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Arithmetic

Arithmetic

Here we will learn about arithmetic, including key terminology and mathematical symbols, using the four operations with positive and negative integers and inverse operations.

There are also arithmetic worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is arithmetic?

Arithmetic is the study of numbers and the operations between them. It is an elementary branch of mathematics. The definition of arithmetic originates from the Greek word arithmos, meaning number, or arithmētikē, the art of counting.

To solve problems using basic arithmetic, we need to understand and use four arithmetic operators. At GCSE, we need to be able to use these operations for both positive and negative integers (whole numbers), fractions and decimals. 

The four types of computations we will be looking at are: addition, subtraction, multiplication and division.

What is arithmetic?

What is arithmetic?

Mathematical symbols

Below is a table showing the four basic operations of arithmetic with their associated symbols.

NEW arithmetic image 1

Each operation has a different function that you should already be confident with using.

Arithmetic with four operations

Addition

Addition combines elements in two or more sets together. For two sets a and b, the new set c is equal to the sum of a and b, written as a+b=c and pronounced a plus b is equal to c.

NEW arithmetic addition image 1

Addition is commutative. The order in which the calculation is carried out does not matter.

For example,

3+4=4+3=7

Addition can be done with positive and negative integers, fractions and decimals. Addition represents a movement up the number line. Here are some examples,

NEW arithmetic addition image 2

In order to perform more difficult additions, we can use the column method.

For example, 347+21

NEW arithmetic addition image 3

It is important to consider the place value of each digit and line up the corresponding digits in each number.

Subtraction

Subtraction removes elements from a set. The set of elements remaining when the elements in b are subtracted from the elements in c would be equal to a, written as c-b=a and pronounced c take away b is equal to a.

NEW arithmetic subtraction image 1

Subtraction is not commutative. If the order of the numbers within the calculation changes, the result will change.

For example,

7-4 {4}-7

Subtraction can also be done with positive and negative integers, fractions and decimals. Subtraction represents a movement down the number line. Here are some examples,

NEW arithmetic subtraction image 2

In order to perform more difficult subtractions, we can use the column method. 

For example, 89.4-3.1

NEW arithmetic subtraction image 3

Multiplication

Multiplication is essentially repeated addition. If we have n copies of set a, we multiply a by n to find how many elements are in the new set, m. This is the same as calculating a+a+a+... \ n times. This would be written as n \times a=m and pronounced n times a is equal to m.

NEW arithmetic multiplication image 1

Multiplication is commutative. The order in which the calculation is carried out does not matter.

For example,

3\times{4}=4\times{3}=12

Multiplication can also be done with positive and negative integers, fractions and decimals. When multiplying positive and negative numbers, the following rules apply,

NEW arithmetic multiplication image 2

To perform more difficult multiplications, we can use the grid method.

For example, 27 \times 35

NEW arithmetic multiplication image 3

600+210+100+35=945

27 \times 35=945

Division

Division shares the elements of a set into a number of distinct groups. If the elements of set m can be shared equally between n groups, with no remainder, then this is written as m\div{n}=a and pronounced m divided by n is equal to a.

NEW arithmetic division image 1

Unlike multiplication, division is not commutative. If the order of the numbers within the calculation changes, the result will change.

12\div{4} {4}\div{12}

Division can also be done with positive and negative integers, fractions and decimals. When dividing positive and negative numbers, the following rules apply,

NEW arithmetic division image 2

To perform more difficult divisions, we can use the method of short division.

For example, 452.1 \div 3

NEW arithmetic division image 3

Inverse operations

Inverse operations reverse an operation that has been carried out. Below is a table outlining some operations along with their inverse operations.

NEW arithmetic inverse operations image 1

Note that we can switch the columns so the inverse operation of subtraction is addition and the inverse operation of division is multiplication, etc.

NEW arithmetic inverse operations image 2

NEW arithmetic inverse operations image 3

BIDMAS

When solving problems involving arithmetic operations, it is important to apply BIDMAS. BIDMAS tells us what order to perform operations in and stands for

Brackets

Indices

Division

Multiplication

Addition

Subtraction

In the calculation 3+4 \times 2, the multiplication should be done before the addition.

3+4 \times 2=3+8=11

Step-by-step guide: BIDMAS (coming soon)

Quotients

The quotient is the answer we get when we divide one number by another. 

The word quotient comes from Latin and means ‘how many times’. When we divide we are finding out ‘how many times’ a number goes into another number.


