Adding fractions
Add fractions with the same bottom number by adding the tops — and when the bottoms differ, make them match first with equivalent fractions.
You eat one quarter of a pizza, then another quarter. How much pizza did you eat? Two
quarters — 2/4, which is half. The slices were the same size, so you just counted
them. That is the whole secret of adding fractions: the parts must be the same size.
After this lesson you can add and subtract fractions that have the same denominator, explain why a shared denominator is required, find a common denominator for fractions that differ, and add them by rewriting each one first.
When the denominators match, add the numerators and keep the denominator. 1/4
and 2/4 are both made of quarter-sized parts. One quarter plus two quarters is three
quarters: 1/4 + 2/4 = 3/4. You are counting parts of the same size, so the
denominator — the size of one part — does not change. Only the count changes.
Subtraction with matching denominators works the same way. Subtract the numerators
and keep the denominator: 3/8 − 1/8 = 2/8. Three eighth-sized parts, take away one,
leaves two. Again the denominator stays put because the parts are all the same size.
When the denominators differ, you cannot add yet. 1/2 + 1/3 mixes half-sized
parts with third-sized parts — different sizes, so you cannot just count them
together. When you see two fractions with different bottoms, your first move is always
the same: find a number that both denominators divide into. First you must rewrite both
fractions so they share one denominator. A denominator that both can be rewritten to is
called a common denominator.
Find a common denominator, rewrite both fractions, then add. A number that both
denominators divide into works. For 1/2 and 1/3, the number 6 works: 2 and 3 both
divide into 6. Rewrite each fraction with 6 on the bottom using equivalent fractions —
then the denominators match and you add the numerators as in Step 1. Simplify the
result at the end if it can be simplified.
Add 1/2 + 1/3.
The denominators 2 and 3 differ, so find a common denominator: 6 works, because 2 and 3 both divide into 6.
Rewrite 1/2 with denominator 6: multiply top and bottom by 3 — 1/2 = 3/6.
Rewrite 1/3 with denominator 6: multiply top and bottom by 2 — 1/3 = 2/6.
Now the denominators match. Add the numerators: 3/6 + 2/6 = 5/6. Since 5 and 6 share
no divider above 1, 5/6 is already in simplest form. So 1/2 + 1/3 = 5/6.
▸Why this works
Why must the denominators match before you add? Because a denominator is the size of
one part. Adding 1/2 + 1/3 directly would be like adding one large slice and one
small slice and calling the answer “2 slices” — but slices of what size? The common
denominator recuts both fractions into parts of one size, and only then can you count
them together honestly.
▸Common mistake
The most common mistake is adding the denominators too: writing 1/2 + 1/3 = 2/5. The
denominator is the size of a part, not a count — it must not be added. Once the
denominators match, add only the numerators and leave the shared denominator alone.
Add 1/5 + 2/5. The sum has denominator 5 — type its numerator.
Subtract 3/8 − 1/8. The result has denominator 8 — type its numerator.
To add 1/2 + 1/4, type a common denominator both 2 and 4 divide into.
1/2 + 1/4 rewritten over 4 is ?/4. Type the numerator of the sum.
To add 1/3 + 1/6, type the smallest common denominator.
Why can't you add 1/2 + 1/3 by just adding tops and bottoms to get 2/5?
To add or subtract fractions with the same denominator, add or subtract the numerators and keep the denominator — you are counting equal-sized parts. When the denominators differ, you cannot add yet: find a common denominator, rewrite both fractions to it with equivalent fractions, then add the numerators. Never add the denominators — a denominator is the size of a part, not a count. Now when you see two fractions to add, check the denominators first: if they match, go straight to the numerators; if they differ, find a common denominator before touching anything else.
Practice
Start at the top. Tasks go easiest → hardest: recall a fact, apply it to a case, then a senior-level stretch. Open one, attempt it, then reveal.
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