**Below are answer explanations to the full-length Math test of the previously released ACT from the 2021-2022 “Preparing for the ACT Test” (form 2176CPRE) free study guide available here for free PDF download.**

The ACT Math test explained below begins on page 22 of the guide (24 of the PDF). Other answer explanations in this series of articles:

- English Answer Explanations from 2021-2022 ACT Practice Test
- Math Answer Explanations from 2021-2022 ACT Practice Test (this article)
- Reading Answer Explanations from 2021-2022 ACT Practice Test
- Science Answer Explanations from 2021-2022 ACT Practice Test
- Writing Answer Explanations from 2021-2022 ACT Practice Test

When you’re finished reviewing this official practice ACT test, start practicing with our own 10 full-length practice ACT tests.

**ACT Practice Test Math Answer Explanations 2021-2022**

**Question 1** **“paper in a jar”** **the answer is E**

This question tests your understanding of probability.

- There are 15 pieces of paper in the jar. Eight of those pieces are less than 9.
- The probability of drawing desired pieces is
- The probability is therefore .

**Question 2 “” the answer is J**

This question tests your understanding of manipulating algebraic expressions.

- First we add the like terms () and ()
- We then end up with . This is the correct answer.

**Question 3 “10 + 3(12 ****÷ (3x))****”** **the answer is 16**

This question tests your understanding of simplifying an expression using PEMDAS.

- The first step is to plug in x = 2.
- This would simplify the equation to 10 + 3(12 ÷(3⋅2))
- Simplifying from inside the parenthesis we would get 10 + 3 (12 ÷(6))
- Which would be equal to 10+3(2) = 10+6 = 16.

**Question 4 “⎪6 − 4⎪ − ⎪3 − 8⎪” the answer is G**

This question tests your understanding of simplifying absolute value expressions.

- The straight brackets | are used to indicate the absolute value of the expression in between two brackets.
- The first step would be to simplify the two expressions inside the straight brackets. This would simplify the equation to |2| – |-5|.
- The next step is to calculate the absolute value of each term. This would simplify the expression to 2 – 5.
- Finally performing the final subtraction would yield an answer of -3.

**Question 5** **“(4c-3d)(3c+d)” the answer is C**

This question is testing that we understand the distributive property.

- To calculate we begin by multiplying all the terms in the second parenthesis with the term 4c. This would yield 4c(3c) + 4c(d). This would simplify to + 4cd.
- The next step is to multiply all the terms in the second parenthesis with the term (-3d). This would yield -3d(3c) – 3d(d). This would simplify to .
- Combine the terms from the first and second steps, we get . Then by combining the like terms we end up with the expression . This is the final answer.

**Question 6** **“Student grades” the answer is F**

This question tests your understanding of applying proportions in the setting of word problems.

- We know that there are 180 students, so if students earned an A, that would mean that 45 students earned an A.
- We also, know that students earned a B, so that would mean of the 180 students, 60 students earned a B.
- Hence, we can calculate the students who earned a C by subtracting those that earned an A or a B from the total. This would be calculated as follows: 180-45-60 = 75.
- So the number of students that earned a C for the course is 75.

**Question 7 “Skipper’s Pond” the answer is B**

This question tests your understanding of applying proportions in the setting of word problems.

- We’re told that the equation to represent the number of fish in the pond is .
- Hence in the year 2006, the value of x is 2006 – 2000 = 6.
- Plugging this value of x into the equation would leave the equation . This would mean that there are 192 fish in the Pond at the beginning of the year 2006.

**Question 8 “Speed” the answer is H**

This question tests your understanding of speed and distance.

- Since we know that Manish was 510 km away from Baton Rouge at 8:00 a.m., and 105 km away from Baton Rouge at 1:00 p.m., we can conclude that Manish covered 510 – 105 km in this time. This is 405 km.
- The amount of time it took to cover 405 km is 5 hours, which is the difference between 1:00 p.m., and 8:00 a.m.
- Based on this we could calculate the speed by dividing 405 km by the 5 hours, which would yield a speed of 81 km per hour.

