# ACT Math : How to find the solution to an equation

## Example Questions

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### Example Question #1 : Linear / Rational / Variable Equations

n/17 = 54/9

What is the value of n?

102

None of the above.

54

93.5

85

102

Explanation:

Cross multiply: 9n = 54 * 17 → n = (54 * 17)/9 = 102.

This also can be solved by reducing the right hand side of the equation, so n/17 = 6.

### Example Question #2 : Linear / Rational / Variable Equations

If 12x + 3 = 2(5x + 5) + 1, what is the value of x?

5

3

7

6

4

4

Explanation:

Starting with 12x + 3 = 2(5x + 5) + 1, we start by solving the parenthesis, giving us 12x + 3 = 10x + 11. We then subtract 10x from the right side and subtract three from the left, giving us 2x = 8; divide by 2 → x = 4.

### Example Question #1 : Linear / Rational / Variable Equations

If you multiply two integers together and then add 5, the result is 69. Which of the following could not be the sum of the two integers?

34

16

24

20

65

24

Explanation:

The equation is xy + 5 = 69, making xy equal to 64. If we factor 64, we see that 1x64, 2x32, 4x16 and 8x8 all equal 16 when the two numbers are added together, so 24 is the only possibility that does not work.

### Example Question #1 : Linear / Rational / Variable Equations

Three consecutive positive numbers have the sum of 15. What is the product of these numbers?

120

45

20

75

30

120

Explanation:

Define the variables as x = the first number, x + 1, the second number, and x + 2 the thrid number.

The sum becomes x + x + 1 + x + 2 = 15 so 3x + 3 = 15.  Subtract 3 from both sides of the equation to get 3x = 12 → 3x/3 = 12/3 → x = 4

The three numbers are 4, 5, and 6 and their product is 120.

### Example Question #5 : How To Find The Solution To An Equation

A company makes toy boats.  Their monthly fixed costs are $1500. The variable costs are$50 per boat.  They sell boats for \$75 a piece.  How many boats must be sold each month to break even?

75

100

50

60

80

60

Explanation:

The break-even point is where the costs equal the revenues

Fixed Costs + Variable Costs = Revenues

1500 + 50x = 75x

Solving for x results in x = 60 boats sold each month to break even.

### Example Question #2 : Linear / Rational / Variable Equations

The length, in meters, of a rectangular fence is 4 more than twice its width. Which of the following gives the length (l)  in terms of width (w) of the rectangular fence?

= 2w * 4

l = (1/2)w – 4

l = 2w + 4

l = 2w – 4

l = (1/2)w + 4

l = 2w + 4

Explanation:

To find the length, we must take twice the width, meaning to multiply the width by 2. Then we must take 4 more than that number, meaning we must add 4 to the number. Combining these, we get:

l = 2w + 4

### Example Question #5 : Linear / Rational / Variable Equations

Given the following equation:

y = –3x – 5

What is y when x = –2?

5

3

–1

2

1

1

Explanation:

Plug in x and evaluate the equation.

### Example Question #3 : Linear / Rational / Variable Equations

Jim got a job trimming trees. If he trims trees from noon to 6pm, with the exception of a 30 minute lunch break, Jim can trim 55 trees. How long does it take him to trim one tree?

5.5 minutes

7 minutes

6 minutes

6.5 minutes

5 minutes

6 minutes

Explanation:

Once you subtract the 30 minute break, you are left with 5 and a half hours. You multiply that by 60 to get 330 minutes. You then divide that by 55 trees, to get 6 minutes per tree.

### Example Question #5 : Linear / Rational / Variable Equations

If 16+ 4 = 5(3+ 1) + 2, what is the value of x?

4

1

2

5

3

3

Explanation:

By the order of operations, we multiply 5 by 3+ 1

Add like terms to get 16+ 4 = 15+ 7

Subtract 15x from both sides, then subtract 4 from both sides.

Then you have = 3.

Solve for x.

4(x + 3) = 16

Undefined

x = 1

x = 4

x = 12

x = 2