Adding Fractions

Goal: add two fractions whose numerator and denominator are each less than 20, and write the answer in lowest terms.

Three cases

Case 1 — same denominator

Add the numerators. Keep the denominator. Simplify.

2/9 + 4/9 = 6/9 = 2/3

That's it. The denominator never changes.

Case 2 — one denominator is a multiple of the other

You only have to scale one fraction.

1/3 + 1/6
   ↑     ↑
  ×2/2   already 6

Multiply the top and bottom of the first by 2:

1/3 = 2/6
2/6 + 1/6 = 3/6 = 1/2

Case 3 — different denominators with no shared factor (or only a small one)

Find the LCM of the two denominators. That's your common denominator. Scale each fraction up.

2/5 + 1/4
LCM(5, 4) = 20
2/5 = 8/20
1/4 = 5/20
8/20 + 5/20 = 13/20  (already in lowest terms)

The recipe

1. Find the common denominator (use LCM if the denominators differ).
2. Scale each fraction so they share that denominator.
3. Add the numerators. Keep the denominator.
4. Simplify by dividing top and bottom by their GCF.

Worked examples

Example A — same denominator

3/8 + 2/8
= (3 + 2)/8
= 5/8         (8 and 5 share no factor; lowest terms)

Example B — multiple denominators

1/4 + 5/12
LCM(4, 12) = 12
1/4 = 3/12
3/12 + 5/12 = 8/12
GCF(8, 12) = 4
8/12 = 2/3

Example C — different denominators

3/5 + 2/7
LCM(5, 7) = 35
3/5 = 21/35
2/7 = 10/35
21/35 + 10/35 = 31/35   (31 is prime; already in lowest terms)

Example D — answer can exceed 1 (improper fractions are fine)

4/5 + 3/5 = 7/5

We accept improper fractions. You don't have to convert to a mixed number. Just keep it in lowest terms.

Common mistakes to avoid

• Adding denominators. 2/3 + 1/3 ≠ 3/6. Wrong. Keep the denominator, add the numerators only.
• Forgetting to simplify. 4/8 is not the answer; 1/2 is.
• Using a common denominator that's bigger than the LCM. It still works, but you'll have huge numbers and more chances to slip.

Try these (answers below)

1. 1/2 + 1/4
2. 2/3 + 1/6
3. 3/7 + 2/7
4. 5/8 + 1/3

1. 3/4
2. 5/6
3. 5/7
4. 23/24