PROPORTION PROBLEMS

PERCENTS (advanced)

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 Percents 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.

Example 1: The percent successes was 76 and made a total of 77 successes. They decreased the successes rate by 13 percent. How many successes do they have now? Round answer to the nearest one.

Step 1 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | Form the first ratio using the percentages given for the known and unknown successes. |
| We were given that the first percent in the problem is 76 and that the second percent is the first percent reduced by 13. This gives us a second percent of 13. |
Step b | Form the second ratio using the data given for the total number of successes, both known and unknown. |
| Form a proportion using the two sets of data. | =
Step 2 | Apply the Cross-product Rule to solve for X. First, multiply 63 by 77. Then, divide 4851 by 76. Finally, round 63.8 to the nearest one. X equals 64. | X = 64
Total number of successes is now 64.

Example 2: If 57 percent (which are 30) of farmers found gold, how many farmers would find gold if the percent were changed to 49 percent? Round answer to the nearest person.

Step 1 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | Form the first ratio using the percentages given for the known and unknown farmers. |
| We were given that the first percent in the problem is 57 and that the second percent is 49. |
Step b | Form the second ratio using the data given for the total amount of farmers that found gold, both known and unknown. |
| Form a proportion using the two sets of data. | =
Step 2 | Apply the Cross-product Rule to solve for X. First, multiply 49 by 30. Then, divide 1470 by 57. Finally, round 25.7 to the nearest person 26. | X = 26
The number of farmers to find gold would be 26.

Example 3: The casket factory had a 49% defect rate which amounted to 129 defects per year. They eliminated 5 defects per month. What is the new defect percentage? Round to the nearest hundredth of a percent.

Step 1 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | Form the first ratio using the defects given for the known and unknown percentages. |
| We are given the first amount of defects per year, 129. The second amount of defects we will have to calculate. Since the problem states that they were able to eliminate 5 defects per month then we are able to calculate how many defects were eliminated per year. There are 12 months in a year and 5 defects eliminated per month: 5 x 12 = 60 defects less per year Therefore, the new total amount of defects per year is the old number of defects minus the new number of defects: 129 – 60 = 69 Now we can form our ratio. |
Step b | Form the second ratio using the data given for the percentages of defects, both known and unknown. |
| Form a proportion using the two sets of data. | =
Step 2 | Apply the Cross-product Rule to solve for X. First, multiply 69 by 49. Then, divide 3381 by 129. Finally, round 26.209 to the nearest hundredth of a percent: 26.21. | X = 26.21
The new defect percentage is 26.21