Adding and subtracting fractions
If you were to add \(\frac{1}{2}\) and \(\frac{1}{3}\), it is hard to picture what the answer would be.
Rewriting the fractions with a common bottom number, or denominator (in this case, \({6}\)), makes it easier.
Remember, you can only add and subtract fractions when the bottom numbers, or denominators, are the same.
So, to add or subtract fractions:
- Change the fractions so they have the same denominator.
- Add or subtract the top numbers, or numerators.
Example
\(\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\)
\(\frac{7}{10} - \frac{2}{5} = \frac{7}{10} - \frac{4}{10} = \frac{3}{10}\)
Question
What is \(\frac{1}{4} + \frac{1}{3} = \)?
Answer
\(\frac{1}{4} + \frac{1}{3}= \frac{3}{12} + \frac{4}{12} = \frac{7}{12}\)
From the diagram it is clear that the denominator remains the same, as the circles have been split into the same number of parts.
Mixed numbers
To add or subtract mixed numbers, it is usually easiest to change them to improper fractions first and then change the answer back into a mixed number (if needed).
Question
\(3 \frac{1}{3} + 4 \frac{1}{2} = \)?
Answer
\(3 \frac{1}{3} + 4 \frac{1}{2} = \frac{10}{3} + \frac{9}{2} = \frac{20}{6} + \frac{27}{6} = \frac{47}{6} = 7 \frac{5}{6}\)
Alternatively, you can add the whole numbers and then the fractions:
\(3 \frac{1}{3} + 4 \frac{1}{2} = 3 + 4 + \frac{1}{3} + \frac{1}{2} = 7 + \frac{2}{6} + \frac{3}{6} = 7 \frac{5}{6}\)
Find out how to add fractions with this short video
Have a go
Image caption, Click to see a step-by-step slideshow.
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Find out how to subtract fractions
Have a go
Image caption, Click here to see a step-by-step slideshow.
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Multiplying and dividing fractions
Multiplying fractions
\(\frac{1}{2}\) of \(\frac{1}{2} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)
\(\frac{2}{3}\) of \(\frac{4}{5} = \frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\)
Multiply the numerators to find the new numerator, multiply the denominators to find the new denominator, then simplify where necessary.
Question
Calculate \(\frac{3}{4} \times \frac{2}{5}\)
Answer
\(\frac{3}{4} \times \frac{2}{5} = \frac{3 \times 2}{4 \times 5} = \frac{6}{20}\)
Now put the the answer into its simplest form:
\(\frac{6}{20}= \frac{3}{10}\)
Dividing fractions
When you divide \({10}\) by \({2}\), you are working out how many \({2}\)s there are in \({10}\).
\(10 \div 2 = 5\), so there are five \({2}\)s in \({10}\).
In a similar way, when dividing \({2}\) by \(\frac{1}{2}\), you are working out how many \(\frac{1}{2}\)s there are in \({2}\).
There are four \(\frac{1}{2}\)s in \({2}\), so:
\(2 \div \frac{1}{2} = 4\)
If you divide \(1 \frac{1}{2}\) by \(\frac{1}{4}\) you are working out how many \(\frac{1}{4}\)s there are in \(1 \frac{1}{2}\).
There are six \(\frac{1}{4}\)s in \(1 \frac{1}{2}\), so:
\(1\frac{1}{2} \div \frac{1}{4} = 6\)
Do you see a pattern?
Let's write out those calculations a different way.
\(2 \div \frac{1}{2} = 4\) and \(2 \times 2 = 4\), so \(2 \div \frac{1}{2}\) is the same as \(2 \times 2\)
\(1\frac{1}{2} \div \frac{1}{4} = \frac{3}{2} \div \frac{1}{4} = 6\), so \(\frac{3}{2} \div \frac{1}{4}\) is the same as \(\frac{3}{2} \times 4 = \frac{12}{2} = 6\)
So, '\(\div\frac{1}{2}\)' is the same as 鈥榎(\times 2\)鈥.
