Joseph an estimate of $2,200.00 to install the fence around his yard. What is the set amount per foot of fence?

$$(3x+6)(2x-1)=6x^2-3x+12x-6=6x^2+9x-6$$

Choice D.

11. Which of the following expressions is equivalent to (3x + 6)(2x — 1) ?

A box of 25 candies at Tamika’s Treat costs $4.25. Thus the money left is $10.00 – $4.25 = $5.75, which is 23 quarters:

$$\frac{$5.75}{$0.25}=23$$

Choice J.

12. Carrie’s Chocolate Shop and Tamika’s Treat Shop both sell candy in boxes. The table below lists the price (the total amount the customer pays) of each box of candy sold at the shops. For each shop, there is a linear relationship between the price of a box of candies and the number of candies in that box. These are the only numbers of candies that can be purchased at the shops. Jeremy has $10.00 in quarters to spend on candy. What is the maximum number of quarters he would have left after paying for a box of 25 candies at Tamika’s Treat Shop? (Note: Each quarter is worth $0.25.)

$$\frac{$3.75}{20}=$0.1875\approx $0.19$$

Choice B.

13. At Tamika’s Treat Shop, what is the average price per candy in a box of 20, to the nearest $0.01 ?

Only the equation in F fulfils all the given prices and the number of candies given in the table.

Choice F.

14. Which of the following equations gives the relationship between the price in dollars, c, and the number of candies, n, in a box of candies at Carrie’s Chocolate Shop?

The opposite of ‘at least 1’ is ‘no one’.

Choice E.

15. Mr. Chang made the true statement “At least 1 student in this class is a section leader in the band.” Which of the following statements must be FALSE?

Choice J.

16. In the figure below, vertices D and F of ADEF lie on CG, the measure of LCDE is 148°, and the measure of EFG is 140°. What is the measure of DEF ?

$$\frac{9\times9\times12}{3\times3\times3}=36$$

Choice E.

17. A company ships notepads in rectangular boxes that each have inside dimensions measuring 9 inches long, 9 inches wide, and 12 inches tall. Each notepad is in the shape of a cube with an edge length of 3 inches. What is the maximum number of notepads that will fit in 1 closed box?

Choice .

18. The function f is defined as f(x) = —4×3 4×2. What is f(-4) ?

$$x+2y=4 \\ -2x+y=7 \\

\Rightarrow x=-2, \ y=3$$

Choice A.

19. Which of the following (x,y) pairs is the solution for the system of equations x + 2y = 4 and —2x + y = 7 ?

$$x=\sqrt{8^2+9^2}\approx12 \\

Perimeter = 8+10+12+19=49\approx50$$

Choice G.

20. The floor plan of Helena’s kitchen is shown in the figure below; the given dimensions are in feet. Helena will install baseboard along the bottom of each wall (shown by solid lines in the floor plan). According to the floor plan, which of the following distances is closest to the perimeter, in feet, of Helena’s kitchen?

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21. A 5-inch-by-7-inch photograph was cut to fit exactly into a 4-inch-by-6-inch frame. What is the area, in square inches, of the part of the photograph that was cut off?

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22. In the figure shown below, AE || DC, AC and DE intersect at B, and the given lengths are in feet. What is the length, in feet, of BC ?

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23. Given that x <= 4 and x + y >= 5, what is the LEAST value that y can have?

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24. The circle shown below has diameter AD, and points B and C lie on the circle. The measure of ZCAD is 300, and the measure of minor arc CD is 60°. What is the measure of minor arc AC ?

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25. One caution sign flashes every 4 seconds, and another caution sign flashes every 10 seconds. At a certain instant, the 2 signs flash at the same time. How many seconds elapse until the 2 signs next flash at the same time?

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26. For all nonzero values of a and b, the value of which of the following expressions is always negative?

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27. Graphed in the same standard (x,y) coordinate plane are a circle and a parabola. The circle has radius 3 and center (0,0). The parabola has vertex (-3,-2), has a vertical axis of symmetry, and passes through (-2,-1). The circle and the parabola intersect at how many

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28. 40% of 250 is equal to 60% of what number?

