Lesson Percentage problems

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Percentage problems


This lesson is an introduction to the percentage problems.
Let us start with simple examples first.

Example 1. One percent (1%) of the number B is one hundredth of this number, or %281%2F100%29%2AB, or 0.01%2AB.

Example 2. Seven percents (7%) of the number B is seven hundredth of this number, or %287%2F100%29%2AB, or 0.07%2AB.

Example 3. Ten percents (10%) of the number B is ten hundredth of this number, or %2810%2F100%29%2AB, or %281%2F10%29%2AB, or 0.1%2AB.

Example 4. 2.5% of the number B is 2.5 hundredth of this number, or %282.5%2F100%29%2AB, or 0.025%2AB.

Example 5. 25% of the number B is 25 hundredth of this number, or %2825%2F100%29%2AB, or %281%2F4%29%2AB, or 0.25%2AB.

Example 6. 225% of the number B is 225 hundredth of this number, or %28225%2F100%29%2AB, or %289%2F4%29%2AB, or 2.25%2AB.

The percentage problems include three numbers.
One number is the base B. It represents the total amount of something or the measure of something.
The 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).
The third number is the part P. It is the amount or the measure of the part.

Note that all examples considered above fall into one general formula
r+=+R%2F100, P+=+r%2AB.                       (*)
This formula converts the percentage rate R to the decimal r and then calculates the part P multiplying the base by this decimal.

There are three major types of percentage problems.
Type 1: Finding the Part
You are given the base B and the rate R (percentage); you are asked to find the part P.

Type 2: Finding the Rate
In Type 2 problems you are given the base B and the part P; you are asked to find the rate R (percentage).

Type 3: Finding the Base
In Type 3 problems you are given the part P and the rate R (percentage); you are asked to find the base P.

Solving Type 1 percentage problems: Finding the Part


In Type 1 percentage problems you are given the base B and the percentage rate R. The part P is unknown you should find.
The basic formula to solve Type 1 percentage problems is
r+=+R%2F100, P+=+r%2AB.                        (1)
This is the same formula as the one above.
The formula converts the percentage rate R to the decimal r and then calculates the part P multiplying the base by this decimal.

Example 7
What is 5% of 80?
Solution
Apply the basic formula (1).
Here B = 80, R = 5%.
Represent the percentage rate as a decimal and then multiply the base by this decimal:
5%2F100+=+0.05, P+=+0.05%2A80+=+4.
Answer. 5% of 80 is 4.

Example 8
Find 12.75% of 133.
Solution
Apply the basic formula (1).
Here B = 133, R = 12.75%.
Represent the percentage rate as a decimal and then multiply the base by this decimal:
12.75%2F100+=+0.1275, P+=+0.1275%2A133+=+16.9575.
Answer. 12.75% of 133 is 16.9575.

Solving Type 2 percentage problems: Finding the Rate


In Type 2 percentage problems you are given the base B and the part P. The percentage rate R is unknown you should find.
From the general formula (*) you can express the decimal rate r as
r+=+P%2FB and then convert it to the percentage rate R=r%2A100.
The basic formula to solve Type 2 percentage problems is
r+=+P%2FB, R+=+r%2A100.                        (2)
The formula calculates the decimal rate r and then converts it to the percentage rate R multiplying by 100.

Example 9
What percent is 4 of 80?
Solution
Apply the basic formula (2).
Here B = 80, P = 4.
Calculate the decimal rate and then convert it to the percentage rate multiplying by 100:
r+=+4%2F80+=+0.05, R+=+0.05%2A100+=+5.
Answer. Number 4 is 5% of number 80.

Example 10
What percent is 16.9575 of 133?
Solution
Apply the basic formula (2).
Here B = 133, P = 16.9575.
Calculate the decimal rate and then convert it to the percentage rate multiplying by 100:
r+=+16.9575%2F133+=+0.1275, R+=+0.1275%2A100+=+12.75.
Answer. Number 16.9575 is 12.75% of number 133.

Solving Type 3 percentage problems: Finding the Base


In Type 3 percentage problems you are given the part P and the percentage rate R. The base B is unknown you should find.
From the general formula (*) you can express the base B as
B+=+P%2Fr. Before applying this formula, you should convert the percentage rate to the decimal dividing by 100: r=R%2F100.
The basic formula to solve Type 3 percentage problems is
r+=+R%2F100, B+=+P%2Fr.                          (3)
The formula converts the percentage rate R to the decimal r and then calculates the base B.

Example 11
Number 4 is 5% of what number?
Solution
Apply the basic formula (3).
Here P = 4, R = 5.
Convert the percentage rate to the decimal dividing it by 100 and then calculate the base:
r+=+5%2F100+=+0.05, B+=+4%2F0.05+=+80.
Answer. Number 4 is 5% of number 80.

Example 12
Number 16.9575 is 12.75% of what number?
Solution
Apply the basic formula (3).
Here P = 16.9575, R = 12.75.
Convert the percentage rate to the decimal dividing it by 100 and then calculate the base:
r+=+12.75%2F100+=+0.1275, B+=+16.9575%2F0.1275+=+133.
Answer. Number 16.9575 is 12.75% of number 133.


My other lessons on percentage problems in this site are
    - Percentage word problems (Type 1 problems, Finding the Part)
    - Percentage word problems (Type 2 problems, Finding the Rate)
    - Percentage word problems (Type 3 problems, Finding the Base)
    - Simple percentage problems
    - More complicated percentage problems
    - Advanced problems on percentage
    - Percentage problems on chains of discounts
    - Problems on percentage that lead to unexpected results
    - Simple interest percentage problems
    - Compound interest percentage problems
    - Buying price, selling price and profit percentage problems
    - Business-related entertainment problem on percentage
    - OVERVIEW of lessons on percentage problems


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