Match the piece sizes first — then just add or subtract the parts.
Adding whole numbers is easy because the things you count are the same: 3 apples and 2 apples are 5 apples. Fractions are no different — once the pieces are the same size. Three eighths plus two eighths is five eighths, because every piece is one eighth and you are simply counting how many you have. The whole job of adding and subtracting fractions is making sure the pieces match before you count: if the bottoms already agree, add or subtract the tops and you are done; if they disagree, recut both fractions into a shared size first. Carry when a fraction sum spills past a whole, borrow when you cannot take enough away, and tidy the answer at the end. That is the entire lesson.
When two fractions have the same denominator, they are built from the very same unit piece. Two sixths and three sixths are both made of sixth-sized pieces, so putting them together is plain counting: two pieces and three more pieces make five pieces — five sixths. The denominator never changes when you join same-size parts; it only records how big each piece is. So the rule is short:
Same bottom → keep the bottom, add (or subtract) the tops. The denominator counts the size of the pieces, and joining pieces of one size never changes that size.
Subtraction is the same picture run backward. From five sixths take away one sixth and four sixths are left — and since 4 and 6 share a factor of 2, that tidies to 23. (Simplifying at the end is the habit we build in §3.3.4.)
What about one half plus one third? These pieces are different sizes, so you cannot just add the tops — a half and a third are not the same thing, the way an apple and an orange are not the same thing. Try it on the picture and you will see the two shaded strips don't line up. The fix is the tool from §3.2: give both fractions a common denominator by recutting each into smaller, equal pieces.
The smallest size that both halves and thirds can be recut into is sixths — the least common multiple of 2 and 3. Recut: one half becomes 36 and one third becomes 26. Now the pieces match, so we are back to §3.3.1:
12 + 13 = 36 + 26 = 56. The bottoms are made equal first; only then do we add the tops 3 + 2 = 5.
Never add the denominators. 12 + 13 is not 25. The picture proves it: two fifths is less than a half, yet one half plus anything must be more than a half. The bottom names the piece size — it does not get added.
A mixed number is a whole plus a fraction, such as 134. To add two of them, the safest move is to turn each into an improper fraction (whole × bottom + top, all over the bottom), add as in §3.3.2, then turn the answer back into a mixed number. Watch a sum that spills over a whole:
134 + 212 = 74 + 52 = 74 + 104 = 174 = 414. The fraction parts added to more than one whole, so a whole was carried.
Subtraction sometimes needs the reverse move. To do 314 − 134, the top fraction one fourth is too small to take three fourths from. So borrow a whole: rewrite 314 as improper 134, and 134 as 74; now 13 − 7 = 6, giving 64 = 112.
An answer is not fully said until it is in lowest terms — and an improper fraction is usually turned back into a mixed number. These two finishing moves cost nothing and make the answer readable. Three eighths plus three eighths is six eighths, but 6 and 8 share a factor of 2, so the tidy answer is 34. And one sixth more than a whole, written 76, reads better as 116.
Finish every answer: reduce to lowest terms (divide top and bottom by their greatest common factor) and turn an improper fraction into a mixed number. Same amount — just clearly written.
| Situation | What to do | Example |
|---|---|---|
| Same denominator | Keep the bottom; add or subtract the tops | 38 + 28 = 58 |
| Different denominators | Common denominator first, then add the tops | 12 + 13 = 56 |
| Mixed numbers | Make improper, add/subtract, carry or borrow | 134 + 212 = 414 |
| The trap | Never add the denominators — the bottom is a piece size, not a count | |
| Finish | Reduce to lowest terms; turn an improper answer back into a mixed number | |
Same bottom, so keep the 9 and add the tops: 4 + 3 = 7, giving 79. Since 7 and 9 share no factor, it is already in lowest terms.
Keep the 6, subtract the tops: 5 − 1 = 4, giving 46. Both 4 and 6 divide by 2, so it tidies to 23.
Different bottoms, so find a common denominator. The least common multiple of 4 and 6 is 12, so recut: 14 = 312 and 16 = 212. Now add the tops: 3 + 2 = 5, giving 512 (already lowest terms).
Common denominator of 3 and 6 is 6, so 23 = 46. Then 46 + 16 = 56 cup of flour.
Make each improper: 134 = 74, 212 = 52 = 104. Add: 7 + 10 = 17, so 174 = 414 — the fractions carried a whole.
One fourth is too small to take three fourths from, so borrow: write both as improper, 134 − 74 = 64. Simplify: 6 and 4 divide by 2, giving 32 = 112.
Six questions to lock it in. Tap the answer you think is right.
This lesson develops 4.NF.B.3.a and 4.NF.B.3.c — adding and subtracting fractions with like denominators by joining and separating parts of the same whole, and doing the same with mixed numbers (including carrying and borrowing). It then reaches 5.NF.A.1, adding and subtracting fractions with unlike denominators by first rewriting them with a common denominator (an equivalent fraction for each). The recurring emphasis — that the denominator names a piece size and is never added — heads off the most common error, "12 + 13 = 25."