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29. Which of the following inequalities is equivalent to —2x— 6y> 2y— 4 ?

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30. For an angle with measure a in a right triangle, sin a = 40/41 and tan a = 40/9. What is the value of cos a ?

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31. The perimeter of rectangle ABCD is 96 cm. The ratio of the side lengths AB:BC is 3:5. What is the length, in centimeters, of AB ?

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32. For ABC shown below, base AC has a length of 16 inches and altitude BD has a length of 8 inches. The area of a certain square is equal to the area of ABC. What is the length, in inches, of a side of the square?

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33. In the figure shown below, ABCD is a rectangle, EFGH is a square, and CD is the diameter of a semicircle. Point K is the midpoint of CD. Point J is the midpoint of both AB and EF. Points E and F lie on AB. The 3 given lengths are in meters. The length of EH is what percent of the length of AD?

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34. What is the length, in meters, of JD?

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35. What is the length, in meters, of arc CD?

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36. The figure will be placed in the standard (x,y) coordinate plane so that K is at the origin, AB is parallel to the x-axis, and 1 meter equals 1 coordinate unit. Which of the following values could be the y-coordinate of H?

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37. What is the length, in coordinate units, of the altitude from C to AB in ABC shown in the standard (x, y) coordinate plane below?

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38. At a local post office, on average, 3 customers are in line when the post office closes each day. The probability, P, that exactly n customers are in line when the post office closes can be modeled by the equation P=. Given that e-3 0.05, which of the following values is closest to the probability that exactly 2 customers are in line when the post office

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39. What is the amplitude of the function f(x) = 1/2 cos(3x + pi)?

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40. A sphere with radius x inches has a volume of 12 cubic inches. What is the value of x ? (Note: The volume of a sphere with radius r is 4/3 Pi r^3.)

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41. For 20 quiz scores in a typing class, the table below gives the frequency of the scores in each score interval. Which score interval contains the median of the scores?

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42. The angle of elevation to the top of a tree from a spot on level ground 50 feet from the base of the tree is 40°. Which of the following is closest to the height of the tree, in feet? (Note: sin 40° = 0.64, cos 40° = 0.77, tan 40° = 0.84)

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43. After visiting Florida State University during spring break, Francisco rents a car for 2 days to travel around Florida. He has $255 to spend on car rental for the 2 days. Sea Horse Car Rental charges $50 per day and $0.25 per mile. Ocean Blue Car Rental charges $60 per day and $0.20 per mile. Which company, if either, allows him to travel more miles for the 2 days, and how many miles more? (Note: Taxes are already included in the rental charges.)

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44. The table below gives experimental data values for variables x and y. Theory predicts that y varies directly with x. Based on the experimental data, which of the following values is closest to the constant of variation? (Note: The variable y varies directly with the variable x provided that y = kx for some nonzero constant k, called the constant of variation.)

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45. During a snowstorm, the relationship between the depth of accumulated snow, y inches, and the elapsed time, x hours, was modeled by the equation 2x — 5y = —5. One of the following graphs in the standard (x,y) coordinate plane models the equation for positive values of x and y. Which one?

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46. Diana is baking bread, and the original recipe calls for 1 1/2 teaspoons of yeast and 2 1/2 cups of flour. Diana will use the entire contents of a packet that contains 2 1/4 teaspoons of yeast and will use the same ratio of ingredients called for in the original recipe. How many cups of flour will Diana use?

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47. For all nonzero values of x,

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48. Four matrices are given bwlow. Which of the following matrix products is undefined?

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49. A circle, 2 chords. and some lengths, in centimeters, are shown in the figure below, which is not drawn to scale. What is the value of x ? (Note: When two chords intersect, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.)

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50. If the area of KLMN in the figure below is 12 square centimeters, what is the length, in centimeters, of LM ?

B.

Adding the 2 largest numbers in this set (18 and 17) yields 35 as the maximum sum, and doing the same with the 2 smallest numbers (1 and 2) yields 3 as the minimum. The only possible perfect squares,therefore, are those that lie between 3 and 35: 4, 9, 16, and 25.