And '\(\div\frac{1}{4}\)' is the same as 鈥榎(\times 4\)鈥.
To divide fractions, turn the second fraction upside down, then multiply.
Question
Calculate \(\frac{3}{4} \div \frac{4}{5}\)
Answer
\(\frac{3}{4} \div \frac{4}{5}= \frac{3}{4} \times \frac{5}{4} = \frac{15}{16}\)
Have a go
Image caption, Click to see a step-by-step slideshow.
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Find out how to divide fractions using a bar model
Have a go
Image caption, Click to see a step-by-step slideshow.
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How do you find a fraction of a quantity?
To find a fraction of a quantity, divide by the number on the bottom of the fraction (denominator) and multiply by the number on the top (numerator).
Example
A teacher has a box containing 54 pencils and gives out \(\frac{2}{3}\) of them to his class.
How many pencils has he given out?
How many pencils are left in the box?
Divide by the (denominator) bottom number (3) to find one third.
\(54 梅 3 = 18\)
Now multiply by the (numerator) top number (2) to find two thirds.
\({18} \times {2} = 36\)
The teacher gives out 36 pencils.
How many pencils are left in the box?
\(54 鈥 36 = 18\) pencils are left.
Question
A school has 950 pupils and \(\frac{1}{5}\) of them walk to school every day.
How many pupils walk to school?
Answer
Divide by the bottom number (denominator).
\(950 梅 5 = 190\)
Now multiply by the top number (numerator).
\({190}\times{1} = {190}\) (no change after multiplying by 1)
190 pupils walk to school
For a unitary fraction (with 1 as the top number), just divide by the bottom number.
Question
Sophie has 拢64 birthday money.
She spends \(\frac{3}{8}\) of this money and saves the rest.
How much does she save?
Answer
Sophie spends \(\frac{3}{8}\) of her money so she has \(\frac{5}{8}\) left.
\(\frac{5}{8}\) of 拢64 \(= {~64} 梅 {8}\times{5} = {40}\)
Sophie has 拢40 left to save.
OR
Sophie spends \(\frac{3}{8}\) of her money.
\(\frac{3}{8}\) of 拢64 \(= {~64}\div{8}\times{3} = {24}\)
\(拢64 - 拢24 = 拢40\)
Sophie has 拢40 left to save.
A one-minute video on how to work out a fraction of an amount using bar models.
How to work out a fraction of an amount
Image caption, WHAT YOU NEED: Whiteboard, pen, ruler and a calculator.
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Converting fractions: The great pizza party
Follow one teenager's attempt to share pizza and bills equally by using fractions and percentages. Learn how to convert fractions into decimals and percentages.
Billy Oh no. I鈥檓 starving.
(DOOR BELL)
Hey guys, the pizza鈥檚 here!
Yeah, coming, coming鈥
Hi there, is it all paid for?
Delivery Man Yeah, It鈥檚 already been paid online.
Billy OK. Where鈥檚 the third one?
Delivery Man There was only two in the order.
Billy Oh no. Thank you
Oh no鈥 how on earth am I going to split this into nine?
One thing they always go on about at school is pizzas, fractions and percentages.
If I can鈥檛 work this out鈥
Friends Pizza, Pizza, Pizza!
Billy Coming! I鈥檓 just cutting it up now.
Right, you can do this.
I have to share two pizzas between nine people.
That鈥檚 two divided by nine. I think I鈥檒l get my calculator for this.
Now, let鈥檚 take this nice and slowly.
Well to convert a fraction to a decimal I need to divide the numerator by the denominator.
So the fraction is two ninths. What鈥檚 two divided by nine?
Two divided by nine equals naught point two, two, two, two, two.
That鈥檚 a recurring decimal.
Now to convert a decimal to a percentage you multiply the decimal by 100.
