Mathematics from zero
Adding 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. 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.