Dividend \div divisor = quotient

For example,

NEW arithmetic quotients image 1

In this calculation, 8 is the dividend

4 is the divisor and 

the answer, 2 , is the quotient.

The quotient will only be 0 if the dividend is 0 but the divisor is not. 

For example,

0 \div 8 = 0

Word problems

At GCSE level, we are often presented with word problems that need to be worked out using arithmetic. The first step is to decide on which operation is required and then the relevant calculations can be performed. It can help to highlight important information in the questions. 

How to use arithmetic with positive and negative numbers

In order to use arithmetic with positive and negative numbers:

  1. Determine which operator you need to use.
  2. Carry out the calculation.

Explain how to use arithmetic with positive and negative numbers

Explain how to use arithmetic with positive and negative numbers

Arithmetic worksheet

Get your free arithmetic worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Arithmetic worksheet

Get your free arithmetic worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Arithmetic examples

Example 1: addition with decimals

Calculate the value of 10.9+34.3.

  1. Determine which operator you need to use.

Here we need to use addition.

2Carry out the calculation.

Rearranging the layout of the calculation into a column addition format, we have

NEW arithmetic example 1 image 1

Adding up the right hand column first we have 9+3=12. The tenths column value of 2 is therefore placed in the answer bar below the sum, and the 1 is carried across below the answer bar so that we can add it to the next column total (the units column).

NEW arithmetic example 1 image 2

Adding 0, 4, and the 1 that was carried across from the tenths column, we have 0+4+1=5 and so 5 goes into the answer bar for the units column.

NEW arithmetic example 1 image 3

The final column requires us to add 1 and 3. \ 1+3=4 and so we put a 4 in the tens column.

NEW arithmetic example 1 image 4

So 10.9+34.3=45.2 .

Example 2: subtraction (worded problem)

96 mathematicians attended a celebration party at a hotel. 38 mathematicians stayed overnight. How many mathematicians did not stay overnight?

Here we have a subtraction. We need to calculate 96-38.

Writing the subtraction using the column method, we have


NEW arithmetic example 2 image 1


Starting from the units column, 6-8 is a negative number and so we have to borrow 1 from the tens column.


NEW arithmetic example 2 image 2


16-8=8 and so the value for the units column is 8.


NEW arithmetic example 2 image 3


Continuing with the tens column, we have 8-3=5.


NEW arithmetic example 2 image 4


58 guests did not stay overnight.

Example 3: multiplication

Each sack contains 36 potatoes. A farmer packs 24 sacks of potatoes with no left over potatoes. How many potatoes does the farmer pack?

As we have 36 potatoes per sack and 24 sacks, we need to multiply these values together.

Using the grid method, we have


NEW arithmetic example 3 image 1


By multiplying each row value by each column value, we get


NEW arithmetic example 3 image 2


Adding the new values in the grid, we have


NEW arithmetic example 3 image 3


The farmer packs 864 potatoes.

Example 4: division

A bar of chocolate is made up of 84 individual cubes. The bar is 6 cubes wide. How many rows does the chocolate bar have?

As each column contains 6 cubes and there are 84 cubes in total, we need to divide 84 by 6 to determine the number of rows in the chocolate bar.

Using long division, we have


NEW arithmetic example 4 image 1


8\div{6}=1 remainder 2 and so we place the 1 above the 8 on the answer line, and carry the 2 across in front of the 4 to determine the units.


NEW arithmetic example 4 image 2


24\div{6}=4 with no remainder and so we write 4 on the answer line.


NEW arithmetic example 4 image 3


The chocolate bar has 14 rows.

Example 5: addition, with negative numbers

Calculate (-3)+(-2).

Here, we are calculating an addition with the two negative numbers.

(-3)+(-2) = -3 + -2


+ and - together make a - therefore


-3 + -2= -3 -2 = -5 .

Example 6: subtraction, with negative numbers

Calculate 5-(-3).

Here we are subtracting -3 from 5.

5-(-3) = 5--3


Two - signs together make a + therefore


5--3=5+3=8 .

Example 7: multiplication, with negative numbers

Calculate 8\times(-2).

Here we are using a multiplication.

8\times(-2)=-16.

Example 8: division, with negative numbers

Calculate \frac{-120}{-3}.

A fraction is the division of two quantities and so here we are calculating a division. The numerator is known as the dividend and the denominator is known as the divisor. The result of a division is called a quotient.

