ACT Form 71H Math Answer Explanations
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Explain to me:
1. Danica is trying to encourage her classmates to donate books for the fund-raiser and book-drive for the local library. Danica will donate $35.00, plus $0.07 for each book donated by her classmates. Which of the following equations gives Danica’s donation, d dollars, if b books are donated by her classmates?
it says that “Danica will donate $35.00, plus $0.07 for each book donated by her classmates.”, and that her classmates donated b books:
d = 35 + 0.07b
Choice A.
2. If z/-4 = —16, then z = ?
z = -16 × (-4) = 64
Choice K.
3. 4b^8 • 5b^3 is equivalent to:
Rule name | Rule | Example |
---|---|---|
Product rules | a^{ n} ⋅ a^{ m} = a^{ n+m} | 2^{3} ⋅ 2^{4} = 2^{3+4} = 128 |
a^{ n} ⋅ b^{ n} = (a ⋅ b)^{ n} | 3^{2} ⋅ 4^{2} = (3⋅4)^{2} = 144 | |
Quotient rules | a^{ n} / a^{ m} = a^{ n}^{–m} | 2^{5} / 2^{3} = 2^{5-3} = 4 |
a^{ n} / b^{ n} = (a / b)^{ n} | 4^{3} / 2^{3} = (4/2)^{3} = 8 | |
Power rules | (b^{n})^{m} = b^{n⋅m} | (2^{3})^{2} = 2^{3⋅2} = 64 |
_{b}n^{m} _{= b}(n^{m}) | _{2}3^{2} _{= 2}(3^{2})_{= 512} | |
^{m}√(b^{n}) = b ^{n/m} | ^{2}√(2^{6}) = 2^{6/2} = 8 | |
b^{1/n} = ^{n}√b | 8^{1/3} = ^{3}√8 = 2 | |
Negative exponents | b^{-n} = 1 / b^{n} | 2^{-3} = 1/2^{3} = 0.125 |
Zero rules | b^{0} = 1 | 5^{0} = 1 |
0^{n} = 0 , for n>0 | 0^{5} = 0 | |
One rules | b^{1} = b | 5^{1} = 5 |
1^{n} = 1 | 1^{5} = 1 | |
Minus one rule | (-1)^{n} = -1 , n odd v (-1)^{n} = 1 , n even |
(-1)^{5} = -1 |
Derivative rule | (x^{n})‘ = n⋅x^{ n}^{-1} | (x^{3})‘ = 3⋅x^{3-1} |
Integral rule | ∫ x^{n}dx = x^{n}^{+1}/(n+1)+C | ∫ x^{2}dx = x^{2+1}/(2+1)+C |
4b^{8}⋅5b^{3 }= 4⋅5⋅b^{8+3}=20b^{11}
Choice D.
4. In the figure below, ∠QPR measures 40°, ∠PQR measures 105°, and points Q, R, and S are collinear. What is the measure of ∠PRS ?
The Exterior Angle Theorem states that, for a triangle:
- The exterior angle d equals the angles a plus b
- The exterior angle d is greater than angle a, or angle b.
Therefore, ∠PRS = 40° + 105° = 145°
Choice K.
5. The sum of 2 numbers is 90. The smaller number is 50 less than the larger number. What is the larger number?
Let the larger number be L, the smaller number be S:
L + S = 90
L = S + 50
⇒ S = 20, L = 70
Choice D.
6. |7 — 5| — |1 – 8| = ?
The absolute value of a number is its distance from zero on a number line.
|7 – 5| – |1 – 8| = 2 – 7 = -5
Choice G.
7. What is the value of x^2 + 2 + y^2 — 3 when x = 3 and y = —3 ?
3^{2 }+ 2 + (-3)^{2} – 3 = 17
Choice E.
8. What is the sum of the 4 binomials listed below? x^2 + 2x, 3x + 5, x^2 + 1, 6x — 4
(x^{2 }+ 2x) + (3x + 5) + (x^{2 }+ 1) + (6x – 4)
= (x^{2 }+ x^{2}) + (2x + 3x + 6x) + (5 + 1 – 4)
=2x^{2} + 11x + 2
Choice G.
9. Bert’s Building Supply receives shipments of only 2 kinds of lawn mowers: Tough Cuts and Easy Pushes. Today’s shipment contains 96 lawn mowers with twice as many Tough Cuts as Easy Pushes. How many of these 96 are Tough Cuts?
Let the number of Tough Cuts be T, the number of Easy Pushes be E,
T + E = 96
T = 2E
⇒ E = 32, T = 64
Choice E.
10. Adina plays in a bowling league. Her bowling scores for last week are listed below. What is the median of Adina’s bowling scores for last week? 142,186,201,191,116,201,175
The median of a set of numbers is the middle number in the set (after the numbers have been arranged from least to greatest) — or, if there are an even number of data, the median is the average of the middle two numbers.
