How to Add and Subtract Fractions, Step by Step

Adding fractions feels mysterious until a child sees the one rule behind it: you can only add pieces that are the same size. Once that clicks, everything else is just bookkeeping.

Fractions are where a lot of confident young mathematicians suddenly stall. The reason is almost always the same — they try to add the tops and the bottoms, getting 1/2 + 1/2 = 2/4, which is obviously wrong. This guide walks through the logic in the order US students meet it: same denominators first (4th grade, CCSS 4.NF.B.3), then unlike denominators (5th grade, 5.NF.A.1), with a fully worked example and clear advice on simplifying.

The one big idea: same-size pieces

A fraction's denominator (the bottom number) tells you the size of the pieces; the numerator (the top) tells you how many of those pieces you have. You can only add or subtract pieces that are the same size. Two quarters plus one quarter makes three quarters — the pieces match, so you just count them. This is why the denominator never changes when you add same-denominator fractions: the piece size stays the same, you are only counting more of them.

Step 1: Same denominators — just add the tops

When the bottoms already match, add or subtract the numerators and keep the denominator exactly as it is.

• 2/7 + 3/7 = 5/7. Two sevenths plus three sevenths is five sevenths.
• 5/8 − 2/8 = 3/8. Take two eighths away from five eighths and three eighths remain.

Say it out loud with the piece name — "five eighths minus two eighths" — and the answer almost announces itself. A common early error is adding the bottoms too (getting 5/14). Naming the pieces prevents it: nobody says "five fourteenths" when the pieces were eighths.

Step 2: Unlike denominators — find a common denominator

When the bottoms are different, the pieces are different sizes, so you cannot add yet. You first rewrite both fractions with the same denominator — the least common denominator (LCD), which is the smallest number both denominators divide into evenly. The fastest way to find it is to list multiples of the larger denominator until one is also a multiple of the smaller one.

Step 3: A full worked example — 1/4 + 2/3

Let's add 1/4 + 2/3 from start to finish.

1. Find the LCD. Multiples of 4: 4, 8, 12. Multiples of 3: 3, 6, 9, 12. The smallest shared multiple is 12.
2. Rewrite each fraction over 12. To turn quarters into twelfths, multiply top and bottom by 3: 1/4 = 3/12. To turn thirds into twelfths, multiply top and bottom by 4: 2/3 = 8/12. Whatever you multiply the bottom by, you must multiply the top by the same amount, so the value doesn't change.
3. Add the numerators. Now the pieces match: 3/12 + 8/12 = 11/12.
4. Simplify. Can 11 and 12 both be divided by the same number? No — 11 is prime and shares no factor with 12. So 11/12 is already in lowest terms. Final answer: 1/4 + 2/3 = 11/12.

Step 4: Simplifying the answer

Whenever the top and bottom of your answer share a common factor, divide both by it to reduce to lowest terms. For example, 4/8 becomes 1/2 (divide both by 4), and 6/9 becomes 2/3 (divide both by 3). If the numerator ends up larger than the denominator — say 7/4 — that's an improper fraction, which you can leave as is or rewrite as the mixed number 1 3/4. Most teachers expect the final answer fully simplified, so make it a habit to ask, "Can this reduce?" every single time.

Subtraction works exactly the same way

Subtracting unlike fractions uses the identical process — find the LCD, rewrite both fractions, then subtract the numerators. For 2/3 − 1/4: the LCD is again 12, so 2/3 = 8/12 and 1/4 = 3/12, giving 8/12 − 3/12 = 5/12, which is already in lowest terms. The only new wrinkle appears with mixed numbers, where a child sometimes has to "borrow" a whole — but the core rule never changes: make the pieces the same size, then work with the tops.

Common mistakes — and how to fix them

• Adding the denominators. Writing 1/2 + 1/2 = 2/4. Fix: remind them the denominator is the piece size — it stays put when the pieces match.
• Changing only the bottom. Turning 1/4 into 1/12 without also scaling the top. Fix: multiply top and bottom by the same number, every time.
• Forgetting to simplify. Leaving 6/12 instead of 1/2. Fix: end every problem with the question "Can this reduce?"
• Picking a common denominator that's too big to manage. Fix: multiplying the two denominators always works, but listing multiples usually finds a smaller, tidier LCD.

How much practice is enough?

Fractions reward steady, spaced practice more than cramming. Aim for a short set of five or six problems most days: a couple with same denominators to stay warm, a few with unlike denominators, and one that needs simplifying. Because every generated worksheet is different, your child meets fresh number pairs each time and practices the actual method — finding the LCD and rewriting — instead of memorizing the answers to one page.