Adding fractions — same and different bottoms
Maths · Key Stage 2 · The School
Same bottoms: just add the tops
2/7 + 3/7 = 5/7 — sevenths plus sevenths make sevenths, the same way 2 apples + 3 apples make 5 apples. The denominator names the unit; the numerators count it. NEVER add the bottoms. Here is why that rule holds, rather than just being a rule. Add the bottoms too and 2/7 + 3/7 would come to 5/14. But 5/14 is smaller than 3/7 on its own, so adding something positive would have made the total go down. Any method that can do that is broken, and one line of checking shows it.
Different bottoms: rename first
1/2 + 1/3 cannot add as written — halves and thirds are different units. Rename both in sixths (the smallest cut both fit): 1/2 = 3/6 and 1/3 = 2/6. Now 3/6 + 2/6 = 5/6. Finding that shared bottom: if one bottom divides the other, use the bigger one — for 1/4 + 3/8, eighths already fit both. If neither divides the other, multiplying them always works: for 1/3 + 1/5, fifteenths. That is not always the smallest choice, but it is never wrong, and a bigger bottom only means simplifying at the end.
Simplify at the end
2/4 + 1/4 = 3/4 — already simplest. But 1/6 + 1/6 + 1/6 = 3/6, and 3/6 simplifies to 1/2 (divide top and bottom by 3). An answer in lowest terms shows you finished the thought. How do you know you are there? When no whole number above 1 divides both top and bottom. Test the small primes in turn: are both even? Do both digit-sums make 3? Do both end in 0 or 5? For 18/24 the answer to the first is yes, giving 9/12, then 3 divides both, giving 3/4, and now nothing does.
When neither bottom fits into the other
1/2 + 1/3 was gentle because 6 swallows both. Try 3/4 + 5/6. Four does not go into six and six does not go into four, so hunt for a number both divide into: 12. Rename each — 3/4 = 9/12 (times top and bottom by 3) and 5/6 = 10/12 (by 2) — and add: 9/12 + 10/12 = 19/12. A top bigger than the bottom is allowed; it just means the answer passed 1, and 19/12 = 1 7/12. Two checks worth doing every time. Estimate BEFORE you start: 3/4 is nearly 1 and 5/6 is nearly 1, so expect something near 2 — and 1 7/12 is. And if your answer came out smaller than the bigger fraction you started with, you have slipped somewhere.