PROPORTION PROBLEMS

GEOMETRY

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 Geometry Proportion Problems

Step 1 | Determine the ratio of one shape to the other from the problem.
Step 2 | Substitute the values given in the problem for the variables.
Step 3 | Use the appropriate mathematical formulas and/or equations to solve the given problem.
Step 4 | Form a proportion using two ratios. Use X for the unknown number.
Step 5 | Apply the Cross-product Rule to solve for X.

Example 1: Circle A is similar to Circle B in the ratio of 2 : 1. Given that “r” is 32, what is the Circumference of B? Round your answer to the nearest hundredth.

Step 1 | Determine the ratio of one shape to the other from the problem. | 2 : 1
Step 2 | Substitute the values given in the problem for the variables. | r = 32
Step 3 | Use the appropriate mathematical formulas and/or equations to solve the given problem.
Remember that the formula for circumference is . Remember also that diameter equals two times radius. Since the radius in this circle is 32, the diameter is 64. Next, substitute the number 3.14159 for . Finally, compute the answer. |
Step 4 | Form a proportion using two ratios. Use X for the unknown number. | Not necessary in this problem.
Step 5 | Apply the Cross-product Rule to solve for X. | Not necessary in this problem.
The circumference of Circle B is 201.06.

Example 2: The ratio between Circle A and Circle B is 5 : 2. Given that “r” is 26, what is the Circumference of A? Round your answer to the nearest hundredth.

Step 1 | Determine the ratio of one shape to the other from the problem. | 5 : 2
Step 2 | Substitute the values given in the problem for the variables. | r = 26
Step 3 | Use the appropriate mathematical formulas and/or equations to solve the given problem.
Remember that the formula for circumference is . Remember also that diameter equals two times radius. Since the radius in this circle is 26, the diameter is 52. Next, substitute the number 3.14159 for . Finally, compute the answer. |
Step 4 | Form a proportion using two ratios. Use X for the unknown number. |
Step 5 | Apply the Cross-product Rule to solve for X. |
The circumference of Circle A is 408.4.

Example 3: Rectangle A is similar to Rectangle B in the ratio of 5 : 2. If the Area of A is 7200 yards2, what is the Area of B? Round your answer to the nearest hundredth.

Step 1 | Determine the ratio of one shape to the other from the problem. | 5 : 2
Step 2 | Substitute the values given in the problem for the variables. | Area of A = 7200 yds2
Step 3 | Use the appropriate mathematical formulas and/or equations to solve the given problem. | Not necessary in this problem.
Step 4 | Form a proportion using two ratios. Use X for the unknown number.
step A | First, since the measure of the area is squared (i.e. 7200 yards2), we must also square the ratio 5 : 2. Therefore, |
step B | We can now form a proportion using two ratios. We will use X for the unknown number. |
Step 5 | Apply the Cross-product Rule to solve for X. |
The Area of B is 1152 yards2.

Example 4: Rectangle A is similar to Rectangle B in the ratio of 5 : 3. Given that the Perimeter of B is 138, what is the Perimeter of A? Round your answer to the nearest hundredth.

Step 1 | Determine the ratio of one shape to the other from the problem. | 5 : 3
Step 2 | Substitute the values given in the problem for the variables. | Perimeter of B = 138
Step 3 | Use the appropriate mathematical formulas and/or equations to solve the given problem. | Not necessary in this problem.
Step 4 | Form a proportion using two ratios. Use X for the unknown number. |
Step 5 | Apply the Cross-product Rule to solve for X |
The Perimeter of A is 230.

Example 5: Triangle A is similar to Triangle B in the ratio of 2 : 1. If the Area of A is 2016 cm2, what is the Area of B? Round your answer to the nearest hundredth.

Step 1 | Determine the ratio of one shape to the other from the problem. | 2 : 1
Step 2 | Substitute the values given in the problem for the variables. | Area of A = 2016cm2
Step 3 | Use the appropriate mathematical formulas and/or equations to solve the given problem. | Not necessary in this problem.
Step 4 | Form a proportion using two ratios. Use X for the unknown number. |
step A | First, since the measure of the area is squared (i.e. 2016 yards2), we must also first square the proportion. Therefore, |
step B | We can now form a proportion using two ratios. We will use X for the unknown number. |
Step 5 | Apply the Cross-product Rule to solve for X |
The Area of B is 504 cm2.

Example 6: Triangle A is similar to Triangle B in the ratio of 3 : 2. If the Area of B is 570 in2 and length “b” is 57, what is the length of “e”? Round each intermediate answer (before using) and the final answer to the nearest hundredth.

Step 1 | Determine the ratio of one shape to the other from the problem. | 3 : 2
Step 2 | Substitute the values given in the problem for the variables. | Area of B = 570 in2 length of b = 57
For this problem, we do not need to know the Area of B. We only need to know the length of b to answer the question.
Step 3 | Use the appropriate mathematical formulas and/or equations to solve the given problem. | Not necessary in this problem.
Step 4 | Form a proportion using two ratios. Use X for the unknown number. |
Step 5 | Apply the Cross-product Rule to solve for X |
The length of “e” is 85.5.