Fractions add/subtract (same denominator)

1Summary 重點摘要

27 + 37 = 57. Denominator stays.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: Adding with the same denominatorIn 27 + 37 both denominators are 7, so the pieces are the same size and the tops can be added directly: 2 + 3 = 5, keep the 7, and the answer is 57. Think of it as two 17 blocks plus three 17 blocks, making five 17 blocks. Why does the denominator stay? The size of each piece has not changed; only the count has. Take care that fractions with different denominators may not be added directly and must first be made to share a denominator; hold this line firmly, for most confusion begins here.
2. Part 2: Subtracting with the same denominatorIn 710 - 310 the denominators match, so subtract the tops: 7 - 3 = 4, giving 410. Both 4 and 10 have the factor 2, so it simplifies to 25. After subtracting with equal denominators, always check whether the answer simplifies. Why subtract directly? The reason matches addition: taking some same-sized blocks away leaves so many blocks, and the size each block stands for is unchanged. Take care that the tops subtract big from small; writing 3 - 7 produces a negative number, and in primary work subtraction always takes the smaller from the bigger.
3. Part 3: Simplify at the end19 + 29 = 39, but 39 is not yet in simplest form. Both 3 and 9 divide by 3: 3 ÷ 3 = 1 and 9 ÷ 3 = 3, so the simplest answer is 13. An answer is complete only once it is fully simplified. Why is simplifying allowed? Dividing top and bottom by the same number keeps the value, so 39 and 13 are equal in size, differing only in form. The usual error is dividing only the top or only the bottom; both must be divided by the same number, and neither may be left out.
4. Part 4: When the top passes the bottom56 + 36 = 86, and the top 8 is bigger than the bottom 6, so this is more than one whole. Since 66 is exactly one whole with 26 left over, 86 can be written as the mixed number 1 26, which simplifies to 1 13. Why past one whole? The pieces of the two parts together outnumber the denominator. Take care that the whole part of the mixed number counts the wholes, and the remaining 26 must still simplify to 13; with both the whole part and the fraction written properly, the answer is complete.
5. Part 5: Subtracting from a wholeFor 1 - 38, first rewrite 1 as 88, because 1 can be written as any fraction whose top and bottom match. The expression becomes 88 - 38 = 58. Turning 1 into a matching fraction lets the subtraction go ahead directly. Take care that the denominator chosen for 1 must equal the other fraction's denominator; it became 88 here because the other denominator is 8, and any other denominator still blocks the subtraction, costing another round of conversion for nothing.
6. Part 6: Adding three fractionsIn 15 + 25 + 15 all three denominators are 5. Add the tops in order: 1 + 2 + 1 = 4, keep the 5 unchanged, and the answer is 45. With chains of fractions too, only the tops are added while the bottom stays the same throughout. Why do chains work? Each step follows the same-denominator rule: adding the first two gives 35, then 15 more gives 45, the same as adding all at once. Take care not to skip a top when there are several; add them one by one in order, then count the tops against the question to confirm none is missing.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Same denominators: keep the bottom, add or subtract the top
• Simplify at the end: 39 = 13
• To subtract from 1, rewrite it as n/n, e.g. 88
• Tops over the bottom can become a mixed number
• Adding three fractions just adds all the tops