Free maths tool · Level: Ages 11 to 13
Adding fractions
Two bars, cut up differently. While the pieces are different sizes you cannot count them together, and that is the whole problem.
To add fractions with different denominators you first give them a common one: find the lowest common multiple of the denominators and rewrite both fractions with it. Then add only the numerators and keep the denominator. For 1/2 + 1/3 that gives 3/6 + 2/6 = 5/6. Finally simplify if you can.
Why you cannot just add them
Look at the two bars. One is cut into halves, the other into thirds, and those pieces are not the same size. Adding pieces of different sizes does not work, any more than 3 metres and 3 centimetres add up to 6.
Switch the common denominator layer on and both bars are recut into pieces of equal size. From then on you are simply counting pieces, and the denominator stays put because the size of a piece has not changed.
The lowest common multiple is enough
You can always multiply the denominators together, but that gives bigger numbers than you need. For 1/4 + 1/6, 24 works while 12 already does. With the lowest common multiple the numbers stay small and there is less to simplify afterwards.
If one denominator already divides into the other you are there straight away: for 1/3 + 1/6, 6 is fine and only the first fraction has to be rewritten.
The mistake everyone makes once
Switch the last layer on and you see the answer you get by adding tops and bottoms. For 1/2 + 1/3 that gives 2/5, and the bar shows it is even smaller than 1/2 on its own. A sum that comes out smaller than one of its parts cannot be right.
That quick check is worth doing on every exercise. Add two positive fractions and the answer has to be bigger than both. If it is not, something has gone wrong.
Are fractions still hard going?
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Frequently asked questions
- How do you add fractions with different denominators?
- Give them a common denominator first. Find the lowest common multiple of the two, rewrite both fractions with it, then add the numerators and keep the denominator. Simplify the answer if you can.
- Why can you not add the denominators?
- Because the denominator says how big a piece is, not how many you have. Adding is about the number of pieces, so it is about the numerators. Putting pieces together does not change their size.
- What is the lowest common multiple of two denominators?
- The smallest number both denominators divide into. For 4 and 6 that is 12. You may use 24, which also works, but then there is more simplifying to do afterwards.
- How do you simplify a fraction?
- Divide the numerator and denominator by their greatest common divisor. For 6/8 that is 2, giving 3/4. A fraction is done once the two share no divisor other than 1.
- Does this work for subtraction too?
- Yes, exactly the same. Give them a common denominator, subtract the numerators and keep the denominator. Only multiplying skips this: there you multiply top by top and bottom by bottom.