Dividing mixed numbers

When you reach for it

You already know how to divide one fraction by another: multiply by the reciprocal of the second one, and Why flipping the second fraction works explains why that gives the right answer. Real measurements rarely arrive as plain fractions, though. A plank is \displaystyle 5\frac{1}{2} meters long, a shelf board \displaystyle 1\frac{1}{2} meters, a floor joist \displaystyle 2\frac{1}{4} meters. As soon as either number in a division is a mixed number, a whole number with a fraction beside it, this move is the one you need.

Reach for it in two kinds of question. How many fit? How many \displaystyle \frac{3}{4} m pieces can be cut from a \displaystyle 5\frac{1}{2} m plank (the “how many fit” meaning). How long is each share? A beam \displaystyle 8\frac{3}{4} m long cut into 5 equal posts (the sharing meaning). Both are a division, and both are done the same way. Where they differ is in how the answer should be written, and that is part of this lesson too.

The procedure

Diego is building a ladder for a treehouse. He has a plank \displaystyle 5\frac{1}{2} m long and wants pieces \displaystyle \frac{3}{4} m long. The question is \displaystyle 5\frac{1}{2} \div \frac{3}{4}.

Estimate first. Before any working, decide roughly what the answer must be. A piece is less than 1 m, so more pieces fit than there are meters: more than \displaystyle 5\frac{1}{2}. And \displaystyle 6 \div \frac{3}{4} = 8, since every 3 m holds 4 pieces. The plank is a little shorter than 6 m, so the answer is a little less than 8: somewhere around 7. The estimate costs a few seconds and catches the biggest slips, such as flipping the wrong fraction, which would give an answer smaller than 1.

Rename each mixed number as an improper fraction. \displaystyle 5\frac{1}{2} is 5 wholes and a half; each whole is 2 halves, so it is 10 + 1 = 11 halves (Mixed numbers and improper fractions: rewriting and comparing):

This changes how the number is written, not its size, so the quotient is unchanged. It is needed because a mixed number is a sum, \displaystyle 5\frac{1}{2} = 5 + \frac{1}{2}, and the reciprocal rule works on a single fraction. You cannot divide the whole parts and the fraction parts separately. Try it on \displaystyle 4\frac{1}{2} \div 1\frac{1}{2}: dividing 4 \div 1 and \displaystyle \frac{1}{2} \div \frac{1}{2} and adding gives 5, yet five pieces of \displaystyle 1\frac{1}{2} m make \displaystyle 7\frac{1}{2} m, far more than \displaystyle 4\frac{1}{2}. The true answer is 3. Dividing by a sum is not the same as dividing by each part and adding, just as 6 \div (1 + 2) = 2 while 6 \div 1 + 6 \div 2 = 9. Renaming turns each sum into one fraction, and then there is nothing to split.

Divide the fractions. Multiply by the reciprocal of the divisor, \displaystyle \frac{3}{4}:

When a numerator and a denominator share a factor, you may cancel it before multiplying (here 4 and 2 share 2). That is simplifying early: the product is the same fraction, just with smaller numbers to carry.

Write the answer in the form the question wants. An improper fraction is a correct answer, but it does not say how many pieces. Divide 22 by 3: 3 goes 7 times with 1 over, so

It agrees with the estimate of about 7. A length, such as the size of each share, also reads best as a mixed number: \displaystyle 1\frac{3}{4} m is easier to measure out than \displaystyle \frac{7}{4} m.

Read what the fraction part means. Diego can cut 7 whole pieces. The \displaystyle \frac{1}{3} is not a third of a meter: it is a third of a piece, because the quotient counts pieces. A third of a \displaystyle \frac{3}{4} m piece is

so \displaystyle \frac{1}{4} m of plank is left over. Check it by building the plank back up from its parts:

This is the same relation as a whole-number division with remainder, 23 = 7 \times 3 + 2, and it always holds for a “how many fit” question:

The leftover is always shorter than one piece. If it were not, one more piece would fit. You can find it either way: as the fraction part of the quotient times the piece length, or as the plank minus the whole pieces. The second is often quicker, and the two must agree.

Worked examples

Boundaries

The divisor must not be 0. Every mixed number is more than 1, so it is never a problem there, but a whole-number divisor of 0 has no reciprocal and the division means nothing.

Rename both numbers before dividing. Splitting the divisor into its whole and fraction parts gives a wrong answer, as \displaystyle 4\frac{1}{2} \div 1\frac{1}{2} showed. Splitting only the first number is allowed: dividing by \displaystyle \frac{3}{4} is multiplying by \displaystyle \frac{4}{3}, and a sum can be multiplied part by part, so \displaystyle (5 + \frac{1}{2}) \div \frac{3}{4} = 5 \div \frac{3}{4} + \frac{1}{2} \div \frac{3}{4}. It only doubles the work.

