# Problem Solving with Decimals and Percentages

## Topic Overview

A decimal number is a number with a decimal point in it. For example: 3.21. The number to the left of the decimal point is an ordinary whole number. The first number to the right of the decimal point is the number of tenths (1/10's). The second is the number of hundredths (1/100's) etc. A decimal can be converted to either a fraction or a percentage.

A percentage is a number or ratio expressed as a fraction of 100. The term 'percent' means 'out of 100' and is denoted by the percentage symbol : %.

Throughout your GCSE Maths exam, you will face several questions which require you to understand the basic principles of decimals and percentages in order to solve problems and convert numerical values. Subsequently, this revision guide will introduce you to the basic principles of decimals and percentages, as well as demonstrating how to use these principles in order to solve mathematical problems.

## Key Concepts

In the new linear GCSE Maths paper, you will be required to recognise certain properties of decimals and percentages and solve related calculations. According to the Edexcel Revision Checklist for the linear GCSE Maths paper, you will be required to:

• Approximate to specified or appropriate degrees or accuracy including a given power of ten, number of decimal places and significant figures
• Interpret decimals and percentages as operators
• Recognise and use percentage increase and decrease
• Understand that percentage means 'number of parts per 100' and use this to compare proportions
• Add subtract, multiple and divide whole numbers, integers and decimals
• Order integers and decimals

## Worked Examples

1 - How to value and order decimals

Valuing decimals

A decimal point is used to separate units from parts of a whole. For example: a tenth is 1/10 of a unit or 0.1.

Therefore, in the number 43.971, the value of the figure 9 is a tenth, the value of the figure 7 is a hundredth and the value of the figure 1 is a thousandth of a unit.

Ordering decimals

When ordering numbers, you must always compare the digits on the left first. For example, during your exam you may be asked to order decimals according to their value:

Example
(a) - Which value is greater: 3.805 or 3.85?

Solution
(a) - To order these numbers correctly, you must examine their different units.

Both numbers have two units and seven tenths, but 3.805 has no hundredths, whereas 3.85 has 5 hundredths.

Therefore, 3.85 is greater than 3.805.

(Note: If you write a zero at the end of 3.85, it is easier to compare their values as they will have the same amount of decimal places. Thus, by looking at 3.805 and 3.850, it is easy to recognise that 3.850 is the greater value).

2 - Adding and subtracting decimals
In order to easily add and subtract decimals, you must align the decimal points of each value.

Example
(a) - Calculate 8.35 + 4.73

Solution
(a) -

$8.35 = 8 + 0.3 + 0.05$
$4.73 = 4 + 0.7 + 0.03$
$8 + 0.3 + 0.05 + 4 + 0.7 + 0.03 = 12 + 1 + 0.08$
$= 13.08$

Example
(b) - Calculate 12.89 - 5.08

Solution
(b) -

$12.89 = 12 + 0.8 + 0.09$
$5.08 = 5 + 0.08$
$12.89 - 5.08 = (12 + 0.8 + 0.09) - (5 + 0.08)$
$= 12 - 5 + 0.8 + 0.09 - 0.08$
$= 7 + 0.8 + 0.01$
$= 7.81$

(Note: If you are asked to add or subtract a decimal number with a whole number, it is useful to write out the decimal places of this whole number in order to clearly align the decimal points. For example, if asked to add 4.51 and 9, it is helpful to write 9 as 9.00).

3 - Multiplying and dividing decimals
When multiplying and dividing decimals, this involves moving the decimal point several places to the left or right.

Examples

Multiply a value by 10 and the number will move one place value to the left:

$12.8*10 = 128$

Multiply by 100 and the number will move two place values to the left:

$12.8*100 = 1280$

Multiply by 1000 and the number will move three place values to the left:

$12.8*1000 = 12800$

Divide a number by 10 and the number will move one place value to the right:

$12.8/10 = 1.28$

Divide a number by 100 and the number will move two place values to the right:

$12.8/100 = 0.128$

Divide a number by 1000 and the number will move three places to the right:

$12.8/1000 = 0.0128$

Multiplying decimals

When multiplying decimals, you follow the same rules as when you are multiplying two whole numbers. Therefore, in order to multiply decimals correctly remember that:

• If there is one digit after the decimal point in the question, there will be one digit after the decimal point in the answer.
• If there are two digits after the decimal point in the question, there will be two digits after the decimal point in the answer etc.

Example
(a) - Calculate 2.3 x 3.4

Solution

Multiply without the decimal points : 23 x34 = 782

Both 2.3 and 3.4 have 1 decimal place so the answer will have 2 decimal places

Therefore 2.3 x 3.4 = 7.82

Dividing decimals

When dividing a decimal by a whole number, divide as usual but, as in previous examples, keep the decimal points correctly aligned.

Example
(a) - Calculate 5.75÷ 5

Solution
(a) -

$5\mathop{\left){\vphantom{1{5.75}}}\right. \!\!\!\!\overline{\,\,\,\vphantom 1{{5.75}}}} \limits^{\displaystyle\ {1.15}} = 1.15$

It is also important to note that if you are dividing a decimal by another decimal, you must use equivalent fractions. Remember to always multiply the numerator and denominator by the same number and make sure that the denominator is a whole number.

