Dividing by Fractions

When you divide a whole number by a unit fraction, you are asking a counting question: how many of those small pieces fit into the wholes? Take 4 ÷ 1/5. One whole fits 5 fifths, so 4 wholes fit 4 × 5 = 20 fifths. That is why the answer, 20, is bigger than the number you started with. Dividing by a small piece means lots of pieces are needed to fill the wholes. A tape diagram helps you see it. Draw 4 bars, split each into 5 equal parts, and count all the small parts. You get 20. The same idea works for any unit fraction. In 6 ÷ 1/4, each whole fits 4 fourths, so 6 wholes fit 24. Try checking your answer with multiplication: 20 × 1/5 = 4 and 24 × 1/4 = 6. Multiplication and division undo each other, so the check always brings you back to the number you started with.

Division by a unit fraction shows up in real life all the time. Say you have 6 cups of flour and each batch of pancakes uses 1/4 cup. How many batches can you make? This is 6 ÷ 1/4. Each cup holds 4 quarter cups, so 6 cups hold 6 × 4 = 24 quarter cups. You can make 24 batches. Notice the answer is much bigger than 6. That is not a mistake. When you divide a positive number by a positive fraction smaller than 1, you are counting how many tiny pieces fit, and lots of them fit. You can even compare without computing. Dividing 4 by 1/8 gives more pieces than dividing 4 by 1/5, because eighths are smaller than fifths. The smaller the piece you divide by, the bigger your answer grows. A quick sketch of wholes split into equal parts makes this easy to see and easy to explain to a friend.

Multiplication and division are like undo buttons for each other. If 4 ÷ 1/5 = 20 is true, then 20 × 1/5 must bring you back to 4. Check it: 20 × 1/5 means 20 fifths, and 20 fifths make exactly 4 wholes. This trick lets you test any division answer. Say a friend says 8 ÷ 1/5 = 40. Multiply 40 × 1/5 and you get 8, so the answer checks out. If you get a different number, something went wrong. You can use the same link to solve too. To find 7 ÷ 1/6, ask yourself: what number times 1/6 equals 7? Seven wholes is 42 sixths, because 7 × 6 = 42. So 7 ÷ 1/6 = 42.