**Question 9 “triangle with parallel lines” the answer is D**

This question tests your understanding of similar triangles and parallel lines.

- Since we know that is constant, and that is parallel to we can say that the ADE is similar to the ABC.
- Hence, we can conclude the ratios of the sides are similar.
- Specifically, the ratio of is equal to the ratio of .
- The value of is which simplifies .
- We know that = = . Solving for we get that = 42.

**Question 10 “Katerina” the answer is G**

This question tests your understanding of speed and how it is calculated.

- Katerina average minutes for running one mile can be calculated by dividing the total number of minutes by the total miles ran.
- The total number of minutes requires us to convert hours into minutes by multiplying by 60. Performing this step we get that the number of minutes ran is 150 minutes.
- The total number of miles is given at 15.
- The average number of minutes per mile can be calculated by dividing 150 by 15. This would yield 10 minutes per mile.

**Question 11 “marbles”** **the answer is B**

This question tests your understanding of probability and algebraic manipulations.

- There are currently 8 red marbles out of 24 total marbles. If we add x new red marbles, the new number of red marbles is 8 + x and the total number of marbles is 24 + x.
- The probability of drawing a red marble is equal to . We know this is equal to .
- Setting the two equations equal to each other, we can simplify to the equation 3 (24+x) = 5 (8 + x).
- Solving for x we end up with 16, which is the number of additional red marbles that must be added to the bag.

**Question 12 “midpoint”** **the answer is G**

This question tests your understanding of probability and algebraic manipulations.

- We know that the midpoint of is (2,1).
- From point C (6,8), we travel 4 units to the left, and 7 units down to get to the midpoint.
- Hence, to get to point D, we have to travel the same distance from the midpoint.
- So point D would be represented by the point (2-4, 1-7). This is equal to (-2,-6).

**Question 13 “overtime”** **the answer is D**

This question tests your understanding of word problems and algebraic manipulations.

- We know that for the first 40 hours, Thomas gets paid $15.
- After the first 40 hours, Thomas gets paid $15 x 1.5 = $22.5.
- Since Thomas worked 46 hours, his total pay prior to deductions is 40 x $ 15 + 6 x $22.5 = $600 + $135 = $735.
- Since $117 is removed for deductions, the amount of money that Thomas is left with is $735-$117 = $618.

**Question 14 “Sweet Stuff Fresh Produce” the answer is J**

This question tests your understanding of money based expressions.

- In 2 trips, Janelle purchased 3 bags at $3 a bag and 4 bags at $2.80 a bag.
- The amount she spent was $9 on Monday and $11.20 on Wednesday, for a total of $20.20.
- If she had bought 7 bags at once, she would have paid $2.60 per bag, for a total of $18.20.
- She would have saved $2.00.

**Question 15 “3% of 4.14 x ”** **the answer is A**

This question tests your understanding of simplifying expressions with percentages.

- 3% of 4.14 x = 0.03 x 4.14 x = 3 x 4.14 x 100 = 12.42 x 100 = 1242.

**Question 16** **“value of x” the answer is K**

This question tests your understanding of solving algebraic equations.

- To find the value of x that satisfies this equation, -3 ( 4x – 5) = 2 (1 – 5x), we begin by distributing the values on both parentheses.
- We end up with the equation -12x + 15 = 2 – 10x. Combining like terms we end up with 13 = 2x.
- x = .

**Question 17** **“right triangle” the answer is D**

This question tests your understanding of sine of an angle in a triangle.

- Sin A = .
- Sin A =

**Question 18 “ ” the answer is J**

This question tests your understanding of simplifying exponents.

- We begin by taking the cube root of the fraction to eliminate the 3 in the exponent. Doing this we end up with the term: .
- To eliminate the negative sign in the exponent, we take the reciprocal of the fraction. We end up with the term .
- Squaring the numerator and denominator we end up with the value: .

