Adding Fractions With Different Denominators
You cannot add halves to thirds any more than you can add centimetres to inches. First make the pieces the same size, then it is ordinary counting.
Why you cannot just add them
Here are a half and a third, drawn on identical bars.
Adding 12 and 13 straight across gives 25, which is less than the half you started with. That answer is not slightly off, it is impossible.
Recutting both bars
The fix is to recut both bars so every piece is the same size. Sixths work here, because 2 and 3 both divide into 6.
The half became 36 and the third became 26. Now the pieces match, so you can count them.
The method
Four steps, always the same
1. Find a number both denominators divide into. Multiplying them together always works.
2. Rewrite each fraction with that denominator, using the equivalent fraction rule.
3. Add the numerators. Keep the common denominator.
4. Simplify, and convert to a mixed number if the question asks.
Step 1 has a shortcut worth knowing. If one denominator divides into the other, use the larger one and you only have to rewrite a single fraction. For 14 + 38, use 8, not 32.
Worked examples
When one denominator divides into the other
- Check first: does 4 divide into 8? Yes, so use 8 and leave the second fraction alone.
- Rewrite 14 with a denominator of 8. Multiply top and bottom by 2, giving 28.
- Now add the tops. 2 + 3 = 5, so the answer is 58.
- Can it be simplified? 5 and 8 share no factor, so no.
When neither divides into the other
- 3 does not divide into 4, so multiply them: 3 × 4 = 12. Use twelfths.
- 23: multiply top and bottom by 4, giving 812.
- 14: multiply top and bottom by 3, giving 312.
- Add the tops. 8 + 3 = 11, giving 1112, which will not simplify.
An answer over 1
- 2 divides into 4, so use quarters.
- 12 becomes 24.
- Add the tops. 3 + 2 = 5, giving 54.
- The top is bigger than the bottom, so this is more than one whole. As a mixed number, 5 ÷ 4 = 1 remainder 1, which is 1 and a quarter.
A two-second check
Before writing the answer down, ask one question: is the answer bigger than the larger fraction I started with?
It has to be, because you added something to it. If a child answers 12 + 13 = 25, this check catches it instantly: two fifths is less than a half.
Second check for the same price: the answer should also be less than the two fractions doubled. Between those two bounds, and you have almost certainly done it right.
Common mistakes
1. Adding straight across
12 + 13 = 25. This is the big one. The pieces are different sizes, so they cannot be counted together. Recut first, always.
2. Changing the bottom but forgetting the top
Turning 23 into 212 instead of 812. Whatever you multiply the bottom by, the top gets the same treatment. This is the equivalent fraction rule and it does not bend for addition.
3. Adding the denominators too, at the end
Getting as far as 36 + 26 and then writing 512. All the hard work was done correctly and the mark is lost on the last line. Once the denominators match, the denominator is finished changing.
Practice questions
Seven questions. One attempt each, and the explanation appears once the answer is in.
Check yourself
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For parents
Most children can carry out the four steps. What they skip is asking whether the answer is sensible. Get them to say the estimate out loud before the working: "a half plus a third is a bit more than a half, so the answer is somewhere near 0.8". A child who estimates first almost never hands in an impossible answer.
What to learn next
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