Lesson Percentage word problems (Type 3 problems, Finding the Base)

Percentage word problems (Type 3 problems: Finding the Base)

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 part P and the rate R as percentage.
The third number, the base B, is unknown, and you should find it.
These problems are Type 3 problems on percentage, as defined in the lesson Percentage problems.

Specifically for the Type 3 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 3.

Below are examples of the Type 3 word problems on percentage.

Problem 1. Test results

The student answered correctly 76 questions on the mathematics test, which was 95% of the total number of questions.
How many questions were in the test?

Solution
You know that 76 questions are 95% of the total number of questions.
So, you are given the part number (P), which is 76, and you are given the rate number (R) expressed in percents, which is 95%.
This is the Type 3 percentage problem.
Apply formula (***).
The total number of questions was
.

Answer. The total number of questions was 80.

Problem 2. Calculator price

Daniel bought a calculator at the store with the tax of 7.5%.
The tax amount was $1.87.
Find the calculator price before tax.

Solution
You are given that 7.5% of the calculator price was $1.87.
Thus, you know the part P and the percentage R.
So, this is the the Type 3 percentage problem.
Apply formula (***) to find the calculator price before tax.
It is equal to
dollars.

Answer. The calculator price before tax was $26.99.

Problem 3. Discount

Brian bought a printer on sale at the store. The discount was 15%, and Brian saved $13.50.
What was the printer original price?

Solution
You are given the discount amount, it is $13.50. This is the part P ot the original price.
You are given that this part is 15% of the original price. This is the rate R, expressed in percents.
So, this is the the Type 3 percentage problem.
Apply formula (***).
The original price was
dollars.

Answer. The original price for the printer before the discount was $90.00.

Problem 4. Population growth

The population of Georgetown was 50000 in 2009. The population increased by 3 percents in 2010, or 1500.
What was the population at the end of 2009?

Solution
You are given the population increase amount of 1500.
You are given that this amount is 3% of the Georgetown population at the end 2009.
So, you are given the part P and the percentage rate R.
Thus, this is the Type 3 percentage problem.
Apply formula (***) to calculate the Georgetown population at the end 2009.
You have
.

In other words, 1500/3 = 500 was the measure of 1% of population, or 1/100 part.
Then 100%, whole population, was 100 times more, or 500 x 100 = 50000.

So, the population of Georgetown was 50000 at the end of 2010.

Answer. The population of Georgetown was 50000 at the end of 2009.

Problem 5. Investment

Kathy invested into the bank account for 2% per year.
After one year, Kathy earned $500.00 in this account.
What was Kathy's initial investment?

Solution
You are given that $500.00 (the part P) are 2% (the rate in percents, R) of the initial sum.
This is the Type 3 percentage problem.
Apply formula (***) to find the initial investment (the base B).
The initial investment was
dollars.

In other words, 500.00/2 = 250 was the measure of 1% of Kathy's original investment, or 1/100 part.
Then 100%, the whole original investment, was 100 times more, or 250 x 100 = 25000.00 dollars.

Answer. Kathy's initial investment was $25000.00.

Problem 6. Alloy (mixture)

An alloy contains 70% of silver. The amount of silver in the alloy is 1.75 pounds.
Find the weight of the alloy?

Solution
You are given that 70% of the alloy mass weights 1.75 pounds.
So you know the part P, which is 1.75 pounds, and percentage rate (70%).
This is the Type 3 percentage problem.
Apply formula (***) to find the weight of the alloy.
It is equal to
pounds.

Answer. The weight of the alloy is 2.5 pounds.

Examples of percentage word problems of the Type 1 and the Type 2 are presented in lessons
Percentage word problems (Type 1 problems: Finding the Part) and
Percentage word problems (Type 2 problems: Finding the Rate)
in the section Word problems of this site.