**Question 19 “Loto’s walk” the answer is A**

This question tests your understanding of the Pythagorean Theorem.

- Adding up the number of yards Loto walked east and north separately, we end up with 20 yards east and 11 yards north.
- To find the distance Loto would walk if he could walk directly, we apply the Pythagorean theorem to get the equation:
- Solving for x we end up with x = 22.83.
- The total difference in distance Loto would have saved is 20 + 11 – 22.83 = 8.17 yards which is approximately 8 yards.

**Question 20 “standard score” the answer is F**

This question tests your understanding of solving algebraic equations.

- We know that z = 2 = .
- Solving for x we get x = 90.

**Question 21 “circle”** **the answer is E**

This question tests your understanding of triangles in the setting of a circle.

- In approaching this we can begin by looking at the answer choices and seeing if they exist in the circle centered at O.
- The ABO is an acute triangle. Since is the radius and is also the radius we know the A = B. Since we know the AOB, is 60 degrees we can deduce that A and B are both 60 degrees.
- Hence, the AOB is also an equilateral triangle.
- Since we know that and are both radii of the circle, we can conclude that the DOC is an isosceles triangle. The DOC is also a right triangle since the DOC is 90 degrees.
- Hence the only triangle that does not appear in the figure is the scalene triangle.

**Question 22 “slope” the answer is G.**

This question tests your understanding of parallel lines and slopes.

- If the line is parallel to x + 5y = 9, we can conclude they have the same slope.
- Solving the equation for y we get, y = . So the slope is .

**Question 23 “y = ” the answer is E**

This question tests your understanding of solving equations.

- Since we know that x > 1, we know that y cannot be negative or 0. Hence, the only possible answer choices that are possible are 0.9 and 1.9
- Plugging 0.9 for y; we get x = – 9. Since, x has to be greater than 1, we can conclude that only 1.9 works for y.
- Plugging in 1.9 for y we get x = 2.111. Since x is greater than 1 we can conclude that this is the correct answer.

**Question 24 “all positive integers”** **the answer is H.**

This question tests your understanding of prime factors.

- For a number to be divisible by 15 and 35 it has to be a multiple of both of the numbers’ factors.
- 15 has the prime factors 5 and 3. 35 has the prime factors 5 and 7.
- The least common multiple is 5 x 3 x 7. This is 105.

**Question 25 “ triangle ABC” the answer is D**

This question tests your understanding of angles.

- If = , then B = C. Since A is 58 degrees, we can calculate the sum of B and C by subtracting 58 from 180.
- 180 – 58 = 122 degrees. Since B is equal to C, both angles have to 61 degrees.

**Question 26 “Earth’s surface” the answer is G**

This question tests your understanding of probability.

- To find the probability of landing on water, we have to find the area covered by water and divide that value by the total area on Earth’s surface.
- The desired equation would be P = . Factoring out and dividing both the numerator and denominator by yields the equation
- P =

**Question 27 “statistical tests” the answer is E**

This question tests your understanding of mean, median and mode of a series of numbers.

- Prior to his 8th test, Jamal’s mean, median, and mode were 79, 80, and 80 respectively.
- Following his score of 90 on his 8th test, Jamal’s mean was 80.13. His median was still 80, and his mode was still 80. Hence, the only value that changed was his mean, which was greater.

**Question 28 “solid rectangular prism” the answer is H**

This question tests your understanding of three-dimensional figures.

- The solid rectangular prism has sides of length 5, 6, and 7 units. Hence, there are a total of 210 cubes.
- Since the black cubes and white cubes are alternating equally, we can conclude that half of the cubes are black and half are white.
- Half of 210 is 105.

**Question 29 “square ABCD” the answer is C**

This question tests your ability to calculate the area of squares and rectangles.

- If one side of the square ABCD has length 12 meters, then the area of the square is 144 .
- Since, we know the width of the rectangle is 8 meters, dividing 144 by 8 meters, yields the length which is 18 meters.

