How to Solve Proportions
Suppose there are two ratios a:b and c:d. They can be written as fractions and , respectively. Now, if we set these two ratios equal to each other then it becomes a proportion.
Ways to Write a Proportion
There are two ways to write a proportion.
Next, let’s identify the parts of a proportion. We need this concept in order to solve problems later on.
In colon form, the extremes are the two outermost values, while the means are the two innermost values.
FRACTION FORM (Standard Form)
In a fraction form, the extremes are the values hit by a diagonal drawn from top left to bottom right, while the means are the values hit by a diagonal drawn from the bottom left to top right.
After getting familiar with the definition and parts of a proportion, we can now talk about the properties of proportions. These are two useful properties that can be used to solve problems.
PROPERTIES OF PROPORTIONS
1) Reciprocal Property
Description: If two ratios are equal, then their reciprocals must also be equal as long as they exist.
2) Cross Product Property
Description: The product of the extremes is equal to the product of the means.
Examples of How to Apply the Concept of Proportions
Example 1: Show that the proportion is true.
In order for a proportion to be true, the fractions on both sides of the equations must be reduced to the same value. The fraction on the left side of the equation has a greatest common divisor of 5. While the fraction on the right has a greatest common divisor of 6.
Since the two fractions on both sides are equal after reducing to lowest terms, we can claim that the given proportion is true!
Example 2: Show that the proportion is true.
We can also show if a proportion is true using the Cross Product Property. Simply put, if the product of their extremes (outer values) equals the product of means (inner values) then the proportion is true.
This shows that the given proportion is true!
Example 3: Solve the proportion .
This problem is a proportion with an unknown value. Our goal is to find the value of “x” that could make the proportion a true statement. We can easily solve this using the Cross Product Property.
You may back substitute x = 2 into the original proportion and verify that it is indeed the correct answer.
Example 4: Solve the proportion .
The only difference of this problem from example #3 is that the unknown variable “x” is found in the denominator. Solving this proportion is as easy as applying the Cross Product Property, and then solving the simple equation that comes out of it.
Alternatively, you may first apply the Reciprocal Property to move the variable “x” from the bottom to top before using the Cross Product Property. The answer should come out the same.
Example 5: Solve the proportion .
The presence of x‘s on both sides of the equation should not intimidate you. The Cross Product Property of Proportionality should take care of this problem quite easily. The only difference here is to make sure that you’re careful in every step because this is simply an exercise of solving a linear equation. Don’t forget to get rid of a parenthesis using the distributive property. Other than that, I am sure that you can follow the rest of the steps below.
Substitute x = -3 back to the original proportion to verify your answer.
Example 6: Solve the proportion .
This is another type of problem that you may encounter when solving proportions. The format of the proportion is using a colon instead of a fraction. To work this out, we need to rewrite the proportion in fractional form, and then solve this as usual.
Since a:b = c:d can be written as , then our original problem becomes . Let’s go ahead and solve this…
Substitute x = 2 back to the original proportion to verify your answer.
Example 7: The exchange rate between the US Dollar and the Indian Rupee is 2 to 106. At this rate, how much US Dollar would you have if you exchanged 901 Indian Rupees?
What we want is to set up a proportion that we can solve. We can do this two ways. One way is to place the dollar values in the numerators while the rupees in the denominators of the proportion. And the other way is to swap their locations. Either of the setups should give us the same answer.
For this exercise, we will put the dollar information on top.
Solve the unknown value of “x” to get the required dollar value.
That means at the time of the exchange, 17 US Dollars is equivalent to 901 Indian Rupees.
Example 8: You want to cut a wood with a length of 72 feet into two pieces such that the ratio of the shorter to the longer piece is 2 to 7. What are their lengths?
Let “x” be the length of the shorter piece. That means ” 72 − x ” will be the longer piece. See diagram below.
It is given that the ratio of the shorter to the longer piece is 2 : 7. Using all these information, we can now setup the proportion to solve for the lengths of both short and longer pieces.
Solving the proportion above using the Cross Product Property of Proportionality…
Since the shorter piece is x =16 feet, that means the longer piece is 72 – x = 72 – 16 = 56 feet.
To perform a check, we were told in the problem that the ratio of the shorter piece to longer piece is 2 to 7. Notice that when we reduce the fraction 16/56 to lowest term, we will get the desired ratio.