Algebra: Proportion Word Problems

Proportion problems are word problems where the items in the question are proportional to each other. They are two main types of proportional problems: Directly Proportional Problems and Inversely Proportional Problems.

Directly Proportional Problems

The question usually will not tell you that the items are directly proportional. Instead, it will give you the value of two items which are related and then asks you to figure out what will be the value of one of the item if the value of the other item changes.

Proportion problems are usually of the form:

If x then y. If x is changed to a then what will be the value of y?

For example,

If two pencils cost $1.50, how many pencils can you buy with $9.00?

The main difficulty with this type of question is to figure out which values to divide and which values to multiply.

The following method is helpful:

Change the word problem into the form:

If x then y. If x is changed to a then what will be the value of y?

which can then be represented as:

For example,

You can think of the sentence:

If two pencils cost $1.50, how many pencils can you buy with $9.00?

as

If $1.50 then two pencils. If $9.00 then how many pencils?

Write the proportional relationship:

Example 1:
Jane ran 100 meters in 15 seconds. How long did she take to run 1 meter?

Step 1: Think of the word problem as:

If 100 then 15. If 1 then how many?

Step 2: Write the proportional relationship:

Answer: She took 0.15 seconds

Example 2:
If of a tank can be filled in 2 minutes, how many minutes will it take to fill the whole tank?

Step 1: Think of the word problem as:

If then 2. If 1 then how many? (Whole tank is )

Step 2: Write the proportional relationship:

Answer: It took 3.5 seconds

Example 3:
A car travels 125 miles in 3 hours. How far would it travel in 5 hours?

Step 1: Think of the word problem as:

If 3 then 125. If 5 then how many?

Step 2: Write the proportional relationship:

Answer: He traveled miles.

Inversely Proportional Problems

Inversely Proportional questions are similar to directly proportional problems, but the difference is that when x increase y will decrease and vice versa - which is the inverse proportion relationship. The most common example of inverse proportion problems would be “the more men on a job the less time taken for the job to complete”

Again, the technique is to change the proportion problems into the form:

If x then y. If x is changed to a then what will be the value of y?

and then write the inverse relationship (take note of the "inverse" form):

Example:
It takes 4 men 6 hours to repair a road. How long will it take 7 men to do the job if they work at the same rate?

Step 1: Think of the word problem as:

If 4 then 6. If 7 then how many?

Step 2: Write out the inverse relationship:

Answer: They will take hours.

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