# ACT Math : How to find a ratio

## Example Questions

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### Example Question #1 : 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

27:10

14:5

3:1

3:5

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 #8 : Proportion / Ratio / Rate

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

5 : 50

1 : 5

50 : 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

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 #1 : Proportion / Ratio / Rate

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 #1 : 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 #21 : 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

### Example Question #11 : Proportion / Ratio / Rate

There are thirty cups and fifteen saucers on a shelf. If three saucers are broken and five cups are added, what will be the ratio of cups to saucers?

Explanation:

First, begin by calculating the total number of each item that there will be at the end of the process.

Cups:

Saucers:

The ratio of cups to saucers will thus be:

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

Joe needs to repair the roof of his house. He finds two companies that can complete the job. Company A charges an initial cost of $120, plus$15 per hour of labor, while Company B charges an initial cost of $95, plus$20 per hour of labor. After how many hours of labor does Company A cost less than Company B to repair the roof?

Explanation:

In order to solve this problem, create an equation that summarizes the roof repair cost for each company. Begin by composing a formula for Company A, which charges 120 dollars upfront and 15 dollars per hour of labor.

Now, Company B charges 95 dollars upfront and 20 dollars per hour of labor. We can write the following equation:

The question asks us to find how many hours of labor that a repair must take in order for Company A to be cheaper than Company B. In other words, we need to compose an inequality in which the cost of Company A is less than the cost of Company B. We will substitute the variable  for hours and solve.

Subtract  from each side of the inequality.

Subtract 95 from both sides of the inequality.

Divide both sides of the inequality by 5.

If the repair will take more than 5 hours, Company A will be cheaper.

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

On her birthday in 2013, Molly was three times older than Steve. On her birthday in 2016, Molly was 2 times older than Steve. How old was Steve on Molly's birthday in 2013?

3

10

9

12

2

3

Explanation:

First, let's assign variables to the names of the individuals to represent their age in 2013.

In 2013, Molly was three times older than Steve; therefore, we can write the following expression:

We are also told that in 2016, Molly will be two times older than Steve; thus, we can write another expression:

.

We can then substitute  in for  in the second equation to arrive at the following:

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

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

9 to 5

2 to 3

27 to 10

10 to 27

3 to 2

27 to 10

Explanation:

Using the given information we can generate the following two proportions:

and

Next, cross-multiply each proportion to come up with the following two equations:

and

Each equation shares a term with the  variable. We need to make this variable equal in both equations to continue. Multiply the first equation by a factor of 3 and the second by a factor of 2, so that the  terms are equivalent. Let's start with the first equation.

Now, we will perform a similar operation on the second equation.

Now, we can set these equations equal to one another.

Drop the equivalent  terms.

The proportion then becomes the following:

or

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