Since 25 is the only perfect square that 16, 17, and 18 can make by combining with 1 other number, they must be paired as follows:

16 + 9 = 25

17 + 8 = 25

18+7 = 25

16 must be added to 9 and no other number, so choice A can be eliminated. That leaves:

This leaves 15 and 3 after the other 3 options are eliminated.

15 can be added to either 1 or 10 to yield a perfect square, but adding it to 10 would leave: 1, 2, 3, 4, 5, 6, 11, 12, 13, and 14.

14+2=16

13+3=16 or 1+3=4 or 6+3=9

12+4=16 or 4+5=9

11+5=16 or 5+4=9

1+3=4

2 would have only one match: 14

3+13

4+12

5+11

This would leave 1 and 6 whose sum is not a perfect square.

If 15 were added to 1, the numbers remaining would be: 2, 3, 4, 5, 6, 10, 11, 12, 13, 14

2+14 or 14+11

3+13 or 3+9

4+5 or 4+12

5+11 or 5+4

6+3 or 6+10

10+6

11+5 or 11+14

12+4 or 12+13

13+3 or 13+12

14+11 or 14+2

10+6

2+14 or 14+11

3+13 or 3+9

4+5 or 4+12

5+11 or 5+4

11+5 or 11+14

12+4 or 12+13

13+3 or 13+12

14+11 or 14+2

2+14, 3+13, 4+12, 5+11

The final set of pairs would have to be as follows based on the information above:

18+7

17+8

16+9

15+1

14+2

13+3

12+4

11+5

10+6

51. A teacher assigns each of her 18 students a different integer from 1 through 18. The teacher forms pairs of study partners by using the rule that the sum of the pair of numbers is a perfect square. Assuming the 9 pairs of students follow this rule, the student assigned which number must be paired with the student assigned the number 1?

If (x+2) and (x-1) are factors of the given quadratic, there will be no remainder. Since multiplying (x+2) and (x-1) would yield a quadratic, the question essentially wants to know what the coefficients are of the quadratic equation in a different form.

The coefficients of both sides of the equation can be set equal to each other to solve for a and b as follows:

so the coefficients are both 1, and do not give us any information related to a and b.

so 1=a+2

when we look at the constants of both sides of the equation.

We derive the following information from which we can find the values of a and b:

Since the second equation has only one variable, we can solve for that variable (a) first:

We can now use the value of a that we found to calculate the value of b:

Choice K

52. Given that (x + 2) and (x — 1) are factors of the quadratic expression below, what are the values of a and b ?

Since the given equation is a quadratic, factored form would be (x+a)(x+b) as follows:

Since h is given in terms of t, factored form would yield t=8,-3. Time cannot be negative, so t must be 8. We know that this is when the ball hits the ground because we are only able to solve for t by setting the quadratic equal to 0 (h=0).

Choice E.

53. The height above the ground, h units, of an object seconds after being thrown from the top of a building is given by the equation h = —2t2 + 10: + 48. An equivalent factored form of this equation shows that the object:

The slopes of two lines that are perpendicular to each other are negative reciprocals. The slope of the given line is:

The slope of the perpendicular line is, therefore, ½.

Choice J.

54. In the standard (x,y) coordinate plane below, the line through the points (-2,0) and (0,-4) is graphed. Which of the following values is the slope of any line that is in this plane and is perpendicular to the graphed line?

The shaded area lies below line B and above line A. Since the question presents the answers in terms of linear inequalities, the most standard method would be to find the equation of each line by identifying exact points. In both lines A and B, the clearest and most definitive points are the x and y-intercepts.

Line A:

● Coordinates: (4,0) and (0,6)

● Y-intercept: 6

● Slope:

Line B:

● Coordinates: (9,0) and (0,-6)

● Y-intercept: -6

● Slope:

Since we are looking for the region below line B:

Since we are looking for the region above line A:

Choice A

55. The shaded portion of the graph in the standard(x,y) coordinate plane below represents the solution set of one of the following systems of linear inequalities. Which one?

