Adding and subtracting in standard form

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In order to access this I need to be confident with:

Adding and subtracting decimals Multiplying and dividing decimals Laws of indices Negative numbers Place value

This topic is relevant for:

GCSE Maths Number Standard Form

Adding and Subtracting in Standard Form

Here we will learn about adding and subtracting numbers in standard form.

There are also adding and subtracting numbers in standard form 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 adding and subtracting in standard form?

Adding and subtracting in standard form works in a similar way to adding and subtracting ordinary numbers. There are two methods we can use.

We can either convert standard form to ordinary numbers then use the column method for addition or subtraction, or we can adjust the numbers so that they have the same power of ten and then use addition or subtraction.

E.g.
$(4\times10^{3})+(6\times10^{2})$

Converting to ordinary numbers first: $4000 + 600 = 4600$
However, this method is not very efficient especially for very large and very small numbers.

To add and subtract numbers in standard we can first convert the numbers so that they have the same power of ten.

E.g.
Using standard form: $(4\times10^{3})+(0.6\times10^{3})=(4.6\times10^{3})$

What is adding and subtracting in standard form?

How to add and subtract with standard form

In order to add and subtract numbers in standard form:

Convert one of the numbers so that both numbers have the same power of ten. Select the number with the lower power of 10.
Add/subtract the decimals.
Write your answer in standard form.

How to add and subtract with standard form

Adding and subtracting in standard form worksheet

Get your free adding and subtracting in standard form worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Adding and subtracting in standard form worksheet

Get your free adding and subtracting in standard form worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Adding and subtracting standard form examples

Example 1: adding numbers in standard form

Work out:

\[ (5\times10^{5})\quad+\quad(2\times10^{4})\]

Convert one of the numbers so that both numbers have the same power of ten. Select the number with the lower power of 10.

$10^{4}$ is the lowest power of ten. Multiply it by $10$ so it also becomes $10^{5}$

$10^{4}$ x $1= 10^{5}$

Divide 2 by 10 to maintain the value:

2 \div 10 = 0.2

2 \times 10^{4} = 0.2 \times 10^{5}

2 Add & subtract the decimals.

5 + 0.2 = 5.2

3 Write your answer in standard form.

( $5$ x $10^{5})$ + ( $0.2$ x $10^{5}) = 5.2$ x $10^{5}$

Example 2: adding numbers in standard form

Work out:

\[(7\times10^{-3})\quad+\quad(6\times10^{-4})\]

Convert one of the numbers so that both numbers have the same power of ten. Select the number with the lower power of 10.

$10^{-4}$ is the lowest power of ten. Multiply it by $10$ so it also becomes $10^{-3}$

\[10^{-4}\times10\quad=\quad10^{-3}\]

Divide 6 by 10 to maintain the value.

\[6\div10 = 0.6\]

\[6\times10^{-4}\quad=\quad0.6\times10^{-3}\]

Add/subtract the decimals.

\[7 + 0.6 = 7.6\]

Write your answer in standard form.

\[(7\times10^{-3})\quad+\quad(0.6\times10^{-3})\quad=\quad7.6\times10^{-3}\]

Example 3: adding numbers in standard form

Work out

\[(8.1\times10^{7})\quad+\quad(2.5\times10^{5})\]

Convert one of the numbers so that both numbers have the same power of ten. Select the number with the lower power of 10.

$10^{5}$ is the lowest power of ten. Multiply it by $100$ so it also becomes $10^{7}$

\[10^{5}\times100\quad=\quad10^{7}\]

Divide 2.5 by 100 to maintain the value.

\[2.5\div100 = 0.025\]

\[2.5\times10^{5}\quad=\quad0.025\times10^{7}\]

Add/subtract the decimals.

Write $8.1 + 0.025 = 8.125$

Write your answer in standard form.

\[(8.1\times10^{7})\quad+\quad(0.025\times10^{7})\quad=\quad8.125\times10^{7}\]

Example 4: subtracting numbers in standard form

Calculate

\[(8\times10^{4})\quad-\quad(6\times10^{3})\]

Write your answer in standard form.

Convert one of the numbers so that both numbers have the same power of ten. Select the number with the lower power of 10.

$10^{3}$ is the lowest power of ten. Multiply it by $10$ so it also becomes $10^{4}$

\[10^{3}\times10\quad=\quad10^{4}\]

Divide 6 by 10 to maintain the value.

\[6\div10 = 0.6\]

\[6\times10^{3}\quad=\quad0.6\times10^{4}\]

Add/subtract the decimals.

\[8 – 0.6 = 7.4\]

Write your answer in standard form.

\[(8\times10^{4})\quad+\quad(0.6\times10^{4})\quad=\quad7.4\times10^{4}\]

Example 5: subtracting numbers in standard form

Calculate

\[(8\times10^{-2})\quad-\quad(3\times10^{-3})\]

Write your answer in standard form.

Convert one of the numbers so that both numbers have the same power of ten. Select the number with the lower power of 10.

$10^{-3}$ is the lowest power of ten. Multiply it by $10$ so it also becomes $10^{-2}$

\[10^{-3}\times10\quad=\quad10^{-2}\]

Divide 3 by 10 to maintain the value.

\[3\div10 = 0.3\]

\[3\times10^{-3}\quad=\quad0.3\times10^{-2}\]

Add/subtract the decimals.

\[8 – 0.3 = 7.7\]

Write your answer in standard form.

\[(8\times10^{-2})\quad-\quad(0.3\times10^{-2})\quad=\quad7.7\times10^{-2}\]

Example 6: subtracting numbers in standard form

Calculate

\[(6.2\times10^{4})\quad-\quad(1.8\times10^{2})\]

Write your answer in standard form.

