# ACT Math Practice Test 5

Difficulty Level – 3: Medium / Difficult

Directions: Solve each problem and then click on the correct answer. You are permitted to use a calculator on this test.

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 Question 1

### A number, $x$, is decreased by 40% then increased by 25%. What is the final result in terms of $x$?

 A $0.55x$ B $0.575x$ C $0.65x$ D $0.7x$ E $0.75x$
Question 1 Explanation:
The correct answer is (E). To solve this, set $x$ = 100 and perform the operations. One hundred is a great number to choose here because 1% of 100 is 1.

$100 − 0.4 \ast 100 = 60$

$60 + 0.25 \ast 60 = 75$

Because $x$ = 100 was our initial value, we want to choose the answer choice that gives us 75 when we substitute 100 for $x$:

$0.75 \ast 100 = 75$
 Question 2
$\left(\dfrac{1}{3}y^\frac{1}{3}\right)^4*(2x^2y)^2=$

 A $\dfrac{4}{81}x^4y^\frac{10}{3}$ B $\dfrac{1}{81}x^2y^3$ C $\dfrac{4}{81}x^5y^\frac{9}{3}$ D $\dfrac{1}{81}x^2y^4$ E $\dfrac{4}{27}x^4y^\frac{8}{3}$
Question 2 Explanation:
The correct answer is (A). Evaluate the expression:

$\frac{1}{3}$ to the 4th power is $\frac{1}{81}$ and $y\frac{1}{3}$ raised to the 4th power is $\frac{4}{3}$ so the first expression is $y^\frac{4}{3}$ divided by 81.

The second expression evaluates to $4 * x^4 * y^2$. Our coefficient becomes $\frac{4}{18}$ and our $x$ is to the 4th power and when multiplying $y^\frac{4}{3}$ and $y^2$, we add the exponents to get $y^\frac{10}{3}$.
 Question 3

### The expression $(x + y) [z − (x − y)]$ is equivalent to which of the following?

 A $xz − x^2 + 2xy + y^2$ B $2xz + x^2 − yz + y^2$ C $xz − x^2 + yz + y^2$ D $xz + x^2 − yz − y^2$ E $xyz + y^2 − xy + yz + y^2$
Question 3 Explanation:
The correct answer is (C). Recall order of operations and the distributive property to answer this question. The second bracket simplifies to $\,z − x + y\,$ and to this we multiply $(x + y)$.

Begin by distributing the $x$ through $(z − x + y)$ and to this add the distribution of y through $(z − x + y)$ to get answer choice (C).
 Question 4

### There are 32 marbles in a jar: 14 blue, 10 red, 5 green, and 3 yellow. Sally pulls one marble, randomly, from the jar. Without replacing the marble, she pulls another marble. What is the probability that both marbles will be red?

 A $\dfrac{40}{426}$ B $\dfrac{41}{456}$ C $\dfrac{43}{476}$ D $\dfrac{44}{476}$ E $\dfrac{45}{496}$
Question 4 Explanation:
The correct answer is (E). The probability of selecting a red marble on the first draw is $\frac{10}{32}$ because there are 10 red marbles and 32 total marbles.

After removing the first red marble there are now 9 red marbles and 31 total marbles left so $\frac{9}{31}$ chance of selecting the second red marble.

To find the probability of both events occurring, we multiply the probabilities to get $\frac{9*10}{32*31}$ which reduces to $\frac{45}{496}$.
 Question 5

### If $x+y=7$ $\text{and}$ $x-y=3$, what is $x^2-y^2= \; ?$

 A $4$ B $21$ C $25$ D $36$ E $40$
Question 5 Explanation:
The correct answer is (B). We can either use the system of equations to solve for $x$ and $y$ individually to find that $x = 5$ and $y = 2$ which gives $25 − 4 = 21$, or we can see that if we multiply $(x+y)$ and $(x-y)$ we will be left with $x^2-y^2$, $7 * 3 = 21$.
 Question 6

### The circumference of a large wedding cake is 60 inches. If the cake is divided evenly into 12 slices, what is the length of the arc, in inches, made by 5 combined slices?

 A $12$ B $13$ C $15$ D $20$ E $25$
Question 6 Explanation:
The correct answer (E). Recall that the circumference is the distance around a circle. If the cake is divided into 12 equal slices, each arc length will be 5. Summing 5 of these slices will give 25.
 Question 7

### A rectangle measures 65 inches by 77 inches. What is the length of the diagonal of the rectangle (rounded to the nearest hundredth)?

 A $100.77$ B $110.77$ C $125.07$ D $134.67$ E $137.77$
Question 7 Explanation:
The correct answer is (A). Use the Pythagorean Theorem to solve this problem. Recall that the sum of the side lengths squared equals the hypotenuse squared, so:

$65^2 + 77^2 = \text{diagonal}^2$

Take the square root of both sides to find the length and round to the nearest hundredth.
 Question 8

### If $f(x) = x^2 + 2x + 2$, what is $f(x + h)?$

 A $2x^2 + 4hx + h^2 − 2x − 2h$ B $x + h^2 + 2xh + 2 + h$ C $x^2 + 2x + 2xh + 2h + h^2 + 2$ D $x^2 + 4xh + 4h^2 + 2x$ E $2x^2 + 2xh + 2h^2 + 2x + 2$
Question 8 Explanation:
The correct answer is (C). To solve this problem we will substitute $(x + h)$ for every $x$ in our function $f(x)$:

$f(x +h) = (x +h)^2 + 2(x +h) + 2$

Evaluate the expression using the distributive property to arrive at:

$x^2 + 2x + 2xh + 2h + h^2 + 2$
 Question 9 ### In the figure above, lines $d$ and $f$ are parallel and the angle measures are as given. What is the value of $x$?

 A $35^\text{o}$ B $60^\text{o}$ C $85^\text{o}$ D $100^\text{o}$ E $120^\text{o}$
Question 9 Explanation:
The correct answer is (C). Since vertical angles are congruent, the angle vertical to the 35-degree angle also has a measure of 35 degrees. The supplement of the 120-degree angle has a measure of 60 degrees, so we then have a triangle with angles measuring 35, 60, and $x$ degrees, as shown in the figure provided. Since the angles of a triangle add to 180 degrees:

$35 + 60 + x = 180$

$x = 180 − 35 − 60$

$x = 85^\text{o}$
 Question 10

### Which of the following represents the equation of the line that passes through the point $(2, 3)$ with a slope of $−\frac{1}{3}$?

 A $y = 2x + 4$ B $y = 4x + 2$ C $y = −\dfrac{1}{3}x + \dfrac{11}{3}$ D $y = −3x + \dfrac{11}{3}$ E $y = −\dfrac{2}{3}x + \dfrac{11}{3}$
Question 10 Explanation:
The correct answer is (C). Recall that when provided with a point and the slope of a line, we can use point-slope formula to write an equation for the line. The point slope formula is $y − y_1 = m(x − x_1)$ where $(x_1, y_1)$ is the point provided and m is the slope. Plug in the point and slope provided and solve for $y$:

$y − 3 = −\dfrac{1}{3} (x − 2)$

$y = −\dfrac{1}{3}x + \dfrac{11}{3}$
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