It is important to recognize at first glance that since 3 and 27, this exponential equation can be solved by converting both sides of the equation so as to have 3 as the base.

The purpose is to be able to equate the exponents to each other, and solve for the value accordingly. It may, at first, seem difficult, as the square root of 3 cannot be put in terms of 3 as the base.

This, however, is untrue since the square root of a number can be indicated by the square root sign and an exponent of ½. Based on this, the square root of 3 is notated by raising 3 to the power of ½.

Similarly, 27 is equivalent to 3 to the power of 3. Once that is done (without losing the outer exponent), the exponent within the parentheses can be multiplied by the one outside to make both the bases identical but raised to different powers. ½ is multiplied by j, simplifying the left side to 3 to the power of j/2, and 3 is multiplied by k, simplifying the right side to 3 to the power of 3k. The next step is simply to set j/2 equal to 3k (keeping the question in mind).

NOTE: The question is not looking for the value of j or k; it is looking for the value of j/k. Now both sides of the equation can be multiplied by 2 leaving us with j=6k. Since the question asks for the value of j/k, dividing both sides by k yields j/k=6.

Choice K

56. Whenever j and k are positive integers such that (square foot of 3)^j = 27^k, what is the value of j/k ?

Since the sequence is arithmetic, let d be the common difference between each term. Each term in an arithmetic sequence is derived by adding the common difference to the previous term. Since we know that the first term is ¾, the sequence is as follows where n is the number of the term:

The mean is:

Since the sequence has 7 terms, the median is the same as the 4th term (half way), which is also:

Choice A

57. A finite arithmetic sequence has 7 terms, and the first term is 3/4. What is the difference between the mean and the median of the 7 terms?

The length of any arc of a circle is in proportion to the corresponding central angle.

Angle CDB can be found by using the sides of the given right triangle. Since the diagram provides the lengths of the hypotenuse and the side opposite the angle, it is not necessary to use the Pythagorean Theorem to calculate the length of side BD. Sine can be used based on the information that is given:

NOTE: The format of the answer choices indicate that the value above does not need to be calculated.

Choice F.

58. In the circle with center D shown below, the length of radius CD is 4 cm, the length of BC is 1 cm, and BC is perpendicular to radius AD at B. When ADC is measured in degrees, which of the following expressions represents the length, in centimeters, of AC ?

The asymptote of a function is a line that approaches but never “touches” the graph of a rational function. There exist two types of asymptotes: vertical (x=point at which the line intersects the x-axis) and horizontal (y=point at which the line intersects the y-axis).

The question specifically asks for the vertical asymptote, which means that the denominator of the function (assuming the equation is in the form y=some function of x) must be set equal to zero (to find the value that the asymptote absolutely cannot be). The vertical asymptote of a rational function can be found by first setting the denominator equal to 0:

As can be seen above, the value that x cannot be is -204/203, as this would make the expression undefined (denominator=0).

Choice D.

59. Which of the following linear equations gives the vertical asymptote for the graph of y = 201x+ 202 / 203x + 204 in the standard (x,y) coordinate plane?

Conic sections are the result of various cross sections of a cone. More specifically, they are the result of the intersection of a cone and a plane. The conic section that this intersection yields is subject to the angle at which the plane intersects or “slices” the cone as follows:

The question specifies that the plane is not parallel to the base, which means it cannot be a circle–choice G can be eliminated. Options J, K, and H can be eliminated as well since the resulting sections cannot have vertices.That leaves option F, which is also valid based on the outcome of the intersection between a cone and a plane as shown in the diagram with the ellipse.

Additionally, the question mentions that the plane does not intersect the vertex or the base. A parabola would result if the plane intersected the base of the cone, and a hyperbola would result if the plane were intersecting two vertically aligned cones whose vertices met at a point.

Additionally, a parabola could also be the result if the plane were intersecting both the base and the vertex at a 90 degree angle from the base.

This answer can also be derived intuitively by visualizing the possibilities.

Choice F.

60. A right circular cone is intersected by a plane that is not parallel to the base and does not intersect the base or vertex, ps shown in the figure below. One of the following figures shows the shape of the intersection. Which figure is it?

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