Convert one of the numbers so that both numbers have the same power of ten. Select the number with the lower power of 10.

$10^{2}$ is the lowest power of ten. Multiply it by $100$ so it also becomes $10^{4}$

\[10^{2}\times100\quad=\quad10^{3}\]

Divide 1.8 by 100 to maintain the value.

\[1.8\div100 = 0.018\]

\[1.8\times10^{2}\quad=\quad0.018\times10^{4}\]

Add/subtract the decimals.

\[6.2 – 0.018 = 6.182\]

Write your answer in standard form.

\[(6.2\times10^{4})\quad-\quad(0.018\times10^{4})\quad=\quad6.182\times10^{4}\]

Common misconceptions

Using the column method

While this is not wrong, it is an inefficient method and can lead to errors when converting the numbers, especially with very large and very small numbers with many place holders.

E.g.
Work out $(8\times10^{4})\quad+\quad(6\times10^{3})$

\[8000 + 600 = 8600 = 8.6\times10^{4}\]

Using standard form:

\[(8\times10^{4})\quad+\quad(0.6\times10^{4})\quad=\quad8.6\times10^{4}\]

Not converting solutions to standard form

After calculating with standard form, a common mistake is not ensuring the number is in standard form.
Remember to be in standard form the number needs to have two parts, the first part should between 1 and 10 (1 ≤ n < 10) and the second part should be a power of 10.

E.g.
$62\times10^{7}$ is not in standard form as 62 is greater than 10.
In standard for this should be written as $6.2\times10^{8}$

Negative powers

A common mistake is to become mixed up when using negative powers.
E.g.
$10^{-3}$ is smaller than $10^{-2}$ because $-3$ is less than $-2$ .
$10^{-3}=0.001$ and $10^{-2}=0.01$ so $10^{-2}$ is greater than $10^{-3}$

Adding and subtracting standard form is part of our series of lessons to support revision on standard form. You may find it helpful to start with the main standard form lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Practice adding standard form and subtracting standard form questions

23 \times 10^{5}

5\times 10^{11}

3.2 \times 10^{5}

2.3 \times 10^{6}

3 \times 10^{5}=0.3 \times 10^{6}\\ (2 \times 10^{6})+(0.3 \times 10^{6}=2.3 \times 10^{6}

4.7 \times 10^{-5}

4.7 \times 10^{-4}

7.4 \times 10^{-4}

47 \times 10^{-6}

7 \times 10^{-6}=0.7 \times 10^{-5}\\ (4 \times 10^{-5}) + (0.7 \times 10^{-5})=4.7 \times 10^{-5}

6.75 \times 10^{8}

643.5 \times 10^{6}

6.435 \times 10^{8}

9.9 \times 10^{14}

3.5 \times 10^{6} = 0.035 \times 10^{8}\\ (6.4 \times 10^{8}) + (0.035 \times 10^{8}) = 6.435 \times 10^{8}

7.4 \times 10^{4}

66 \times 10^{3}

6.6 \times 10^{4}

3 \times 10^{4}

4 \times 10^{3} = 0.4 \times 10^{4}\\ (7 \times 10^{4})-(0.4 \times 10^{4})=6.6 \times 10^{4}

4.8 \times 10^{-3}

1.5 \times 10^{-3}

4.8 \times 10^{-4}

3 \times 10^{1}

2 \times 10^{-4} = 0.2 \times 10^{-3}\\ (5 \times 10^{-3})-(0.2 \times 10^{-3}) = 4.8 \times 10^{-3}

7.57 \times 10^{5}

7.823 \times 10^{5}

777.7 \times 10^{3}

7.777 \times 10^{5}

2.3 \times 10^{3} = 0.023 \times 10^{5}\\ (7.8 \times 10^{5})-(0.023 \times 10^{5}) = 7.777 \times 10^{5}

Adding and subtracting standard form GCSE questions

1. The table below shows the population of several cities

City	Population
London	$8.9$ x $10^6$
Manchester	$5.5$ x $10^5$
Birmingham	$1.1$ x $10^6$
Oxford	$1.1$ x $10^5$

Work out the total population of London and Manchester. Give your answer in standard form.

(3 marks)

Show answer

Simplifying by writing the two populations as an ordinary number:
London: $8900000$
Manchester: $550000$

converting $5.5\times10^{5}$ to $0.55\times10^{6}$

(1)

Adding the numbers:
$8900000 + 550000 = 9450000$

8.9\times10^{6} +0.55\times10^{6}  = 9.45 \times 10^{6}

(1)

9.45 \times 10^{6}

(1)

2. Work out $7\times10^{6}\quad-\quad4\times10^{5}$ .
Give your answer in standard form.

(3 marks)

Show answer

Simplifying by writing the two numbers as an ordinary numbers:

7000000 - 400000

converting $4\times10^{5}$ to $0.4\times10^{6}$

(1)

Subtracting the numbers:
$7000000 - 400000 = 6600000$

7\times10^{6} – 0.4\times10^{6}  = 6.6 \times 10^{6}

(1)

6.6\times10^{6}

(1)

3. Show that
$2.4\times10^{2}\quad+\quad3.7\times10^{3}\quad=\quad3.94\times10^{3}$

(2 marks)

Show answer

$240 + 3700$ or $0.24\times10^{3}$ or $39.4\times10^{2}$

(1)

Correct working shown

(1)

Learning checklist

You have now learned how to:

Add numbers in standard form

Subtract numbers in standard form

The next lessons are

Still stuck?

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