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Multiplying And Dividing Decimals

Multiplying And Dividing Decimals

Here we will learn about multiplying and dividing with decimals, including calculations involving integers and decimals, or just decimals using formal written methods.

There are also multiplying and dividing decimals 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 multiplying and dividing decimals?

Multiplying and dividing decimals is the skill of carrying out a calculation involving multiplication and division where one or more of the values is a decimal.

For this skill the ability to multiply and divide by powers of ten is essential.

What is multiplying and dividing decimals?

What is multiplying and dividing decimals?

Multiplying with decimals

When multiplying with decimals we change any decimals in the calculation into integers (whole numbers) by multiplying by powers of katex is not defined This gives us a scaled calculation which is easier to carry out. Remember when multiplying by katex is not defined the decimal point doesn’t move, rather the digits move up changing their place value.

For example,

katex is not defined

katex is not defined

katex is not defined

We now need to scale this answer down to find the solution to the original calculation. We do this by dividing by the same powers of katex is not defined which we multiplied by in the first step. Remember when dividing by katex is not defined the decimal point doesn’t move, rather the digits move down changing their place value.

katex is not defined

Another method of decimal multiplication, or as a way of checking your answer, is to count the number of decimal places. The combined number of decimal places in the question should be the same as the number of decimal places in the answer.

katex is not defined

Dividing with decimals

  • When dividing a decimal number by an integer we can use the standard short division method.

For example, katex is not defined

Multiplying And Dividing Decimals Image 1

Sometimes trailing zeros may need to be written when the solution has more decimal places than the decimal number being divided.

For example,

katex is not defined

Multiplying And Dividing Decimals Image 2

  • When dividing a number by a decimal we can use equivalent fractions to find an equivalent calculation which divides by an integer.

For example, katex is not defined

First we write this division as a fraction katex is not defined

Then we decide what to multiply katex is not defined by in order to make it an integer. Most often we will multiply by a power of ten. We then write the equivalent fraction which has a denominator that is an integer.

Step-by-step guide: Equivalent fractions

Multiplying And Dividing Decimals Image 3

This tells us that katex is not defined is equivalent to katex is not defined and therefore the answers to each of these calculations will be the same.

katex is not defined

Therefore katex is not defined

Sometimes a combination of equivalent fractions and short division is needed.

For example,

katex is not defined

Multiplying And Dividing Decimals Image 4

Multiplying and dividing decimals is used in many real life contexts.

For example,

  • Money – Share katex is not defined equally between two people. katex is not defined
  • Speed / Distance / Time – It takes a vehicle katex is not defined hours to travel katex is not defined miles.
    What is the average speed of the vehicle in miles per hour? katex is not defined
  • Mass – katex is not defined slices of shortbread is made using katex is not defined of brown sugar. How much brown sugar is needed for katex is not defined slices? katex is not defined
  • Area – A rectangular field measures katex is not defined by katex is not defined What is the area of the field? katex is not defined
  • Converting units – Convert katex is not defined millilitres into litres. katex is not defined

How to multiply with decimals

In order to multiply with decimals:

  1. Change any decimals in the calculation into integers (whole numbers) by multiplying by powers of katex is not defined.
  2. Calculate the answer to this scaled calculation.
  3. Divide the answer to step 2 by the same powers of ten which you multiplied by in step 1.
  4. Check your answer by ensuring that the combined number of decimal places in the question is the same as the number of decimal places in the answer.

How to multiply with decimals

How to multiply with decimals

Multiplying and dividing decimals worksheet

Multiplying and dividing decimals worksheet

Multiplying and dividing decimals worksheet

Get your free multiplying and dividing decimals worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Multiplying and dividing decimals worksheet

Multiplying and dividing decimals worksheet

Multiplying and dividing decimals worksheet

Get your free multiplying and dividing decimals worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Related lessons on decimals

Multiplying and dividing decimals is part of our series of lessons to support revision on decimals. You may find it helpful to start with the main decimals 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:

Multiplying decimals examples

Example 1: multiplying an integer by a decimal

Calculate katex is not defined

  1. Change any decimals in the calculation into integers (whole numbers) by multiplying by powers of katex is not defined

Multiply katex is not defined by katex is not defined to make the integer katex is not defined

katex is not defined

katex is not defined

katex is not defined (This is the scaled calculation.)

