Percentages are used in any science, in any job, and even in everyday communication. A person who knows percentages well gives the impression of being intelligent and educated. In this lesson we will learn what a percentage is and what actions can be performed with it.

## What are percentage?

In everyday life, fractions are the most common. They even got their names: half, one-third, and one-quarter, respectively. But there is another fraction that also occurs frequently. This fraction is 1/100 (one hundredth). This fraction is called a percentage. What does the fraction one hundredth mean ? It means that something is divided into a hundred parts and one part is taken from that. So a percentage is one hundredth of something.

A percentage is one hundredth of something

For example, of one meter is 1 cm. One meter divided into a hundred parts, and took one part (remember that 1 meter is 100 cm). And one part of those hundred parts is 1 cm. So one percent of one meter is 1 cm. of one meter is already 2 centimeters. This time one meter was divided into a hundred parts and from there we took not one, but two parts. And two parts of a hundred are two centimeters. So two percent of one meter is two centimeters.

Another example, of one dollar is one cent. A dollar is divided into a hundred parts, and one part is taken from that. And one part of those hundred parts is one cent. So one percent of one dollar is one cent.

Percentages were so common that people replaced the fraction with a special sign that looks like this: This reads like this: "one percent". It replaces the fraction . It also replaces the decimal 0.01 because if we convert the regular fraction into a decimal, we get 0.01. So we can put an equal sign between these three expressions:

1% = = 0.01

Two percent in fractional form will be written as , in decimal form as 0.02 and with a special symbol two percent is written as 2%.

2% = = 0.02

## How to calculate percentage?

The rule of finding a percentage is the same as finding a fraction of a number.

To find the percentage of something, divide it by 100 parts and multiply the resulting number by the desired percentage.

For example, find 2% of 10 cm.

What does the entry 2% mean ? The entry 2% replaces the entry . If we translate this assignment into more comprehensible language, it would look like this:

Find of 10 cm

And we already know how to solve such problems. To find a fraction of a number, divide the number by the denominator of the fraction, and multiply the result by the numerator of the fraction.

So, we divide the number 10 by the denominator of the fraction  We got 0.1. Now multiply 0.1 by the numerator of the fraction 0.1 × 2 = 0.2

The answer is 0.2. So 2% of 10 cm is 0.2 cm. And if you convert 0.2 centimeters to millimeters, you get 2 millimeters:

0.2 cm = 2 mm

So 2% of 10 cm is 2 mm.

Example 2. To find 50% of \$300.

To find 50% of \$300, you need to divide \$300 by 100, and multiply the result by 50.

So, divide \$300 by 100.

300 : 100 = 3

Now multiply the result by 50

3 × 50 = \$150

So 50% of \$300 is \$150.

If at first it is difficult to get used to the entry with the % sign, you can replace this entry with the usual fractional notation.

For example, the same 50% can be replaced by the entry . Then the problem would look like this: Find of \$300, and solving such problems is so far easier for us

300 : 100 = 3

3 × 50 = \$150

There is nothing complicated here. If difficulties arise, we advise you to stop and revise fractions and how they can be applied.

Example 3. A garment factory has produced 1,200 suits. Of these, 32% are new suit styles. How many new suit styles did the factory produce?

Here you should find 32% of 1200. The resulting number will be the answer to the question. Use the rule of finding the percentage. Divide 1200 by 100 and multiply the result by the desired percentage, i.e. 32

1200 : 100 = 12

12 × 32 = 384

Answer: 384 suits of the new style were produced by the factory.

## The second way to calculate percentage

The second way to find the percentage is much easier and more convenient. It consists in the fact that we should multiply the number (from which we are looking for the percentage) by the desired percentage, expressed as a decimal.

For example, let's solve the previous problem this way. Find 50% of \$300.

The entry 50% replaces the entry , and if we convert that to a decimal we get 0.5

Now to find 50% of 300, it will be enough to multiply the number 300 by the decimal 0.5

300 × 0.5 = 150

By the way, the same principle is used to find percent on calculators. To find the percentage using a calculator, you need to enter into the calculator the number from which you are looking for the percentage, then press the multiplication key and enter the desired percentage. Then press the percent key % ## How to find a number given its percent

If you know the percentage of a number, you can find out the whole number.

For example, a business paid us \$60000 for a job, and that is 2% of the total profit the business made. Knowing our share, and how much percentage it is, we can find out the total profit.

First you need to find out how many dollars is one percent. How to do it? Try to guess by carefully studying the following picture: If 2 % of the total profit is \$60,000, then it's easy to guess that one percent is \$30,000. And to get that \$30,000, you have to divide \$60,000 by two.

60 000 : 2 = 30 000

We found one percent of the total profit, i.e., . If one part is 30 thousand, then to determine one hundred parts, you must multiply 30 thousand by 100

30 000 × 100 = 3 000 000

We found out the total profit. It is three million.

Let's try to learn a rule for finding a number by its percentage.

To find a number by its percentage, you need to divide the known number by the percentage and multiply the result by 100.

Example 2. The number 35 is 7% of some unknown number. Find this unknown number.

Read the first part of the rule:

To find a number by its percentage, you need to divide the known number by the percentage

We have a known number, 35, and a given percentage, 7. Let's divide 35 by 7

35 : 7 = 5

Read the second part of the rule:

and multiply the result by 100

Our result is number 5. Multiply 5 by 100

5 × 100 = 500

500 is the unknown number you wanted to find. You can do a test. To do this, we find 7% of 500. If we did everything correctly, we should get 35

500 : 100 = 5

5 × 7 = 35

We got 35. So the exercise was solved correctly.

The method of finding a number by its percentage is the same as the usual way of finding a whole number by its fraction. If percentages are confusing and difficult at first, you can replace percentages with a fractional form.

For example, the previous problem can be stated as follows: The number 35 is of some unknown number. Find this unknown number. We already know how to solve such problems. This is finding a number by a fraction. To find a number with a fraction, we divide the number by the numerator of the fraction, and multiply the result by the denominator of the fraction. In this example, we divide the number 35 by 7 and multiply the result by 100.

35 : 7 = 5

5 × 100 = 500

In the future we will be solving exercises based on percentages, some of which will be complicated. In order to learn it easily at first, it is enough to be able to find a percentage of a number, and a number by a percentage.

## Exercises

Task 1. Find 20% of the number 200
200 : 100 = 2
2 × 20 = 40
Task 2. Find 34% of the number 1050
1050 : 100 = 10.5
10.5 × 34 = 357
Task 3. Find 25% of the number 80
80 : 100 = 0.80
0.8 × 25 = 20
Task 4. Find 185% of a number 1.5
1,5 : 100 = 0.015
0.015 × 185 = 2.775
Task 5. Find 150% of the number 1150
1150 : 100 = 11.50
11.50 × 150 = 1725
Task 6. Express the expression 15% as a fraction Task 7. Express the expression 25% as a fraction Task 8. Express the expression 125% as a fraction Task 9. The number 12 is 60% of some number. Find this number.
12 : 60 = 0.2
0.2 × 100 = 20
Task 10. The number 40 is 20% of some number. Find this number.
40 : 20 = 2
2 × 100 = 200

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