In descending order, the set of numbers is:
116, 142, 175, 186, 191, 201, 201
Median = 186
Choice H.
11. The table below shows Shannon’s height, in inches, on her birthday from the day she was born (birth) to age 5. What was the average rate of change in Shannon’s height, in inches per year, from birth to age 5 ?
(50 – 20) inches / 5 years = 6 inches
Choice C.
12. The side lengths, in centimeters, of right triangle △DEF are given in the figure below. What is the area, in square centimeters, of △DEF ?
The area of a triangle is given by the equation:
Area_{△ }= (1/2) × base × height
Therefore,
Area_{△ }= (1/2) × 16 × 12 = 96
Choice G.
13. Christopher bought 4 cans of soup for a total of $3.36, which included sales tax of $0.16. At the same per-can cost, what is the cost before the sales tax is added for 6 cans of the same soup?
($3.36 – $0.16) ÷ 4 × 6= $4.8
Choice C.
14. A map of the locations of Marville, Sun Town, and Bright City is shown in the standard (x,y) coordinate plane below, where coordinates for x and y are given in miles. The coordinates of Marville and Bright City are given, and Sun Town is located along a straight line exactly halfway between Marville and Bright City. Highway H from Marville to Bright City is 390 miles long. Highway K from Sun Town to Bright City is 200 miles long. What are the coordinates of Sun Town?
Midpoint of a Line Segment
The midpoint is halfway between the two end points:
- Its x value is halfway between the two x values
- Its y value is halfway between the two y values
To calculate it:
- Add both “x” coordinates, divide by 2
- Add both “y” coordinates, divide by 2
x = (1/2) × (0 + 300) = 150
y = (1/2) × (0 + 160) = 80
Choice H.
15. The straight-line distance, in miles, from Marville to Bright City must be:
Distance between two points P(x1, y1) and Q(x2, y2) is given by:
$$d(P,Q) = \sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}$$
Distance of a point P(x, y) from the origin is given by:
$$d(0,P) = \sqrt{x^2+y^2}$$
Therefore,
$$d(0,Bright\ City) = \sqrt{300^2+160^2}= 340$$
which is between 300 and 390.
Choice C.
16. Chelsea’s car travels an average of 32 miles per gallon of gas used, and she pays an average of $2.25 per gallon of gas. Chelsea will drive her car along Highway H from Marville to Bright City. To the nearest $1, what will be the total cost of gas that her car uses for the drive?
390 ÷ 32 × $2.25 = $27.42 ≈ $27
Choice J.
17. Let 0 <a<b<c
For any fraction, the greater its numerator and the lessor its denominator, the greater the value of the fraction.
Among the choices, d / a has the greatest numerator d and the least denominator a.
Choice A.
18. Maurice earns $12.50 per hour for the first 40 hours he works each week. For each hour beyond 40 that he works in 1 week, Maurice earns $18.75 per hour. Last week Maurice earned $931.25. How many hours did Maurice work last week?
Let the number of extra hours Maurice worked beyond 40 hours be E:
$12.5 × 40 + $18.75 × E = $931.25
⇒ E = 23
⇒ Total hours = 40 + 23 = 63 hours
Choice J.
19. An apartment building is 3 stories tall. Each story has 4 identical apartments. Each apartment’s living space consists of 4 rectangular rooms: a bathroom 8 feet by 10 feet, a kitchen 10 feet by 12 feet, a bedroom 12 feet by 12 feet, and a living room 12 feet by 14 feet. What is the area, to the nearest 1,000 square feet, of living space in the 3-story apartment building?
8 × 10 + 12 × 14 + 10 × 12 + 12 × 12 =
Choice B.
20. The perimeter of a parallelogram is 80 inches, and the length of 1 side is 18 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?
The perimeter of a parallelogram can be found by the formula:
perimeter = 2(w+h)
where:
w is the base length of the parallelogram
h is the side length
80 = 2 (18 + h)
⇒ h = 22
The other sides are: 18, 22, 22.
Choice H.
21. The side lengths of right triangle ALMN are given in centimeters in the figure below. What is tan N ?
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22. The empty container shown below is a right rectangular prism with dimensions given in inches. How many cubic inches of liquid are needed to fill the container to 75% of its capacity?
The volume of a right rectangular prism is given by the formula:
V = length × width × height
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23. The length of a rectangle is 4 inches longer than the width. The perimeter of the rectangle is 28 inches. What is the width of the rectangle, in inches?
The perimeter of a rectangle can be found by the formula:
Perimeter = 2 ( w + l )
where:
w is the width of the rectangle
l is the length of the rectangle
…
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24. The figure below shows 3 different views of the same fair cube. Each face of the cube has 1 design drawn on it. The cube will be rolled once. What is the probability that a face showing a star (*) will be on top?
