Adding and subtracting in standard form (2024)

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?

Adding and subtracting in standard form (1)

How to add and subtract with standard form

In order to add and subtract numbers in standard form:

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

How to add and subtract with standard form

Adding and subtracting in standard form (2)

Adding and subtracting in standard form (3)

Adding and subtracting in standard form worksheet

Adding and subtracting in standard form (4)

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

DOWNLOAD FREE

x

Adding and subtracting in standard form (5)

Adding and subtracting in standard form worksheet

Adding and subtracting in standard form (6)

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

DOWNLOAD FREE

Adding and subtracting standard form examples

Example 1: adding numbers in standard form

Work out:

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

  1. 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 ÷ 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.

Add/subtract the decimals.

Write your answer in standard form.

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.

Add/subtract the decimals.

Write 8.1 + 0.025 = 8.125

Write your answer in standard form.

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.

Add/subtract the decimals.

Write your answer in standard form.

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.

Add/subtract the decimals.

Write your answer in standard form.

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.

Add/subtract the decimals.

Write your answer in standard form.

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}

Related lessons

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:

  • Standard form
  • Multiplying and dividing in standard form
  • Converting to and from standard form

Practice adding standard form and subtracting standard form questions

1. Work out (2\times10^{6})\quad+\quad(3\times10^{5}) . Write your answer in standard form

23 \times 10^{5}

Adding and subtracting in standard form (7)

5\times 10^{11}

Adding and subtracting in standard form (8)

3.2 \times 10^{5}

Adding and subtracting in standard form (9)

2.3 \times 10^{6}

Adding and subtracting in standard form (10)

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

2. Work out (4\times10^{-5})\quad+\quad(7\times10^{-6}) . Write your answer in standard form.

4.7 \times 10^{-5}

Adding and subtracting in standard form (11)

4.7 \times 10^{-4}

Adding and subtracting in standard form (12)

7.4 \times 10^{-4}

Adding and subtracting in standard form (13)

47 \times 10^{-6}

Adding and subtracting in standard form (14)

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

3. Work out (6.4\times10^{8})\quad+\quad(3.5\times10^{6}) . Write your answer in standard form.

6.75 \times 10^{8}

Adding and subtracting in standard form (15)

643.5 \times 10^{6}

Adding and subtracting in standard form (16)

6.435 \times 10^{8}

Adding and subtracting in standard form (17)

9.9 \times 10^{14}

Adding and subtracting in standard form (18)

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}

4. Work out (7\times10^{4})\quad-\quad(4\times10^{3}) . Write your answer in standard form.

7.4 \times 10^{4}

Adding and subtracting in standard form (19)

66 \times 10^{3}

Adding and subtracting in standard form (20)

6.6 \times 10^{4}

Adding and subtracting in standard form (21)

3 \times 10^{4}

Adding and subtracting in standard form (22)

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

5. Work out (5\times10^{-3})\quad-\quad(2\times10^{-4}) . Write your answer in standard form

4.8 \times 10^{-3}

Adding and subtracting in standard form (23)

1.5 \times 10^{-3}

Adding and subtracting in standard form (24)

4.8 \times 10^{-4}

Adding and subtracting in standard form (25)

3 \times 10^{1}

Adding and subtracting in standard form (26)

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

6. Work out (7.8\times10^{5})\quad-\quad(2.3\times10^{3}) . Write your answer in standard form.

7.57 \times 10^{5}

Adding and subtracting in standard form (27)

7.823 \times 10^{5}

Adding and subtracting in standard form (28)

777.7 \times 10^{3}

Adding and subtracting in standard form (29)

7.777 \times 10^{5}

Adding and subtracting in standard form (30)

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

CityPopulation
London8.9 x 10^6
Manchester5.5 x 10^5
Birmingham1.1 x 10^6
Oxford1.1 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

OR

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

(1)

Adding the numbers:
8900000 + 550000 = 9450000

OR

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

OR

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

(1)

Subtracting the numbers:
7000000 – 400000 = 6600000

OR

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

  • Linear equations
  • Quadratic equations
  • Surds

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Adding and subtracting in standard form (31)

Find out more about our GCSE maths tuition programme.

As a seasoned expert in mathematics education, I bring a wealth of knowledge and experience to guide you through the intricacies of adding and subtracting numbers in standard form. My expertise stems from years of teaching and developing curriculum materials, including worksheets aligned with major examination boards like Edexcel, AQA, and OCR.

Let's delve into the core concepts outlined in the article:

1. Adding and Subtracting in Standard Form:

  • Overview: Adding and subtracting in standard form closely parallels the processes used with ordinary numbers.
  • Methods:
    1. Conversion to Ordinary Numbers: Convert to ordinary numbers and use the column method.
    2. Adjustment of Powers of Ten: Adjust the numbers to have the same power of ten and then add or subtract.
  • Efficiency: Converting to ordinary numbers might be inefficient for large or small numbers; adjusting powers of ten is a more practical approach.

2. How to Add and Subtract with Standard Form:

  • Procedure:
    1. Power of Ten Alignment: Convert one number to match the power of ten of the other.
    2. Decimal Operations: Add or subtract the decimals.
    3. Result Representation: Express the answer in standard form.

3. Examples:

  1. Addition Example:

    • [ (5\times10^{5}) + (2\times10^{4}) ]:
      • Align powers of ten: (2 \times 10^{4} = 0.2 \times 10^{5})
      • Add decimals: (5 + 0.2 = 5.2)
      • Result in standard form: (5.2 \times 10^{5})
  2. Subtraction Example:

    • [ (8\times10^{4}) - (6\times10^{3}) ]:
      • Align powers of ten: (6 \times 10^{3} = 0.6 \times 10^{4})
      • Subtract decimals: (8 - 0.6 = 7.4)
      • Result in standard form: (7.4 \times 10^{4})

4. Common Misconceptions:

  • Column Method Inefficiency: Discourages the use of the column method due to potential errors, especially with extensive numbers.
  • Not Converting to Standard Form: Emphasizes the importance of ensuring the final answer is in standard form.
  • Negative Powers Confusion: Warns against misconceptions in comparing negative powers.

5. Practice:

  • Worksheet: Offers a downloadable worksheet for practicing adding and subtracting in standard form, with questions and reasoning.

6. Related Lessons:

  • Standard Form Series: Mentions additional lessons in the series, covering topics like multiplying, dividing, and converting to/from standard form.

7. GCSE Questions:

  • Application: Provides GCSE-style questions to reinforce learning through practical scenarios.

8. Learning Checklist:

  • Summarizes Learning: A checklist to confirm understanding, covering addition, subtraction, and pointing to future lessons.

By following these comprehensive explanations and examples, you will develop a solid understanding of adding and subtracting numbers in standard form, setting the stage for success in more advanced mathematical concepts.

Adding and subtracting in standard form (2024)
Top Articles
Latest Posts
Article information

Author: Errol Quitzon

Last Updated:

Views: 6423

Rating: 4.9 / 5 (79 voted)

Reviews: 86% of readers found this page helpful

Author information

Name: Errol Quitzon

Birthday: 1993-04-02

Address: 70604 Haley Lane, Port Weldonside, TN 99233-0942

Phone: +9665282866296

Job: Product Retail Agent

Hobby: Computer programming, Horseback riding, Hooping, Dance, Ice skating, Backpacking, Rafting

Introduction: My name is Errol Quitzon, I am a fair, cute, fancy, clean, attractive, sparkling, kind person who loves writing and wants to share my knowledge and understanding with you.