\frac{-120}{-3}=40

Common misconceptions

  • Brackets are not an arithmetic operator

Operators allow us to perform calculations between two or more numbers. Brackets dictate the order in which this should be completed and so they are technically not an operator.

  • Multiplying by \bf{0}

It is important to remember that multiplying anything by 0 gives the answer 0.

  • Dividing or multiplying by a fraction

Mistakes can easily be made when multiplying and dividing fractions. When dividing a number by a fraction, a commonly seen error is that the value is multiplied by the fraction instead. For example, 12\div\frac{1}{2}=6 is incorrect. Instead, the correct answer is 12\div\frac{1}{2}=12\times{2}=24.

Practice arithmetic questions

1. Calculate 8.4+10.7.

18.11
GCSE Quiz False

19.1
GCSE Quiz True

18.74
GCSE Quiz False

11.54
GCSE Quiz False

NEW arithmetic practice qustion 1 explanation

2. I bought an item for £25.13. How much change did I get from £30?

£4.87
GCSE Quiz True

£4.97
GCSE Quiz False

£5.87
GCSE Quiz False

£5.97
GCSE Quiz False

NEW arithmetic practice qustion 2 explanation

3. Calculate the 7th multiple of 9.

2
GCSE Quiz False

16
GCSE Quiz False

63
GCSE Quiz True

0.78 \ (2dp)
GCSE Quiz False

We need to multiply 9 by 7.

 

9\times{7}=63

4. 120 students are divided into small research groups. How many students are in each group if there are 20 groups?

2400
GCSE Quiz False

60
GCSE Quiz False

6
GCSE Quiz True

5
GCSE Quiz False

We need to calculate 120 divided by 20.

 

120\div{20}=12\div{2}=6

5. A hotel comprises 67 floors above ground and 4 floors below ground. A guest parks on floor -3, in the basement, and is staying on the 43rd floor of the hotel. How many floors must he go up to get from his car to his hotel room?

27
GCSE Quiz False

43
GCSE Quiz False

44
GCSE Quiz False

46
GCSE Quiz True

This is a subtraction of 43 and -3.

 

43- -3=43+3=46

6. Anna buys a car worth £9,000. She pays a deposit of £2400 and pays the rest off in monthly instalments of £110 per month. For how many months will Anna be paying for her car?

104 months

GCSE Quiz False

22 months

GCSE Quiz False

60 months

GCSE Quiz True

66 months

GCSE Quiz False

This calculation requires a subtraction to calculate the remaining amount to pay for the car, and then a division of the remaining amount by 110.

 

9000-2400=6600

 

6600\div{110}=60

Arithmetic GCSE questions

1. Below is the advert board for a range of speciality teas.

 

NEW arithmetic gcse qustion 1

 

(a) Calculate the total price of 2 green teas, 1 english breakfast tea, and 1 chamomile tea.

 

(b) Calculate the change from £10 for the order in part (a).

 

(4 marks)

Show answer

(a)

 

2+2+2.10+2.25

(1)

£8.35

(1)

 

(b)

 

10-\text{their } 8.35

(1)

£1.65

(1)

2. (a) Simon says “ 16 divided by a half is 8 ”. Is Simon correct? Explain your answer.

 

(b) Stephanie says “ 7 \times 0=7 ”. Is Stephanie correct? Explain your answer?

 

(4 marks)

Show answer

(a)

 

No

(1)

16\div{1}{2}=16\times{2}=32

(1)

 

(b)

 

No

(1)

Any number multiplied by 0 is equal to 0 .

(1)

3. The number of worms in a population doubles every 3 weeks. The number of worms in a garden now is 120. How many worms will be in the garden in 12 weeks?

 

(3 marks)

Show answer

Doubles 4 times.

(1)

120\times{2}\times{2}\times{2}\times{2}

(1)

1920

(1)

Learning checklist

You have now learned how to:

  • Combine knowledge of number facts and rules of arithmetic to solve mental and written calculations
  • Solve a wider range of problems, including increasingly complex properties of numbers and arithmetic
  • Recognise arithmetic sequences

The next lessons are

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FREE GCSE Maths Practice Papers - 2022 Topics

Practice paper packs based on the advanced information for the Summer 2022 exam series from Edexcel, AQA and OCR. 



Designed to help your GCSE students revise some of the topics that will come up in the Summer exams.

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