The cube has 2 stars (★), 2 circles (●), and 2 squares (■).
Thus the probability that a star (★) will show up is:
…
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25. For functions f and g defined by f(x) = 2x^2 + x and g(x) = 3x — 1, what is the value of f(g(3)) ?
First off, we need to to find g(3) which equals to
g(3) = 3 × (3) – 1 = 8
…
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26. What is the value of √(a2 + b2) when a = √5 and b = √10 ?
Plugging the numbers into the expression, we have:
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27. All of the following statements concern triangles that are similar, congruent, or both. Which statement is FALSE ?
Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion . In other words, similar triangles are the same shape, but not necessarily the same size.
The triangles are congruent if, in addition to this, their corresponding sides are of equal length.
…
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28. Distinct lines l and m intersect, forming 4 pairs of adjacent angles. Which of the following statements must be true about these 4 pairs of angles?
The measure of a straight angle is 180°.
As shown in the graph above, each pair of adjacent angles form a straight angle of 180°.
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29. In the standard (x,y) coordinate plane, what is the slope of the line through (3,7) and (-2,4) ?
Given any two points on the line, its slope is given by the formula
$$slope = \frac{A_{y}-B_{y}}{A_{x}-B_{x}}$$
where:
A_{x} the x coordinate of point A
A_{y} the y coordinate of point A
B_{x} the x coordinate of point B
B_{y} the y coordinate of point B
Thus,
…
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30. Which of the following expressions is equivalent to (2x -3)(—x — 7) ?
(2x – 3)(-x – 7) = (2x – 3)(-1)(x + 7) = …
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31. Tomas plans to construct a circular fishpond with a diameter of 9 ft. Which of the following is closest to the length, in feet, of the decorative fencing that Tomas needs to enclose the fishpond along its perimeter?
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32. Ursula will order boxes of red pens and boxes of blue pens for her company. The table below gives the number of pens in each box and the price per box. Ursula will order a total of 50 boxes of pens for a total price of $280. Which of the following systems of equations gives a true relationship between the r boxes of red pens and b boxes of blue pens that Ursula will order?
Since there are a total of 50 boxes of pens, thus:
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33. Which of the following inequalities is an equivalent algebraic expression for the statement below?
The expression ” the product of 4 and a number n” is written numerically as “4n”;
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34. The quadratic formula gives the 2 roots for the equation axe + bx + c = O. What are the 2 roots of the equation 3×2 — x = 10 ?
The quadratic formula for the roots of the equation is way too complicated to apply under test conditions.
Thus we will just get the two roots by factoring the equation:
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35. The shaded sector of the circle shown below is bounded by radius AO, radius BO, and minor arc AB . The length of AO is 4 meters, and the area of the shaded sector is 6π square meters. What is the measure of ∠A0B ?
Generally, The area of a sector of a circle is ½ π r² ⋅ (θ°/360°), where r is the radius and θ the angle in degrees subtended by the arc at the centre of the circle. So in the below diagram, the shaded area is equal to ½ r² θ°/360° .
Thus,
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36. The rental fee for the use of a reception hall is $500. To encourage use of the hall’s catering service, the hall’s rental fee is reduced by $100 for each group of 20 meals ordered through the hall’s catering service. All 20 meals in a group must be ordered for each reduction in the rental fee. One of the following graphs shows the hall’s rental fee when 0 to 100 meals are ordered. Which graph is it?
It says that “rental fee is reduced by $100 for each group of 20 meals”, meaning that the rental fee will be $500 if you order 1 to 19 meals, but will be reduced by $100 to $400 once you order 20 meals. …
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37. A line in the standard (x,y) coordinate plane is parallel to the x-axis and 6 coordinate units above it. Which of the following is an equation of this line?
Any line parallel to the x-axis will be in the form of y = k where k is a real number. …
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38. The figure below shows a 10-foot ladder leaning against a vertical wall. The base of the ladder makes a 65° angle with the level ground. Which of the following expressions gives the distance, in feet, between the base of the ladder and the wall?
Let’s refresh the definitions of the 6 trig ratios:
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39. The only factor that is common to 200 and an integer n is 1. When n/210 is written as a decimal number, what is the minimum number of digits to the right of the decimal point?
Such integers can be:
1, where 1/200 = 0.005;
3, where 3/200 = 0.015;
7, where 7/200 = 0.035;
…
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40. Equilateral triangle △ABC is shown below with each side length given in meters. What is the value of k ?
By definition, each side of an equilateral triangle is equal in length. Thus, …
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41. If (3x-y)/(x+y)=3/5, then x/y=?