**Question 30 “average” the answer is H**

This question tests your understanding of averages for a series of numbers.

- We know that the average of the numbers w, x, y, and z is 92.0. Hence the sum of w, x, y, and z can be represented by the equation. w + x + y + z = 4 x 92 = 368.
- We know that z is 40, so the sum of w, x and y is 368 – 40 = 328.
- We also know that the 4th number of the new list is 48, so the sum of the numbers of the new list is equal to the expression w + x + y + 48. Plugging in 328 for the sum of w, x and y gives the expression 328 + 48 = 376.
- Dividing 376 by 4, yields the average 94.0.

**Question 31 “vector”** **the answer is B**

This question tests your understanding of velocity and vectors.

- Since the vector j represents 1 mile per hour north, -j would represent 1 mile per hour south.
- Maria’s velocity is 12 miles per hour south which would be represented by -12j.

**Question 32** “**glasses”** **the answer is K.**

This question tests your understanding of adding fractions.

- To begin we add the volume of water in each of the glasses.
- To distribute this water equally among the 4 glasses, we divide by 4. This would result in each glass being full.

**Question 33 “carpet tiles” the answer is D**

This question tests your understanding of area of a surface.

- To begin we convert his living room floor length and width to inches. feet = 100 inches. Likewise, 10 feet = 120 inches.
- Hence the area in square inches of the living room floor is 100 inches x 120 inches = 12,000
- Each carpet tile has an area of 20 inches x 20 inches = 400 . Dividing the area of the living room floor by the 400 of each carpet tile, would yield 30 carpet tiles needed to cover this area.

**Question 34 “coordinate plane” the answer is F**

This question tests your understanding of the equation of a circle.

- A circle with a center of () and a radius r would have be mapped out by the equation .
- Plugging the relevant () = (8,5) and r = 9, into the equation we end up with the answer .

**Question 35 “total cost”** **the answer is E**

This question tests your understanding of applying algebra to word problems.

- Since it costs $2,500 per each Yq test and there were 1000 tests administered, the cost of the Yq test was $2,500,000.
- Likewise, it costs $50 pear each Sam77 test and there were 1000 tests administered the total cost of the Sam77 test was $50,000.
- Adding these two numbers yields a total cost of $2,550,000.

**Question 36 “percent of the volunteers” the answer is J**

This question tests your understanding of percentages.

- The number of people who carry the Yq77 gene can be calculated by adding up all the people who had a positive Yq test. This is equal to 590 + 25 = 615.
- Since there were 1000 people tested total, or 61.5% carried the Yq77 gene.

**Question 37 “how many volunteers” the answer is C**

This question tests your understanding of understanding tables.

- To calculate how many volunteers had an incorrect result from the Sam77 test, we have to add up the people who had a Negative Sam77 test but had a Positive Yq test and the people who had a positive Sam77 test but a negative Yq test.
- These two numbers are 25 and 10 respectively, so the total number of volunteers who received an incorrect result from the Sam77 test are 25 + 10 = 35.

**Question 38 “does NOT possess Yq77”** **the answer is F**

This question tests your understanding of probability.

- The number of volunteers who had a positive Sam77 test was 590 + 10 = 600.
- Of these volunteers, only 10 did not have the Yq77 gene.
- Hence the probability is .

**Question 39 “matrices”** **the answer is B**

This question tests your understanding of matrix manipulations.

- To calculate the product of X and Y, we have to multiply the matrices.
- This can be done as follows .

**Question 40 “symmetry” the answer is F**

This question tests your understanding of lines of symmetry.

- A scalene triangle is one where all three sides have different lengths. This would not have a vertical line of symmetry.
- A line would have a verticle line of symmetry. A square would have a verticle line of symmetry. A pentagon would have a verticle line of symmetry. A parallelogram would have a verticle line of symmetry.

