Calculating percentages

What does per cent mean?

Look at the given square. It is divided into 100 smaller squares. The shaded squares will always be a fraction of 100.  There are 44 shaded small squares in this square. 44 out of 100 are shaded. You can also say that 44/100 is shaded.  You can also say that 44 squares per hundred are shaded. When a fraction has 100 as the denominator, we call it a percentage. 44/100 can be called 44 per cent. Per cent means per hundred, as cent means hundred. You can say that 44 per cent of the squares are shaded. 

For per cent, we use the symbol “%”. Thus 44 per cent will be written as 44%.

In simple words, percentages are numerators of fractions with denominators 100. The percentage is also used to compare results.

Examples

Express each of the following as a percentage

Finding Per cent of a number

To find the per cent of a number, we write the per cent as a fraction with denominator 100 and then multiply by the number.

Example 1

Find 75% of 20

Solution

75% of 20

 = (75/100)  x 20

 = 15

Example 2

Find 5% of £300

Solution

5% of £300

= (5/100) x 300

= £15

Writing per cent as a Decimal

In the first step, write the per cent as a fraction with the denominator 100.

Then, write the fraction as a decimal.

Example

Write each per cent as a decimal.

Comparison of Results using Percentage

Suppose, Henry scored 240 marks out of 300 and Peter scored 210 marks out of 250. Whose performance is better or what is the percentage of marks scored by them?

Solution

To compare, first put the marks in the form of a fraction, i.e. 240/300 or 210/250 respectively.

Now find the percentage of marks by multiplying the numerator and the denominator each by 100 as per cent means out of 100.

As Peter’s percentage is more than Henry’s, his performance is better.

Everyday Life Problems Involving Use of Percentage

Example 1

There are 600 students in a school. 60% are boys. How many girl students are there in the school?

Solution

If 60% are boys in the school, then 40% are girls.

60% of 600 = (60/100) x 600 = 60 x 6 = 360

Therefore, number of boys in the school = 360

And

Number of girls in the school = 600 – 360 = 240

Example 2

Silvia had 24 pages to write. By the evening, she had completed 25% of her work. How many pages were left?

Solution

Silvia had completed 25% of 24 pages by the evening.

Number of pages that were left = 75% of 24

 = (75/100) x 24

 = 18

Hence, 18 pages were left to be written.

Example 3

The regular price of a computer was £25,500. Alice bought it from an online shopping site, which was selling it at a 20% discount on the regular price. How much discount did Alice get? How much did she pay for the computer?

Solution

Regular price of notebook computer = £25,500

Discount = 20% of £25,500

 = (20/100) x £25,500

 = £5100

Amount paid by Alice = Regular Price – Discount

 = £25,500 – £5,100

= £20,400

Hence Alice paid £20,400 for the computer.

Example 4

Jack earns £12000 per month. Out of this, he spends 60% of food and other items of daily need, 10% on rent and 5% on petrol for his scooter. How much does he save every month?

Solution

Amount spent on food and other items = £(60% of 12000)

                                                                                = £(60/100  x 12000)

                                                                                 = £(60 x 120)

                                                                                 = £7200

Rent paid = £(10% of £12000) = £10/100  x 12000 = £(10 x 120) = £1200

Amount spent of petrol = £(5% of 12000)

                                                = £(5/100  x 12000) = £(5 x 120) = £600

Total monthly expenses = £(7200 + 1200 + 600)

                                                 = £9000

Hence, his saving every month = £12000 – £9000 = £3000

Therefore, Jack saves £3000 every month.

Converting a Fraction or a Ratio into a Percentage

Ratios when expressed as fractions can have different denominators in their simplest form. to compare them first of all, a common denominator is needed. The comparison becomes more convenient if the denominator is 100. This implies that ratios are being converted into percentages.

Example

In a class of 50 students, the ratio of the number of boys to the number of girls is 3: 2. Find the percentage of boys and girls in the class.

Solution

Unitary Method

The word unitary is derived from the word “unit” which means “one.

A unitary method is a method in which the value of a unit quantity is first derived to find the value of any required quantity. In solving problems through the unitary method, two different things are compared as explained below:

  1. When the total cost is compared with the number of items purchased, cost per item is obtained.
  2. When the distance traveled is compared with the total time, the distance traveled per unit time is obtained.

Example

A train travels 1000 km in 5 hours. Find the percentage of distance travelled by train in 3 hours.

Solution

The distance travelled by a train in 5 hours  = 1000 km

The distance travelled by the train in 1 hour = 1000/5 km = 200 km

The distance travelled by the train in 3 hours = 200 km x 3 = 600 km

Percentage = 600/1000 x 100 = 60%

A graphical representation of calculating percentages has been presented below for better understanding.

The guidance below will help you work through percentage calculation problems including those found on the percentage worksheets page.

As you guide your child you should also take the opportunity to explain the importance and relevance of percentage calculations: pay rises, allowance rises, interest rates, discounts on sale items etc. Learning is always improved when the relevance of what is being learned is appreciated.

What is a percentage?

Percent means “for every 100” or “out of 100.” The (%) symbol as a quick way to write a fraction with a denominator of 100. As an example, instead of saying “it rained 14 days out of every 100,” we say “it rained 14% of the time.”

Percentages can be written as decimals by moving the decimal point two places to the left:

Decimals can be written as a percentages by moving the decimal point two places to the right:

Formula for calculating percentages

The formulas for calculating percentages or for converting from percentages are relatively simple.

To convert a fraction or decimal to a percentage, multiply by 100:

To convert a percent to a fraction, divide by 100 and reduce the fraction (if possible):

Examples of percentage calculations

The following two examples show how to calculate percentages.

1) 12 people out of a total of 25 were female. What percentage were female?

2) The price of a $1.50 candy bar is increased by 20%. What was the new price?

3) The tax on an item is $6.00. The tax rate is 15%. What is the price without tax?

Similar types of problems to those in the examples above are solved in a series of three mini-lessons on Calculating with Percent. These are listed below.

 #1: Introduction

 #2: Calculating with Percent e.g. 12% of 80 is?

 #3: Calculating with Percent e.g. 6 out of 8 is what % and 15 is 30% of what?

Percentage Chart

This Percentage Chart shows what 15% of $1 through $100 is although it is customizable so you can set the percentage and the numbers to whatever you want.

Find 1% – The Unitary Method

Handy Tip: A good way of finding percentages is to start by finding what 1% is.Example: What is 6% of 31?

Find 1%.
Divide by 100 (or move the decimal
point two places to the left)
31 ÷ 100 = .31
We now know what 1% is. We just need to
multiply it by 6 to find 6%
.31 x 6 = 1.86

6% of 31 is 1.86

You can practice calculating percentages by first finding 1% (and/ or finding 10%) and then multiplying to get your final answer using this Calculating Percentages in Two Steps Worksheet. There are also more percentage worksheets here too.

Common error when finding a percentage

Since percentages are often thought of as parts of a larger whole thing, there can be a tendency to divide instead of multiply when faced with a problem such as “find 35% of 80.” As the example below shows, after converting the percent to a decimal, the next step is to multiply, not divide.

error whening finding percentages by dividing instead of multiplying the decimal conversion

An understanding of percent allows students to estimate to check whether their answer is reasonable. In this example, knowing that 35% is between one-quarter and one-half would mean the answer should be somewhere between 20 and 40.