Simplifying the equation, we have:
5(3x – y) = 2(x + y)
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42. Jin’s Office Supply sells different styles of notebooks. The sale price and number of sheets per notebook for 3 styles of notebooks are given in the table below. The sale price is the amount a customer pays for that notebook. Which of the following amounts is closest to the average sale price per sheet for a Style B notebook?
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43. A bin contains 100 Style A notebooks, 100 Style B notebooks, and 100 Style C notebooks. Antoine will select 3 notebooks from the bin. How many different selections of 3 notebook styles are possible? (Note: The order in which the notebooks are selected does not matter.)
Let’s divvy up all the possible selections into the following cohorts:
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44. Last week, customers bought 5 times as many Style A notebooks as Style C notebooks, and customers bought 2 times as many Style B notebooks as Style C notebooks. Last week, customers bought 392 notebooks. What total amount did customers pay for the 392 notebooks?
Let the number of Style A notebooks be A;
Let the number of Style B notebooks be B;
Let the number of Style C notebooks be C;
…
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45. The notebook supplier is offering Jin’s a Style D notebook that has 60% more sheets than the Style C notebook has. How many sheets does the Style D notebook have?
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46. Given that 3x + 5y = 17 and 2x + 3y = 11, what is the value of x + y ?
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47. The least common multiple (LCM) of 2 numbers is 108. The greater of the 2 numbers is 54. What is the maximum value of the other number?
The least common multiple is a number which is both a multiple of A and B. And it is the smallest number like that.
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48. In the standard (x,y) coordinate plane, what is the slope of a line that is perpendicular to 2x — 3y = 9 ?
The slopes of two lines perpendicular to each other will have the following relationship:
Slope_{1} × Slope_{2} = -1
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49. Which of the following number line graphs is that of the solution set to the inequality —2x + 12 >= 20 ?
Graphing an inequality on a number line, is very similar to graphing a number. For instance, look at the top number line x = 3. We just put a little dot where the ‘3’ is.
Now an inequality uses a greater than, less than symbol, and all that we have to do to graph an inequality is find the the number, ‘3’ in this case and color in everything above or below it.
Please note:
- if the symbol is (≥ or ≤) then you fill in the dot, like the top two examples in the graph below
- if the symbol is (> or <) then you do not fill in the dot like the bottom two examples in the graph below
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50. A right prism with triangular bases and with dimensions given in inches is shown below. What is the prism’s total surface area, in square inches?
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51. If # represents the operation defined by a # b = a + b^a, then (1 # 2) # 3 =?
1 # 2 = 1 + 2 = 3
⇒ (1 # 2) # 3 = 3 # 3 = 3 + 3³ = 30
…
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52. In the standard (x,y) coordinate plane below, ADEF will be translated 12 coordinate units down and 2 coordinate units right. What will be the coordinates of E after the translation?
When a point is translated down for n units, its y coordinate will be deducted by n; when a point is translated right for n units, its x coordinate will be added by n.
…
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53. The product of 2 numbers is 25. If 1 of the numbers is the complex number 4 + 3i, what is the other number?
What you need to know about complex numbers is that i² = -1.
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54. Experimental data is represented in the standard (x,y) coordinate plane by a scatterplot consisting of 6 points: (1, 1.4), (2, 2.9), (3, 3.4), (4, 3.1), (5, 1.9), and (6, 0.2). When all possible real values for a, b, and c are considered, which of the following functions best fits the experimental data?
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55. In the figure below, AB is parallel to DE, the measure of ∠ABC is 49°, and the measure of ∠BCD is 137°. What is the measure of ∠CDE ?
When parallel lines get crossed by another line (which is called a Transversal), the sum of the measures of the consecutive interior angles are 180 degrees, the alternate interior angles are congruent.
…
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56. As shown below, 2 circular pulleys with centers 8 inches apart are connected with a tight belt. The belt wraps 2/3 of the way around the larger pulley, which has a radius of 5 inches, and 1/3 of the way around the smaller pulley, which has a radius of 1 inch. What is the exact length of the belt, in inches?
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57. Given that sin A = 5/13 and 0° A < 360°, what are all possible values of cos A ?
It should be remembered that:
sin^{2} A + cos^{2} A = 1
⇒ cos A = ± 12/13
…
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58. The lengths of the sides of the triangle shown below are given in meters. Which of the following equations gives the degree measure 0 ? (Note: For any triangle, c^2 = a^2 + b^2 — 2ab cos C, where a, b, and c are the lengths of the sides opposite angles with measures A, B, and C, respectively.)
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59. The sum of the first 30 positive integers is 465. Which of the following is the sum of the first 60 positive integers?
An arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
The sequence here goes:
1, 2, 3, … 60
This sum can be found quickly by …
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60. What is the minimum degree possible for the polynomial function whose graph is shown in the standard (x,y) coordinate plane below?
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