In the previous problems, you were given the values of the percent and the part. You had to work backwards to calculate the value of the whole group.

As we covered in section 1 of this lesson, a percent problem can be modeled by the proportional relationship:

Part over Whole Part Whole = percent over 100 Percent 100

This means that we can use proportional reasoning to find the value of the whole.

The interactive above describes two different plans that you can use as you apply proportional reasoning in order to solve percent problems.

Let’s look at an example, and see how both plans might look.

Mandy bought a blouse that was marked at 25% off. The amount of the discount Mandy received at the register was $15. What was the original price of the blouse?

1. Set up a proportion relating the percent of the discount, the amount of the discount, and the original price of the blouse.

   Interactive popup. Assistance may be required.

   Check Your Answer

   25 over 100 25 100 = $15 over original price $15 original price

   Let's use Plan A to solve the problem.

2. What factor relates the percent to 100?

   Interactive popup. Assistance may be required.

   Check Your Answer

   Factor = 100 over 25 100 25

3. Use the same factor to convert the amount of the discount (the part) to the oeriginal price (the whole).

   Interactive popup. Assistance may be required.

   Check Your Answer

   Original Price = 100 over 25 100 25 x $15 = $60
   So that we can compare the two plans, let’s use Plan C to solve the problem.
4. What factor related the percent to the part?

   Interactive popup. Assistance may be required.

   Check Your Answer

   Factor = $15 over 25 $15 25
5. Use the same factor to convert 100 to the original price (the whole).

   Interactive popup. Assistance may be required.

   Check Your Answer

   Original Price = $15 over 25 $15 25 x 100 = $60

Now that you’ve seen an example with both plans of solving a problem, let’s generalize how each plan could be used to solve percent problems. Remember that percent problems can be modeled using the proportion:

Part over Whole Part Whole = percent over 100 Percent 100

Plan A:

What factor relates the percent to 100?

Interactive popup. Assistance may be required.

Check Your Answer

Percent * = 100
= 100 over percent 100 Percent

Use the same factor to convert the part to the whole.

Interactive popup. Assistance may be required.

Check Your Answer

Whole = * part
Whole = 100 over percent 100 Percent * part

Plan C:

What factor could be used relate the percent to the part?

Interactive popup. Assistance may be required.

Check Your Answer

Percent * = part
= part over percent Part Percent

Use the same factor to convert 100 to the whole.

Interactive popup. Assistance may be required.

Check Your Answer

Whole = * 100
Whole = part over percent Part Percent * 100

How do the last lines of each plan compare to each other?

Check Your Answer

The last lines are the same because "100 * part" is the same as "part * 100."

Watch the following animation to see how the proportional relationship becomes a y = kx equation.

Based on what you saw in the video, how does the proportional relationship Part over Whole Part Whole = Percent over 100 Percent 100 relate to the equation y = kx? Interactive popup. Assistance may be required.

Check Your Answer

y → Whole
k → 100 over Percent 100 Percent
x → Part

Mike got a test back from his teacher. He got 19 of the problems correct, and his score was a 76%. How many questions were on the test?

Write an equation that you could use to solve this problem for q, the number of questions on the test.

Check Your Answer

Two different equations would work:
q = 100 over 76 100 76 (19)
q = 19 over 76 19 76 (100)

How many questions were on the test?

Check Your Answer

q = 25
There were 25 questions on the test.