Adding fractions with different denominators

1Summary 重點摘要

Different denominators: first make the denominators the same, then add the numerators.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Look at the denominatorsIn 13 + 14 the denominators are 3 and 4, different, so direct adding is impossible: a third is bigger than a quarter, so the pieces differ in size, and counts of unequal pieces cannot simply be added. That is the difference from like fractions.
2. Step 2: Find the least common multipleTo make the denominators match, find their least common multiple: multiples of 3 are 3, 6, 9, 12; multiples of 4 are 4, 8, 12; the smallest shared one is 12. It makes a common denominator with the smallest possible numbers.
3. Step 3: Expand the first fractionExpand 13 to denominator 12: the bottom must be multiplied by 4, so the top is multiplied by 4 too, 1 × 4 = 4, giving 412. Expanding multiplies the top and bottom by the same number and keeps the value unchanged.
4. Step 4: Expand the second fractionExpand 14 to denominator 12: the bottom is multiplied by 3, so the top is multiplied by 3 too, 1 × 3 = 3, giving 312. After both expansions, every piece is exactly the same size.
5. Step 5: Add as like fractionsNow the denominators match, so add directly: 412 + 312 = 712, keeping the denominator 12 and adding the numerators to 7. At this point, unlike-denominator addition has become the familiar like-denominator addition.
6. Step 6: Simplify if neededCheck 7 and 12 for common factors: 7 is prime, and the factors of 12 are 1, 2, 3, 4, 6 and 12, so they share only 1. The fraction 712 is already in simplest form and needs no more work.
7. Step 7: Estimate with decimalsA decimal check helps: 13 is about 0.33 and 14 is 0.25, so the sum should be about 0.58; and 712 is also about 0.58, matching the estimate. Estimates need not be exact, only close enough to show there is no big mistake.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Make denominators the same first (use LCM)
• Expand: multiply both parts
• Add numerators only, then simplify
• If one denominator is a multiple of the other, use the bigger one (e.g. 2 and 6 → 6)
• Estimate with decimals to check your answer