2Calculate the answer to this scaled calculation.

katex is not defined

3Divide the answer to step 2 by the same powers of ten which you multiplied by in step 1.

In step katex is not defined we multiplied by katex is not defined so now we divide by katex is not defined

katex is not defined

4Check your answer by ensuring that the combined number of decimal places in the question is the same as the number of decimal places in the answer.

katex is not defined

Example 2: multiplying two decimals

Calculate katex is not defined

Change any decimals in the calculation into integers (whole numbers) by multiplying by powers of katex is not defined

Show step

Calculate the answer to this scaled calculation.

Show step

Divide the answer to step 2 by the same powers of ten which you multiplied by in step 1.

Show step

Check your answer by ensuring that the combined number of decimal places in the question is the same as the number of decimal places in the answer.

Show step

Example 3: multiplying with decimals when long multiplication is needed

Calculate katex is not defined

Change any decimals in the calculation into integers (whole numbers) by multiplying by powers of katex is not defined

Show step

Calculate the answer to this scaled calculation.

Show step

Divide the answer to step 2 by the same powers of ten which you multiplied by in step 1.

Show step

Check your answer by ensuring that the combined number of decimal places in the question is the same as the number of decimal places in the answer.

Show step

How to divide with decimals

In order to divide with decimals:

  1. Check if the divisor is an integer or a decimal.
    (a) Integer – go to step 3.
    (b) Decimal – write the calculation as a fraction.
  2. Write an equivalent fraction which has a denominator that is an integer.
    Note: The best option is often to multiply the numerator and denominator by a power of ten. Then check if the fraction simplifies before moving onto step 3.
  3. Use short division to calculate the answer.

How to divide with decimals

How to divide with decimals

Dividing decimals examples

Example 4: dividing a decimal by an integer

Calculate katex is not defined

Check if the divisor is an integer or a decimal.
(a) Integer – go to step 3.
(b) Decimal – write the calculation as a fraction.

Show step

Use short division to calculate the answer.

Show step

Example 5: dividing a decimal by an integer

Calculate katex is not defined

Check if the divisor is an integer or a decimal.
(a) Integer – go to step 3.
(b) Decimal – write the calculation as a fraction.

Show step

Use short division to calculate the answer.

Show step

Example 6: dividing a decimal by a decimal

Calculate katex is not defined

Check if the divisor is an integer or a decimal.
(a) Integer – go to step 3.
(b) Decimal – write the calculation as a fraction.

Show step

Write an equivalent fraction which has a denominator that is an integer.

Show step

Use short division to calculate the answer.

Show step

Example 7: dividing a decimal by a decimal

Calculate katex is not defined

Check if the divisor is an integer or a decimal.
(a) Integer – go to step 3.
(b) Decimal – write the calculation as a fraction.

Show step

Write an equivalent fraction which has a denominator that is an integer.

Show step

Use short division to calculate the answer.

Show step

Common misconceptions

  • Getting mixed up between the ‘scaled calculations’ method when multiplying with decimals and ‘equivalent fractions’ method when dividing with decimals
    When we multiply with decimals we can use a scaled calculation. This means the answer we get must be scaled back at the end to find the final answer. When we use equivalent fractions to help us divide with decimals, the fractions we write are equivalent (they have exactly the same value) and do not need adapting at the end.

    Students can sometimes get these methods mixed up and forget to scale back when multiplying decimals, or attempt to adapt an answer that was already correct when dividing with decimals.

    Remember: Multiplying with decimals uses scaled calculations and therefore the answer to this needs scaling back to find the final answer.
    Dividing by a decimal uses equivalent fractions which are totally equal calculations and therefore the answer does not need changing.

