# ACT Math : How to find a ratio

## Example Questions

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### Example Question #3 : Proportion / Ratio / Rate

The ratio of r to s is 4 to 9, and the ratio of s to t is 5 to 6. What is the ratio of t to r?

27 to 10

10 to 27

2 to 3

3 to 2

9 to 5

27 to 10

Explanation:

Given the proportions r/s = 4/9 and s/t = 5/6, cross-multiply to come up with the two equations 9r = 4s and 6s = 5t. Multiply the first by 3 and the second by 2 so that the s terms are equivalent (27r = 12s and 12s = 10t) and drop out (27r = 10t). The proportion becomes r/t = 10/27,  or t/r =27/10.

### Example Question #5 : Proportion / Ratio / Rate

The ratio of a to b is 9:2, and the ratio of c to b is 5:3. What is the ratio of a to c?

20:3

3:5

3:1

27:10

14:5

27:10

Explanation:

Set up the proportions a/b = 9/2 and c/b = 5/3 and cross multiply.

2a = 9b and 3c = 5b.

Next, substitute the b’s in order to express a and c in terms of each other.

10a = 45b and 27c = 45b --> 10a = 27c

Lastly, reverse cross multiply to get a and c back into a proportion.

a/c = 27/10

### Example Question #4 : Proportion / Ratio / Rate

In order to fix your leaky roof, you choose between one of two competing roofing services. Service A charges you an upfront cost of $120, plus$15 per hour of labor. Service B charges an upfront cost of $95, plus$20 per hour of labor. At least how many hours of labor must your leaky roof need before it’s worth considering service A over service B?

6 hours

5 hours

3 hours

2 hours

4 hours

5 hours

Explanation:

You want to figure out at what number of hours h will solve the inequality 120+15h<95+20h. This gives us that 25 < 5h, meaning that h>5.

### Example Question #2 : How To Find A Ratio

There is a shipment of 50 radios; 5 of them are defective; what is the ratio of non-defective to defective?

50 : 5

5 : 50

1 : 5

9 : 1

1 : 9

9 : 1

Explanation:

Since there are 5 defective radios, there are 45 nondefective radios; therefore, the ratio of non-defective to defective is 45 : 5, or 9 : 1.

### Example Question #1 : How To Find A Ratio

In January of 2013, Molly was 3 times older than Steve. In January 2016, Molly will be 2 times older than Steve. How old was Steve in 2013?

9 years old

12 years old

3 years old

4 years old

3 years old

Explanation:

We let Molly’s age in 2013 be M and Steve’s age in 2013 be S. From the given information, we can write the expressions, M = 3S and M + 3 = 2(S +3).

We use M = 3S to plug into the second equation to arrive at:

3S + 3 = 2(S + 3)

3S + 3 = 2S + 6

3S = 2S + 3

S = 3

### Example Question #2 : How To Find A Ratio

Joey loves reading. He went to a bookstore where science books cost $10.00 each and comic books cost$5.50 each. He bought twice as many comic books as science books, and spent a total of \$42.00. How many comic books did he buy?

5

3

4

2

4

Explanation:

Twice as many comic books as science books can be written as 2b where b is the number of science books bought.  The equation for the cost of the books is given by

Cost1 x Number1 + Cost2 x Number2 = Total Cost

10b + 5.50(2b) = 42

And then we can solve for b to find that b = 2

Two science books and four comic books were bought.

### Example Question #4 : How To Find A Ratio

A bag contains 3 green marbles, 5 red marbles, and 9 blue marbles.

What is the ratio of green marbles to blue marbles?

Explanation:

The ratio of green to blue is .

Without simplifying, the ratio of green to blue is  (order does matter).

Since 3 and 9 are both divisible by 3, this ratio can be simplified to .

### Example Question #19 : Calculating Ratio And Proportion

A small company's workforce consists of store employees, store managers, and corporate managers in the ratio 10:3:1. How many employees are either corporate managers or store managers if the company has a total of  employees?

Explanation:

Let  be the number of store employees,  the number of store managers, and  the number of corporate managers.

, so the number of store employees is .

, so the number of store managers is .

, so the number of corporate managers is .

Therefore, the number of employees who are either store managers or corporate managers is .

### Example Question #3 : How To Find A Ratio

The ratio of the number of financial employees who remained in the same role for 2 to 9 years to the number of construction employees who remained in the same role for 0 to 4 years is closest to which of the following?

Explanation:

For this problem, we need to find the number of employees who fall into the categories described, keeping in mind that multiple portions of the pie chart must be accommodated for. Then, we can fit them into a ratio:

For the "2 to 9 years" portion of the financial industry, include

(0.2 + 0.18)(12,000,000) = 4,560,000 workers.

For the "0 to 4 years" portion of the construction industry, include

(0.15 + 0.2)(8,000,000) = 2,800,000 workers.

Now divide and simplify to find the ratio:

4,560,000/2,800,000 = 8/5.

### Example Question #9 : How To Find A Ratio

The ratio of  to  is  to , while the ratio of  to  is  to .

What is the ratio of  to ?

Explanation:

Since the ratios are fixed, regardless of the actual values of , , or , we can let  and

In order to convert to a form where we can relate  to , we must set the coefficient of  of each ratio equal such that the ratio can be transferred. This is done most easily by finding a common multiple of  and  (the ratio of  to  and , respectively) which is

Thus, we now have  and .

Setting the  values equal, we get , or a ratio of

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