RATIO AND PROPORTION PROBLEMS

WORD PROBLEMS

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 ˝.

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.

Example1: When the spelunkers entered the abominable snowman’s den, 1 out of 2 got excited. If there were 110 spelunkers in all, how many did not get excited? Round answer to the nearest whole number.

Use the Steps for Solving Simple Proportion Problems

Step 1 | Form a proportion using two ratios. Use X for the unknown number.
We know that for every two spelunkers, one spelunker got excited. That ratio is written .
We know that there were a total of 110 spelunkers. Set up a proportion using the ratio above to determine how many spelunkers got excited.
Step 2 | 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 55.
We now know that out of 110 spelunkers, 55 spelunkers got excited. To find the number of spelunkers who did not get excited, which is what this problem asks, simply subtract the number of spelunkers who got excited from the total number.
When the ratio of spelunkers who got excited to the total number of spelunkers is 1 : 2, then the number of spelunkers who did NOT get excited is 55.

Example 2: The souse chef found that it took 9 huge measures of sprinkles to make 7 poached bacon balls. The souse chef needed to make 153 poached bacon balls. How many huge measures of sprinkles were needed? Round the answer to the nearest whole number.

Use the 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.
To find the missing number in a proportion, cross multiply,
and divide the product by the third number. Round the answer to the nearest whole number. 196.7 rounded to the nearest whole number is 197.
In the proportion, X is equal to 197.
When it takes 9 huge measures of sprinkles to complete 7 poached bacon balls, it will take 197 huge measures of sprinkles to complete 153 poached bacon balls.

Example 3: The number of cheerleaders in opera increased 3% this year. If there are 19 more cheerleaders in opera this year than there were last year, how many cheerleaders are in the opera this year? Increase fractional parts to the next whole number.

Use the Steps for Solving Simple Proportion Problems

There are 19 more cheerleaders in opera this year than last year, which represents a 3% increase. The total number of cheerleaders in opera last year will be considered 100% in this problem. The total number of cheerleaders in opera this year will be 3% more than last year, so that number will be considered 103% in this problem.
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.
To find the missing number in a proportion, cross multiply,
and divide the product by the third number. Increase fractional parts to the next whole number. 652.333 increased to the nearest whole number is 653.
In the proportion, X is equal to 653.
There are 653 cheerleaders in opera this year.

Example4: Peter has 136 fleas, 76 of which are un-trained and 80 of which are trained. Debbie has 39 fleas, all of them are either un-trained or trained. If the ratio of the un-trained fleas to the trained fleas is the same for both Peter and Debbie, then Peter has how many more trained fleas than Debbie? Round answer to the nearest whole number.

Use the Steps for Solving Simple Proportion Problems

Step 1 | Form a proportion using two ratios. Use X for the unknown number.
We know that Peter has a ratio of 76 un-trained fleas for every 80 which are trained. We take that ratio and reduce it: That ratio is written
We know that Debbie has a total of 39 fleas, both un-trained and trained. We also know that her ratio of fleas is the same as Peter’s. We’ll start off by stating that Debbie has an x amount of un-trained fleas. Then that would leave her with 39 minus the x amount for trained fleas. Putting that into equation format: Since the ratio is the same for Peter and Debbie then we can set up a proportion using the ratio above to determine how many more trained fleas Peter has then Debbie.
Step 2 | Apply the Cross-product Rule to solve for X.
To find the missing number in a proportion, cross multiply,
and combine like terms,
And then solve for x by dividing both sides by 39.
Therefore, Debbie has 19 un-trained fleas. To arrive at the number of trained fleas that she has, we get our equation from above and solve:
We now know that Peter has 80 trained fleas and that Debbie has 20 trained fleas. To find out how many more trained fleas Peter has over Debbie, which is what this problem asks, simply subtract the number of trained fleas Debbie has from Peter’s total amount.
When the ratio of trained fleas to untrained fleas is 19 : 20 and Peter has 80 trained fleas and Debbie has a total of 39 fleas, both trained and un-trained, then the difference between Peter’s and Debbie’s trained fleas is 60.