What is a Ratio?
A ratio is the mathematical relationship between two values that indicates how many times the first value contains the second value, and can be expressed in one of three ways.
- 3 : 4
- 3 to 4
- 3/4
So in the above example, since 3 divided by 4 is equal to 0.75, the first value contains the second value 0.75 times. Or, the first value is 0.75 of the second value.
Or, as a practical example, suppose you have a class of 28 students consisting of 12 boys and 16 girls. In this case, the ratio of boys to girls is 12 to 16 (12:16 or 12/16). Since you can reduce the fraction 12/16 to 3/4, you could say that for every three boys in the class, there are four girls.
How to Simplify a Ratio
Since a ratio can be stated as a fraction (3:4 = 3/4), you simplify ratios the same way you simplify fractions, which is to find the greatest common factor (GCF) of the two numbers, and then divide both sides by the GCF.
For example, to simplify the ratio 12:16, you find that 12 and 16 have a greatest common factor 4. So dividing both numbers by 4 leaves you with a reduced ratio of 3:4.
| 12 | = | 12 ÷ 4 | = | 3 |
| 16 | 16 ÷ 4 | 4 |
How to Determine if Two Ratios Are Equal
To determine if two ratios are equal, for each ratio, divide the first number by the second number. Then compare the two results to see if they are equal.
For example, if checking to see if 12:16 is equal to 54:72, dividing 12 by 16 equals 0.75, and 54 divided by 72 equals 0.75. Therefore the two ratios are equal.
| 12 : 16 | = | 12 | = | 0.75 |
| 16 |
| 54 : 72 | = | 54 | = | 0.75 |
| 72 |
The ratio 12:16 is equal to the ratio 54:72.
Or, if checking to see if 12:16 is equal to 20:25, dividing 12 by 16 equals 0.75, and 20 divided by 25 equals 0.80. Therefore the two ratios are not equal.
| 12 : 16 | = | 12 | = | 0.75 |
| 16 |
| 20 : 25 | = | 20 | = | 0.8 |
| 25 |
The ratio 12:16 is not equal to the ratio 20:25.
How to Solve for Ratio Equality
If you are given a ratio equality problem where one of the values is missing (A:B = C:D), you follow these steps:
- Convert the ratios to fractions.
- Multiply both sides of the equality by the unknown variable.
- Multiply both sides of the equality by the inverse of the fraction having both values.
- Solve for the unknown variable once you have it on one side of the equality all by itself.
Depending on which of the four values is unknown, getting the unknown variable by itself on one side of the equality will leave you with one of the following formulas to solve:
- a = C/D X B
- b = D/C X A
- c = A/B X D
- d = B/A x C
To illustrate how to solve for missing value, suppose you are presented with the following problem to solve:
| A | = | C |
| 12 | 60 | |
| 16 | d | |
| B | D |
Since we know from the above formula list that d will be equal to B/A x C, we can solve for d like this:
| d = B/A x C |
| d = 16/12 x 60 |
| d = 1.3333333333333333 x 60 |
| d = 80 |
Now that we know d is equal to 80, we can complete the equality.
| A | = | C |
| 12 | 60 | |
| 16 | 80 | |
| B | D |
How to Scale a Ratio Up or Down
To scale up a ratio, multiply both sides of the ratio by the scale factor. For example, if you want to scale up the ratio 1:3 by a factor of 3, multiplying both sides by 3 leaves you with a scaled-up ratio of 3:9 ([1 x 3 = 3] : [3 x 3 = 9]).
To scale down a ratio, divide both sides of the ratio by the scale factor. For example, if you want to scale down the ratio 3:9 by a factor of 3, dividing both sides of the ration by 3 leaves you with a scaled-down ratio of 1:3 ([3 ÷ 3 = 1] : [9 ÷ 3 = 3]).
Use the following calculator to scale up or scale down ratios.