### Answers to the Example Problems

Here are the answers and explanations to our free ACT math sample test.

Before reviewing these answers and explanations, you should first have a look at our math formulas for the exam.

Look at the equation and the problem and substitute the values for S and N:

(N!) ÷ [(N − S)! × S!] =

(6 × 5 × 4 × 3 × 2 × 1) ÷ [(6 − 3) ! × (3 !)] =

(6 × 5 × 4 × 3 × 2) ÷ [(3 !) × (3 × 2 × 1)] =

(6 × 5 × 4 × 3 × 2) ÷ [(3 × 2 × 1) × 6] =

720 ÷ (6 × 6) =

720 ÷ 36 = 20

In other words, 20 different three-letter combinations can be made from a set of six different letters.

Calculate for midpoint x:

(−6 + 2) ÷ 2 =

(−4) ÷ 2 =

−2 = midpoint x

Calculate for midpoint y:

(8 + −2) ÷ 2 =

(−6) ÷ 2 =

−3 = midpoint y

The midpoints are represented as (x, y) coordinates for your final answer:

(−2, −3)

a = logbc is always the same as: ba = c

Answers – Question 4:  The correct answer is 25x2 − 20xy + 4y2.

(5x − 2y)2 = (5x − 2y)(5x − 2y)

Using the FOIL method, we see that the first terms are 5x and 5x: (5x − 2y)(5x − 2y)

FIRST: 5x × 5x = 25x2

The outside terms are 5x and −2y: (5x − 2y)(5x2y)

OUTSIDE: 5x × −2y = −10xy

The inside terms are −2y and 5x: (5x2y)(5x − 2y)

INSIDE: −2y × 5x = −10xy

The last terms are −2y and −2y: (5x2y)(5x2y)

LAST: −2y × −2y = 4y2

25x2 − 10xy − 10xy + 4y2 =

25x2 − 20xy + 4y2

Remember that the formula for area of a circle is: π × radius2

Area of circle B: 0.72π = 0.49π

Area of circle C: 0.42π = 0.16π

To determine how much larger the area of circle B is than circle C, we then subtract:

0.49π − 0.16π = 0.33π

The formula to calculate the circumference of a circle is: π × diameter

Remember that the diameter is equal to the radius × 2.

π × 0.9 × 2 = 1.8π

Answers – Question 7:  The correct answer is x intercept = (3, 0) and y intercept = (0, 27).

Substitute 0 for x and y.

Solution for x intercept:

6x2 + 2y = 54

6x2 + (2 × 0) = 54

6x2 + 0 = 54

6x2 ÷ 6 = 54 ÷ 6

x2 = 9

x = 3

x intercept = (3, 0)

Solution for y intercept:

6x2 + 2y = 54

(6 × 0) + 2y = 54

0 + 2y = 54

2y ÷ 2 = 54 ÷ 2

y = 27

y intercept = (0, 27)

Remember the formula for permutations: N! ÷ (N − S)!

When you see the exclamation point in equations such as (N!) you have to multiply that number by each number that is lower than it.

For instance, 4! = 4 × 3 × 2 × 1

In this question, N = 3 and S = 2

N! ÷ (N − S)! =

(3 × 2 × 1) ÷ (3 − 2)! =

(3 × 2 × 1) ÷ 1 =

6 ÷ 1 = 6

In other words, 6 two-character permutations can be made from this 3 letter set.

Slope formula: y = mx + b

Substitute the values and solve.

15 = m2 + 6

15 − 6 = m2 + 6 − 6

9 = m2 + 6 − 6

9 = m2

9 ÷ 2 = m2 ÷ 2

4.5 = m

Volume of a cone: (π × radius2 × height) ÷ 3

You will know by now that the diameter is twice the radius. In this problem, the radius is therefore 8.

So we substitute the values to get the volume:

(π82 × 9) ÷ 3 =

(64π × 9) ÷ 3 =

576π ÷ 3 =

192π

The formula for the area of a rectangle is: length × width

Substitute the values for the length and the width, which are stated in the question, in order to determine the area:

5 × 7 = 35 units

Formula to calculate the area of a triangle: (base × height) ÷ 2

triangle area = (3 × 6) ÷ 2 = 18 ÷ 2 = 9

Here is the formula for calculating the perimeter of squares and rectangles:

(length × 2) + (width × 2)

So the perimeter is calculated as follows:

(length × 2) + (width × 2) =

(4 × 2) + (5 × 2) =

8 + 10 = 18 units

(x2 − 4) ÷ (x − 2) = ?

Look at the term and integer in the first set of parentheses.

The 4 in the first set of parentheses becomes −4 because the integer is preceded by the minus sign.

We have to divide −4 by −2, because −2 is the integer in the second set of parentheses.

−4 ÷ −2 = 2

Therefore the result is expressed as (x + 2).

To solve this type of problem, do multiplication of the items in parentheses first.

If 6x − 3(x + 4) = 0, then x = ?

6x − 3(x + 4) = 0

6x − 3x − 12 = 0

3x − 12 = 0

Then move the integer to the other side of the equation.

3x − 12 = 0

3x − 12 + 12 = 0 + 12

3x = 12

x = 4