RATIO PROBLEMS

RATE

Concept: A ratio is a comparison of two numbers. It can be written in several ways. For example, when comparing the numbers 1 and 2, we can express the comparison as 1 to 2, or 1 : 2 or ½.

Always write the terms of the ratio in the same order as the problem compares them.

Just as we can reduce fractions to lowest terms, we can simplify ratios to their lowest terms, although a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.

Many ratio problems require more than one step.

You may not be given both of the numbers that you need to write a ratio. In this case, you may need to solve another equation to find the missing number.

Finally, when two ratios are written as equal ratios, the equation is called a proportion. For example, is a proportion. In many proportion problems, one of the four terms is missing.

The proportion can be solved using the Cross-product Rule.

Cross-product Rule
To find the missing number in a proportion, cross multiply, and divide the product by the third number. For example, in the proportion , first multiply . Then divide . In this problem .

Steps for Solving Simple Ratio Problems

Step 1 | Set up the ratio, writing the terms in the same order that the problem compares the terms.
Step 2 | Simplify the ratio to its lowest terms. (Remember that a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.)

Steps for Solving Multiple Step Ratio Problems

Step 1 | Set up the first ratio, writing the terms in the same order that the problem compares the terms.
Step 2 | Set up an equation to solve for the unknown number needed for the ratio.
Step 3 | Set up a ratio, writing the terms in the same order that the problem compares the terms.
Step 4 | Simplify the ratio to its lowest terms. (Remember that a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.)

Steps for Solving Simple Proportion Problems

Step 1 | Form a proportion using two ratios. Use X for the unknown number.
Step 2 | Apply the Cross-product Rule to solve for X.

Steps for Solving Multistep Proportion Problems

Step 1 | Set up the first ratio using given information.
Step 2 | Set up an equation, if necessary, to solve for any unknown information.
Step 3 | Set up another ratio, if necessary, using the new information.
Step 4 | Form a proportion using two ratios. Use the ratio from either Step 1 or Step 3 for the first ratio in the proportion, and another ratio, set up in the same order, while substituting X for the unknown number.
Step 5 | Apply the Cross-product Rule to solve for X.

Example 1: The ratio of gleeps to blurbledurps was 1 : 2. If the total number of gleeps and blurbledurps was 2046, then how many gleeps were there? Round the answer to the nearest whole number.

Use the Steps for Solving Multipstep Proportion Problems

Step 1 | Set up the first ratio using the given information. |
Step 2 | Set up an equation, if necessary, to solve for any unknown information. |
| We already know how many gleeps and blurbledurps there are. We need to know the total number of both gleeps and blurbledurps. Add the numbers in the ratio together to find a total number of gleeps and blurbledurps. |
Step 3 | Set up another ratio, if necessary, using the new information. |
| We are asked to compare the number of gleeps to the total number of gleeps and blurbledurps. |
Step 4 | Form a proportion using two ratios. Use the ratio from either Step 1 or Step 3 for the first ratio in the proportion, and another ratio, set up in the same order, while substituting X for the unknown number. |
| We are asked to find the number of gleeps there are when the total number of gleeps and blurbledurps is 1155. |
Step 5 | Apply the Cross-product Rule to solve for X. |
| To find the missing number in a proportion, cross multiply, |
| and divide the product by the third number. |
| In the proportion, X is equal to 682. The problem instructs us to round the answer to the nearest whole number, so our answer is 682 |
| When the ratio of gleeps to blurbledurps is 1 : 2, and the total number of gleeps and blurbledurps is 2046, then there are 682 gleeps. |

Example 2: If 15 bales are needed to feed 4 moose, how many bales are needed to feed 2457 moose? Round the answer to the nearest whole number.

Use the Steps for Solving Multistep Proportion Problems

Step 1 | Set up the first ratio using given information. |
Step 2 | Set up an equation, if necessary, to solve for any unknown information. |
| Not necessary in this problem. |
Step 3 | Set up another ratio, if necessary, using the new information. |
| Not necessary in this problem. |
Step 4 | Form a proportion using two ratios. Use the ratio from either Step 1 or Step 3 for the first ratio in the proportion, and another ratio, set up in the same order, while substituting X for the unknown number. |
| We know that the ratio of bales to moose is 15 : 4, as stated in Step 1. We use that ratio as the first ratio in the proportion. The second ratio in the proportion is set up in the same order as the first, substituting X for the number of bales and entering 2457 as the number of moose. |
Step 5 | Apply the Cross-product Rule to solve for X. |
| To find the missing number in a proportion, cross multiply, |
| and divide the product by the third number. |
| In the proportion, X is equal to 9213.75. |
| When the ratio of bales to moose is 15 to 4, 9213.75 bales are needed to feed 2457 moose. |

Example 3: Given that Olga’s team’s won : lost ratio was 4 : 5, what was her team’s won : played ratio?

Use the Steps for Solving Multiple Step Ratio Problems

Step 1 | Set up the first ratio, writing the terms in the same order that the problem compares the terms. |
Step 2 | Set up an equation to solve for the unknown number needed for the ratio. |
| To determine the number of games played, we must add the number of games won to the number of games lost. |
Step 3 | Set up a ratio, writing the terms in the same order that the problem compares the terms. |
Step 4 | Simplify the ratio to its lowest terms. (Remember that a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.) Cannot be simplified. |
| The won : played ratio for Olga’s games is . |