Fraction ÷ Fraction

1Summary 重點摘要

Divide by a fraction = multiply by its reciprocal.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The mnemonicFor a fraction divided by a fraction, the mnemonic is: keep the first fraction, change the division sign to multiplication, and flip the second fraction upside down. For 12 ÷ 13, keep 12, change ÷ to ×, and flip 13 into 31, so the expression becomes 12 × 3. Once the three moves are done, the division has become a multiplication and proceeds by the rules of fraction multiplication. Why flip and then multiply? Because dividing by a number equals multiplying by its reciprocal, a core fact of division that the mnemonic turns into actions. A common error is flipping the first fraction as well, or flipping without changing the sign; all three moves are required, with none skipped.
2. Step 2: Find the reciprocalFlipping a fraction top to bottom gives its reciprocal: the reciprocal of 13 is 31, which is 3; the reciprocal of 34 is 43; and the whole number 5 can be seen as 51 with denominator 1, so its reciprocal is 15. The hallmark of a reciprocal is the swap of numerator and denominator, and a number times its reciprocal always gives exactly 1, as in 34 × 43 = 1212 = 1, which doubles as a check that the reciprocal was found correctly. A common error is confusing the reciprocal with the opposite sign, writing the reciprocal of 13 as -13; a reciprocal only swaps top and bottom, never the sign, and only a product of 1 makes the pair reciprocals.
3. Step 3: Change to multiplicationOnce the division has become a multiplication, work by the rules of fraction multiplication: 12 × 3 = 32. Keep the dividend, multiply by the divisor's reciprocal, and the product is the quotient of the original division. Before multiplying, look for chances to cancel: cancelling first keeps the numbers small, so 34 × 83 lets the 3 on top cancel the 3 below and 8 reduce with 4 to 2, giving 2 directly. Why cancel first? Because multiplying into large numbers and simplifying afterwards is heavy work and invites slips. A common error is multiplying out large numerators and denominators and cancelling late, losing a factor on the way; scan for cancellations before the first stroke.
4. Step 4: Write as a mixed numberThe quotient 32 has a numerator larger than its denominator, so it is an improper fraction and can be written as the mixed number 1 and 12. The method splits 32 into 22 plus 12, and since 22 is exactly one whole, the result is one whole plus 12. Why bother converting? Because mixed numbers are easier to size up at a glance — 1 and 12 plainly sits between 1 and 2, while 32 needs a mental conversion; answers to word problems are usually given as mixed numbers and read more naturally. A common error is dividing the wrong way, treating 3 ÷ 2 as 2 ÷ 3; remember numerator divided by denominator, quotient for the whole part, remainder for the new numerator, denominator unchanged.
5. Step 5: Why it works1 ÷ 14 = 4, because one whole holds exactly 4 quarters. Dividing by a small fraction asks how many of those small portions fit inside the whole; the smaller the portion, the more of them fit, so the quotient grows instead of shrinking. This is the same truth as in whole-number division, where a smaller divisor gives a larger quotient: 12 ÷ 2 = 6 while 12 ÷ 3 = 4. With this understood, a growing quotient no longer seems strange. A common error is trusting the instinct that division always makes numbers smaller, then doubting the result of 1 ÷ 14 = 4; think in portions — cut the whole into small shares and count them, and the reasoning becomes clear.
6. Step 6: Whole numbers tooThe mnemonic works when the dividend is a whole number: for 3 ÷ 12, keep 3, change the sign to multiplication, and flip 12 into 2, so 3 × 2 = 6. Why 6? Because each whole holds 2 halves, so three wholes hold 6 halves; drawing three circles each cut in two and counting confirms it. A whole number can be seen as a fraction with denominator 1, here 31, and then the divisor's reciprocal is found and multiplied exactly as before, keeping one unified method. A common error is forgetting to bring the whole number into the multiplication, producing something like 32; treat the whole as a fraction over 1 and follow the mnemonic step by step, and nothing is missed.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Dividing by a fraction = multiplying by its reciprocal
• Mnemonic: Keep–Change–Flip (KCF)
• The reciprocal swaps top and bottom: 34 → 43
• Dividing by a number under 1 makes the answer bigger
• Check: answer × divisor = dividend