# Lesson Percentage word problems (Type 2 problems, Finding the Rate)

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## Percentage word problems (Type 2 problems: Finding the Rate)

This lesson presents solution examples of word problems on percentage.
It is a continuation of the lesson Percentage problems in this site.

Let me remind you that 1% of some number is one hundredth (1/100) part of the number.
10% of some number is one tenth (10/100 = 1/10) part of the number.
20% of some number is one fifth (20/100 = 1/5) part of the number.
25% of some number is one fourth (25/100 = 1/4) part of the number.

The percentage problems include three numbers.
One number is the base B. It represents the total amount of something or the measure of something.
Second number is the rate R. It is a measure of the part relatively to the whole thing, expressed in percents, like 3%, 7.5%, 12.75% (percentage).
Third number is the part P. It is the amount or the measure of the part.

The base, the part and the percentage are connected by the formula .
For example, if the base B is equal to 80 and the percentage R is 25%, then the part P is .

Problems considered in this lesson are all of the same type: you are given two of three numbers, namely the base B and the part P.
The third number, the percentage R, is unknown, and you should find it.
These problems are Type 2 problems on percentage, as defined in the lesson Percentage problems.

Specifically for the Type 2 percentage problems the general formula above can be re-written in the form
.                       (**)
This formula is the basic to solve the percentage problems of the Type 2.

Below are examples of the Type 2 word problems on percentage.

### Problem 1. Test results

The student answered 80 questions on the mathematics test.
What was the percent of correct answers?

Solution
Since 76 of 80 answers were correct, the percent of correct answers was equal to .

### Problem 2. Tax calculation

Daniel bought a calculator on-line. The calculator was advertised at \$24.99 before the tax.
When the tax was included, Daniel paid \$26.86.
What percent of his money did he spend on tax?

Solution
The tax amount was \$26.86 -\$24.99 = \$1.87.
You are asked what percent is this of \$24.99.
Apply formula (**).
It was percents.

### Problem 3. Discount

Brian bought a printer on sale at a store. The original price was \$90.00.
After a discount was applied, Brian paid \$76.50 only.
What percent the discount was?

Solution
First, find the amount of the discount.
It is equal to \$90.00 - \$76.50 = \$13.50 dollars.
You should calculate what percent this is of \$90.00.
Apply formula (**).
It is percents.

### Problem 4. Population growth

The population of Georgetown was 50000 at the end of 2009. The population had increased to 51500 at the end of 2010.
What percent of increase is this?

Solution
First, find the amount of the population increase.
Since the population had increased from 50000 to 51500 in 2010, the amount of the population increase is 51500-50000 = 1500.

You are asked what percent is 1500 of 50000.
Apply formula (**).
It is .

### Problem 5. Investment

Kathy invested \$25,000.00 into the bank account.
One year later her account had increased to \$25,500.00.
What was the annual interest rate?

Solution
Kathy's account had increased by \$25500.00 - \$25000.00 = \$500.00 dollars after one year.
To calculate the annual interest rate in percents, apply formula (**).
It is equal to percents.
Answer. The annual interest rate was 2%.

### Problem 6. Alloy (mixture)

An alloy contains of 2.5 pounds contains 1.75 pounds of silver.
What is the percent of silver in the alloy?

Solution
Apply formula (**).
The percent of silver in the alloy is in the alloy.
Answer. The alloy contains 70% of silver.

Examples of percentage word problems of the Type 1 and the Type 3 are presented in lessons
Percentage word problems (Type 1 problems: Finding the Part) and
Percentage word problems (Type 3 problems: Finding the Base)
in the section Word problems of this site.

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