The size of the divisor fixes the direction. Dividing by a number more than 1 gives less than you started with; dividing by a number less than 1 gives more. Every mixed number is more than 1, so dividing by one always shrinks. The estimate rests on this.

The fraction part of a “how many fit” answer counts pieces, not meters. To turn it into a length, multiply by the piece length, or subtract the whole pieces from the plank.

The question chooses the form. A count of pieces cut from a plank is the whole-number part. A count of boards needed to cover a length is the next whole number up, unless the quotient is already whole. A share of a length is the full mixed number. An improper fraction is the right answer to a bare computation, but it is rarely how a length is spoken of.

This lesson stays with positive numbers: lengths, counts and shares. Dividing negative mixed numbers follows the same steps with the sign rules added.

Where this goes next

The same renaming step comes before multiplying mixed numbers, and the check in every example above was exactly that. With division of any positive fractions now in hand, Word problems that divide fractions turns to deciding from a situation whether it asks how many fit or how much each gets, and Finding the whole when only a fraction of it is known runs a division backwards to recover a whole from a part of it. Later, a speed such as \displaystyle 2\frac{1}{2} km in \displaystyle \frac{3}{4} of an hour becomes a unit rate, which is a division of exactly this kind.

Exercises

Rename and divide

Write each answer as a whole number, a fraction less than 1, or a mixed number.

1. \displaystyle 2\frac{1}{4} \div 1\frac{1}{2}
2. \displaystyle 3\frac{1}{3} \div 1\frac{2}{3}
3. \displaystyle 5\frac{1}{4} \div 1\frac{3}{4}
4. \displaystyle 4\frac{2}{5} \div 2\frac{3}{4}
5. \displaystyle 1\frac{7}{8} \div 2\frac{1}{2}
6. \displaystyle 6\frac{1}{8} \div 1\frac{3}{4}

Whole numbers and plain fractions

7. \displaystyle 7\frac{1}{2} \div 3
8. \displaystyle 9 \div 2\frac{1}{4}
9. \displaystyle 3\frac{3}{5} \div \frac{9}{10}
10. \displaystyle \frac{5}{6} \div 1\frac{1}{9}
11. \displaystyle 8 \div 1\frac{3}{5}

Estimate first

Estimate each quotient, then compute it and compare.

12. \displaystyle 11\frac{7}{8} \div 2\frac{3}{8}
13. \displaystyle 2\frac{4}{5} \div 4\frac{2}{3}
14. Without computing, say whether each quotient is more or less than \displaystyle 6\frac{1}{2}, then compute it: (a) \displaystyle 6\frac{1}{2} \div 1\frac{1}{10} \qquad (b) \displaystyle 6\frac{1}{2} \div \frac{9}{10}

In the treehouse

15. Maren has a plank \displaystyle 6\frac{3}{4} m long and cuts shelf boards \displaystyle 1\frac{1}{8} m long. How many boards does she get?
16. Diego cuts ladder rungs \displaystyle \frac{2}{3} m long from a length of timber \displaystyle 4\frac{1}{2} m long. How many whole rungs can he cut, and how much timber is left?
17. Chidi cuts a beam \displaystyle 10\frac{1}{2} m long into 4 equal posts. How long is each post?
18. Astrid has a plank \displaystyle 8\frac{1}{3} m long and needs pieces \displaystyle 1\frac{3}{4} m long. How many whole pieces can she cut, and how much is left?
19. Kenta covers a walkway \displaystyle 12\frac{1}{2} m long with boards \displaystyle 1\frac{1}{3} m long, laid end to end. How many boards must he buy, and how much is cut off the last one?
20. Farah has two planks, \displaystyle 5\frac{1}{2} m and \displaystyle 3\frac{1}{2} m long, and needs pieces \displaystyle 1\frac{1}{4} m long, each cut from a single plank. How many whole pieces can she cut? Explain why dividing the total length 9 m by \displaystyle 1\frac{1}{4} gives the wrong count.

Reasoning

21. Maren works out \displaystyle 4\frac{1}{2} \div 1\frac{1}{2} as 4 \div 1 plus \displaystyle \frac{1}{2} \div \frac{1}{2}. What does she get, what is the true answer, and why does her way fail?
22. Diego divides \displaystyle 5\frac{1}{4} \div \frac{7}{8} and gets \displaystyle \frac{1}{6}. Show with an estimate that this cannot be right, find the correct answer, and say what he divided instead.
23. Cutting pieces from a plank gives a quotient of \displaystyle 3\frac{2}{5}. Explain what the 3 and the \displaystyle \frac{2}{5} each mean, and why the leftover is not \displaystyle \frac{2}{5} m unless each piece is 1 m long.
24. Write one treehouse question whose answer is found from \displaystyle 5 \div 1\frac{1}{2} and is 3, and another whose answer is found from the same division and is 4.