Example
(b) - Calculate 450 ÷ 0.25

Solution
(b) -

$= \frac{{450}}{{0.25}}$

Multiply both the numerator and denominator by 100 in order to use equivalent fractions:

$\frac{{450}}{{0.25}} = 25\mathop{\left){\vphantom{1{45000}}}\right. \!\!\!\!\overline{\,\,\,\vphantom 1{{45000}}}} \limits^{\displaystyle\,\, {1800}}$

4 - Working out percentages
As mentioned at the beginning of this guide, 'percent' means 'out of 100' and is denoted by the percentage symbol : %.

Therefore a percentage is a fraction of 100.

Example

20% means 20 out of 100

As a fraction 20% is 20/100

As a decimal 20% is 0.2

During your GCSE Maths exam, you may be asked to calculate the percentage of a quantity. In order to work out this percentage, you must first write the percentage as a fraction or a decimal, then multiply this value by the quantity.

Example
(a) - Calculate 20% of £60

Solution
(a) -

20% as a fraction is 20/100= 2/10= 1/5

20% as a decimal is 0.2

Therefore to calculate 20% of £60, you must multiply £60 by this value

0.2 x 60 = 12

1/5 x 60/1= 60/5 =12

Therefore 20% of £60 is £12

5 - Percentage increase and decrease
During your GCSE Maths exam, you may also be asked to find the percentage increase or decrease of the price of a certain item. In order to calculate these values, you must find one amount as a percentage of another. To do this, you must form a fraction from the two amounts and multiply this by 100.

Example
(a) - £500 is borrowed for 6 years at an interest rate of 5% per annum (i.e. each year). Calculate the interest generated.

Solution
(a) - This type of question is referred to as a simple interest problem, because the amount of money borrowed remains fixed.

First you must calculate the amount of interest generated after one year, which is 5% of £400

5% of £500= 5/100× 500/1= 2500/100= 25

Therefore the amount of interest generated after one year is £25

The question is asking you to generate the amount of interest after 6 years, so you must multiply this value by 6

25 x 6 = 150

Therefore the amount of interest generated after 6 years = £150

This simple interest process can be memorised using the formula:

Interest = P × R × T

• P (principal) is the amount borrowed.
• R is the rate of interest per year.
• T is the time in years.

You may also be asked to solve a profit and loss question. This type of percentage based question involves an item being bought at one price and sold for another.

If the selling price is greater than the buying price, a profit is made.

If the selling price is less than the buying price, a loss occurs.

Example
(b) - Sarah buys a TV for £90 and sells it for £120. What is her percentage of profit made?

Solution
(b) - First you must work out the amount of profit made. The cost price is £90 and the selling price is £120, so the profit made is:

£120-£90 = £30

To calculate Sarah's percentage profit, you must calculate this profit value as a percentage of the original price of the TV. To do this, you must divide the profit by the original price and multiply the value by 100:

30 ÷120 = 0.25

0.25 x 100 = 25

Therefore the percentage profit made is £25%.

Example
(c) - Darren buys a games console in a sale for £90. The original cost of the console was £250. What is the percentage decrease of the games console?

Solution
(c) - Darren bought the games console for £90 and the original price was £250. Therefore the loss which has occurred is:

£250 - £90 = £160

To calculate the percentage decrease, you must divide the actual decrease by the original price and multiply this value by 100:

160/250= 0.64

0.64 x 100= 64

Therefore the percentage decrease which has occurred is 64%.

6 - Reverse percentages
You may be asked to find the original price of an item after the price has increased or decreased. If you are given a quantity after a percentage increase or decrease, you can calculate the original price of the item by following this method:

Example
(a) - A stereo sells for £120 after a 25% increase in the cost price. Calculate the original cost price of the stereo.

Solution
(a) - Regard the original cost price of the stereo as 100%.

The selling price is a 25% increase of this cost price. Therefore the selling price is:

100% + 25% = 125% of the cost price

The question tells you that the selling price is £120, so 125% = £120

To calculate the original price, first calculate 1% of the cost of the stereo:

125% = £120, so 1% will be 120/125= 0.96

1% = £0.96

The original cost price is 100%, so multiply £0.96 by 100

Therefore the original cost price of the stereo = 0.96 × 100 = £96.

Example
(b) - The cost of a new DVD decreases in price by 20% in a year. After a year the DVD is worth £10. Calculate the original cost price of the DVD when it was new.

Solution
(b) - The original cost price of the DVD is 100%.

The decreased price = 100%- 20% = 80%

So £10 = 80% of the original cost price of the DVD

From this value you can work out that 1% of original price = £10 ÷ 80

Therefore the original cost price of the DVD = 100% = 100 x 1% = 100 x (£10 ÷ 80)

Therefore the original cost price of the DVD = £12.50.

## Exam Tips

1. When ordering numbers, you must always compare the digits on the left first
2. When adding, subtracting, multiplying or dividing decimals, ensure you have correctly aligned the decimal points of all the values in the question
3. Remember that percentages are simply fractions out of 100, and decimals are simply tenths, hundredths and thousandths of 1
4. When calculating percentage increase or decrease, double check your answer and working out in order to make sure your answer is realistic

## Topic Summary

When solving decimal and percentage related problems, it is extremely important that you clearly demonstrate your working out and double check your method. Otherwise, a small mistake or misplaced decimal could cost you vital marks. However, with care and attention, you can practise and perfect how to order, value and calculate a wide range of decimal and percentage related problems!

## Related Topics

• Arithmetic with Fractions
• Indices
• Accuracy of Measurement Problems
• Arithmetic with Positive and Negative Integers
• Standard Form
• Rational and Irrational Numbers
• Combinations