So x 100 equals 22.222222222%
But how does that help me cut up pizzas?
Hang on, if I just stick to fractions.
I need two ninths of two pizzas.
That鈥檚 one ninth of each pizza.
Hey that鈥檚 so cool.
Nine. That鈥檚 easy. That鈥檚 divisible by three.
Cut them into thirds.
Then cut each third into three, that鈥檚 great.
Here we go guys - it鈥檚 two pieces each!
Wow that鈥檚 a disaster.
Having shown off about how good I was at fractions,
they鈥檝e now all challenged me to work out what percentage of the bill they all have to pay.
So without Kevin the birthday boy, there鈥檚 eight of us.
That鈥檚 an 8th each and to get the decimal divide the numerator by the denominator.
That鈥檚 one divided by eight.
To go to a percentage you multiply by 100.
Let鈥檚 just work it out.
I鈥檓 so good at this.
Oh no!
Mum, I need your lipstick.
So what鈥檚 the simplest way of doing this.
So from an eighth as a fraction to a percentage is one divided by eight, times 100, the same as 100 divided by eight.
So, eight into 100. Eight goes into 10 once, with two left over so I carry that to the units.
Eight goes into 20 twice, and there鈥檚 four left over and I need to carry that to the first decimal place.
Eight into 40 is exactly five.
So I can see that an 1/8th is the same as 12.5%.
What鈥檚 12.5% of the bill which was 拢28.
I can use my phone, why didn鈥檛 I think of that.
Now I don鈥檛 know what this percentage button does but I know that if I鈥檓 changing a percentage to a fraction or a decimal then I need to divide by 100.
So 12.5% is the same as 12.5 divided by 100, which is 0.125.
So that鈥檚 0.125 multiplied by 28.
That鈥檚 拢3.50.
So eight of us pays 拢3.50 each.
Now, where am I going to find some change!
Test section
Question 1
A laptop usually costs \(拢420\) but is reduced by \(\frac{1}{3}\) in a sale.
What is the sale price?
Answer
The correct answer is \(拢280\).
Question 2
For a science experiment, a 1 litre cylinder is \(\frac{7}{10}\) full of solution.
How many millilitres of solution is in the cylinder?
Answer
The correct answer is \(700 ml\).
Question 3
A year group of \(120\) pupils voted for one of \(3\) options for a school trip.
\(\frac{5}{12}\) voted for a trip to Portrush and \(\frac{1}{6}\)voted to go to the cinema.
The rest voted to go to W5.
How many pupils voted to go to Portrush?
Answer
The correct answer is \(50\) pupils went to Portrush.
Question 4
Using the information from Question 3, how many pupils voted to go to W5?
Answer
The correct answer is \(50\) pupils went to W5.
Question 5
\(\frac{3}{4}+\frac{1}{8}=\)?
Answer
You have to convert the first fraction to \(\frac{6}{8}\) before adding the two numerators to get \(\frac{7}{8}\).
Question 6
\(\frac{3}{5}-\frac{1}{9}=\)?
Answer
Fractions should have the same denominator before you can subtract.
So \(\frac{27}{45}-\frac{5}{45}=\frac{22}{45}\).
Question 7
Write the answer to the following in its simplest form: \(\frac{5}{6}\times\frac{2}{3}=\)
Answer
You have to multiply the two numerators and both denominators, then simplify: \({5}\times{2}={10}\), \({6}\times{3}={18}\), \(\frac{10}{18}=\frac{5}{9}\).
Question 8
Write the answer to the following as a mixed number: \(\frac{7}{10}\div\frac{1}{4}=\)
Answer
Remember to multiply \(\frac{7}{10}\) by \(\frac{4}{1}\) to get \(\frac{28}{10}\).
Then, you need to convert the improper fraction into a mixed number: \(\frac{28}{10}=2\frac{8}{10}=2\frac{4}{5}\).
More on Fractions
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