**Question 41 “” the answer is A.**

This question tests your understanding of solving a quadratic equation.

- To solve this equation, we can begin by subtracting 15 from both sides. We get the equation .
- We can factor this to yield (6x+5)(4x-3) = 0. Setting 6x+5 = 0 and 4x -3 = 0, we get the two solutions for this equation as and .
- The greater of these two solutions is .

**Question 42 “” the answer is J**

This question tests your understanding of sine and cosine of common angles.

- To solve this we have to know that . Hence, we can use this to replace the with .
- Likewise, we can replace the from the second term with .
- Hence the original equation simplifies to = 1.
- Of the answer choices, the only one whose value = 1 is .

**Question 43 “area of circle” the answer is D**

This question tests your understanding of circumference and area of a circle.

- We know the circumference is = . Solving for r we get r = 6.
- The area of circle is . Plugging in r = 6, we get the area as .

**Question 44 “solvent mixture” the answer is G.**

This question tests your understanding of percentages and mixtures.

- We know that of the 25 liters, 40% is solvent and 60% is water. Hence, we can conclude that there is 0.40 x 25 liters of solvent = 10 liters.
- So if x liters of solvent is added, the amount of solvent would be 10 + x. Furthermore, the new mixture would have a volume of 25 + x.
- So the percent of solvent in the new mixture would be equal to .
- Since we know this is 50% or 0.50 we can set these two values equal to each other and solve for x.
- Simplifying, we get 10 + x = (25 + x )(0.5). Further simplification shows us that 0.5x = 2.5, so x = 5.

**Question 45 “” the answer is E**

This question tests your understanding of simplifying expressions with fractions.

- To simplify this equation we find the common multiple of the two denominators which would be .
- Hence, we would have to multiple the numerator of the first term by (x-y) and the numerator of the second term by (x+y).
- The new expression after combining the two terms would be .
- Factoring out the numerator and denominator we get .

**Question 46 “page advertisem*nts” the answer is F.**

This question tests your understanding of solving word problems with algebra.

- If Carlos sold c advertisem*nts, Mary sold 2c advertisem*nts and James sold 6c advertismeents.
- In total, they sold c + 2c + 6c = 9c advertisem*nts.
- Hence Carlos sold of all the advertisem*nts.

**Question 47 “flower shop” the answer is D**

This question tests your understanding of permutations.

- For the first spot, Emily has 6 plants to choose from.
- After planting the first plant, Emily only has 5 plants to choose from for the second spot.
- Finally, after planting the first two plants, Emily only has 4 plants to choose from for the third spot.
- Multiplying these options out we get 6 x 5 x 4 = 120 different arrangements for the plants.

**Question 48 “quadratic function f” the answer is G**

This question tests your understanding of the area of a triangle.

- To calculate the area of we begin by using the formula for the area of a triangle = .
- Based on this, the base of the triangle is the distance between point M and point P which is 6a-2a = 4a.
- The height of the triangle is the distance from point Q to the line segment MP, which is 5b.
- Hence the area is .

**Question 49 “Point M” the answer is D**

This question tests your understanding of slope of a line on a coordinate plane.

- The slope of can be represented by the equation .
- We know that the numerator is negative because point Q has a smaller y value than point M.
- As point Q moves to the right the difference gets larger.
- Hence, the slope will be negative but will increase as the value of Q gets larger.

**Question 50 “f(5a)” the answer is K**

This question tests your understanding of a function plotted on a coordinate plane.

- To find the value of
*f(5a)*we look at the quadratic function and find where it crosses the line x = 5*a.* - Looking at the curve, we see that this happens at 8b which is the correct answer.

**Question 51 “Jurors” the answer is D**

This question tests your understanding of fractions and decimals.

- We know that if we invite x people only 0.4x of the people actually appear.
- Of this, one-third are excused so two-thirds remain. Specifically, of the initial x people invited, remain.
- We know, that this expression should be equal to 60 people for the jury pool. Hence we get the equation .
- Solving for x we get x = 225 people to summon.