  • Not accounting for zero digits when multiplying with decimals
    When multiplying with decimals we can check our answer by ensuring the combined number of decimal places in the question is equal to the number of decimal places in the answer. However, when zero digits are involved we have to be careful using this rule and remember to account for them.

    katex is not defined
    Here we can see the rule works.

    katex is not defined
    Here it might appear that the rule doesn’t work.

    However, because katex is not defined the original answer would be katex is not defined and then the trailing zero has been removed.

    Watch out for integers that multiply to give a number that ends in zero when using the counting digits rule for checking your answer.

  • Putting the numbers the wrong way round when using short division
    Let’s describe a short division set up as a ‘bus stop’. One number goes inside the bus stop and the other goes outside the bus stop. The answer to the short division goes on top of the bus stop. Students can sometimes get the numbers the wrong way round, particularly when dividing with decimals.

    Remember that the divisor (the number you are dividing by) goes on the outside of the bus stop, and the number you are dividing goes on the inside of the bus stop.
    For example, katex is not defined should be written like this,

    Multiplying And Dividing Decimals Image 5
    Carrying out the division, we have

    Multiplying And Dividing Decimals Image 6

    So katex is not defined

    When writing a fraction as a short division the denominator goes on the outside of the bus stop and the numerator goes on the inside of the bus stop.

    katex is not defined

    Multiplying And Dividing Decimals Image 7
    Carrying out the division, we have

    Multiplying And Dividing Decimals Image 8

Practice multiplying and dividing decimals questions

1) Calculate katex is not defined

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined

 

katex is not defined

 

katex is not defined

 

katex is not defined

2) Multiply together katex is not defined and katex is not defined

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

Multiply katex is not defined by katex is not defined to get katex is not defined

 

Multiply katex is not defined by katex is not defined to get katex is not defined

 

katex is not defined

 

katex is not defined

 

Dividing katex is not defined by katex is not defined we have

 

katex is not defined

 

This means that katex is not defined

3) Calculate katex is not defined

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined

 

katex is not defined

 

katex is not defined

 

katex is not defined

4) Divide katex is not defined by katex is not defined

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

Multiplying And Dividing Decimals practice question 4

5) Calculate the value of katex is not defined

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

Write the division as a fraction katex is not defined

 

Write an equivalent fraction which has a denominator that is an integer.

 

Multiplying And Dividing Decimals practice question 5 image 1

 

Now using short division,

 

Multiplying And Dividing Decimals practice question 5 image 2

 

Answer is katex is not defined

6) Calculate the value of katex is not defined

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

Write the division as a fraction katex is not defined

 

Write an equivalent fraction which has a denominator that is an integer.

 

Multiplying And Dividing Decimals practice question 6 image 1

 

We could now simplify the fraction.

 

Multiplying And Dividing Decimals practice question 6 image 2

 

Lastly using short division,

 

Multiplying And Dividing Decimals practice question 6 image 3

 

Answer is katex is not defined

Multiplying and dividing decimals GCSE questions

1. Calculate

(a) katex is not defined

 
(b) katex is not defined

(3 marks)

Show answer

(a) katex is not defined

(1)

 

(b) katex is not defined

(1)

katex is not defined

(1)

2. The piece of ribbon is katex is not defined metres long. The ribbon is going to be cut into katex is not defined pieces of equal length. How long will each piece of ribbon be?

(2 marks)

Show answer

katex is not defined

(1)

katex is not defined

(1)

3. A sandpit is katex is not defined metres long and katex is not defined metres wide. Calculate the area of the sandpit giving your answer in metres squared.

(2 marks)

Show answer

katex is not defined

(1)

katex is not defined

(1)

4. Two positive numbers less than katex is not defined are multiplied together to give katex is not defined Give one possible pair of values.

 

Multiplying And Dividing Decimals GCSE question 4

(1 mark)

Show answer

Either of the following pairs of numbers (in any order),
 

katex is not defined

(1)

katex is not defined

(1)

Learning checklist

You have now learned how to:

  • Carry out multiplication and division in calculations involving decimals using written methods to show your working out

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