**Question 52 “275th digit” the answer is K**

This question tests your understanding of repeating digits for decimals.

- Since we know the decimal 0.6295 repeats every 4 digits, we should divide 275 by 4.
- Dividing 275 by 4 we end up with 68 with a remainder of 3.
- This would mean that the the 272nd digit would be a 5. Then the next three digits would be a 6, 2 and 9 in that order.
- Based on this, the 275th digit would be a 9.

**Question 53 “ ” the answer is A**

This question tests your understanding of factoring a quadratic function.

- Since we know that
*f(x)*can be found by factoring out*f(x). f(x) = (x+2) (x-2).* - Hence, the two solutions for
*f(x)*are x = 2, and x = -2. - Since we know that these two solutions of x make
*f(x)*equal to 0, we can set*g(x)*equal to these values. - Hence we end up with two equations for
*g(x).**x +3 = 2*and*x + 3 = -2.* - Solving both equations for x we end up with x = -1, and -5.

**Question 54 “p and n” the answer is G**

This question tests your understanding of integers and absolute values.

- Since we know that p is a positive number, n is a negative number and |p| > |n|, we can assign values for both p and n that satisfy these conditions.
- Specifically we can assign p = 5, and n = -3. Then we can solve all of the equations;
- ;
- ;
- Of these answer choices, the greatest expression is

**Question 55 “” the answer is B**

This question tests your understanding of imaginary numbers and calculations using *i*.

- Since
*,*then*, , and .* - Plugging in those values into the equation , we get
- . Simplifying this equation we have which is equal to -1.

**Question 56 “coordinate plane” the answer is K**

This question tests your understanding of slopes and lines in a coordinate plane.

- The first equation is x + 2y
**≤**6. Rearranging this equation to solve for y, we end up with y**≤**-0.5x + 3. This would mean that the slope of the line is negative, and the shaded area has to be below this line. - Of all the answer choices only F and K have a line with a negative slope with the shaded area below the line.
- The second equation is . Rearranging this equation and dividing by 3, we end up with the equation .
- To choose between F and K we can find the two y-intercepts of the circle and if they lie under the line y
**≤**-0.5x + 3, we know the answer is K. - The two y intercepts of the circle can be calculated by setting x = 0 in the equation of the circle.
- We end up with the equation , so y = 2 or y = -2 .
- Plugging in the positive y-intercept (x,y) = (0,2) into the equation for the line we have 2
**≤**-0.5(0) + 3. Simplifying we have 2**≤**3, which is true so the correct answer is K.

**Question 57 “real numbers” the answer is D**

This question tests your understanding of your understanding of multiplying and dividing by 0.

- Since , we can conclude that d = 0.
- Since, abc = d, we know that abc = 0.
- Since ac = 1, we know that b(1) = 0. So, b = 0.

**Question 58** “**cosine function”** **the answer is K**

This question tests your understanding of the cosine function and functions.

- We know that the y-intercept is 3, so that means if x = 0, y has to be 3. Since cos(0) = 1, we know that the constant multiplying the cosine has to be 3.
- We also know that
*,*based on the graph. Plugging in for all the values of x we end up with y = 3*cos(2x)*as the solution that yields a y of -3.

**Question 59 “flying kite” the answer is B**

This question tests your understanding of triangles, their sides and angles.

- In the triangle shown, the missing angle would be 60 degrees.
- We then could set an equation such that . Solving this for the length of the string we end up with = length of string. Plugging in values of and , we can simplify to .

**Question 60 “publisher” the answer is F**

This question tests your understanding of using algebra to solve word problems.

- We know that the publisher charges $15 first the first book and $12 for every additional copy.
- If there are y books ordered, the publisher charges $15 for the first book and $12 for y-1 books.
- The equation for the cost of the books becomes $15 + $12(y-1). Simplifying this we get cost = 15-12 + 12y = 12y + 3.