Math 6 · 5.NF.B.7C · 6.NS.A.1

Word problems that divide fractions

A mill has flour, and the bakeries want it in bags, bins and batches. When does the order call for dividing, and which number goes first?

You need

Dividing fractions: how many fitGrade 5 Dividing fractions: sharing into equal partsGrade 5

Where you are 6 in Dividing fractions

Fractions, ratio and percent3

Fractions15

Dividing fractions6

Dividing fractions: sharing into equal parts Dividing fractions: how many fit Why flipping the second fraction works

Word problems that divide fractions

Dividing by multiplying by the reciprocal Dividing mixed numbers

When you reach for it

At Catalina’s flour mill, flour leaves in fractional kilograms: bags of \displaystyle \frac{3}{4} kg, scoops of \displaystyle \frac{3}{8} kg, a bin that is \displaystyle \frac{2}{3} full. You already know how to compute a fraction division: multiply by the reciprocal, with mixed numbers renamed first, and Why flipping the second fraction works says why. What a bakery’s order does not tell you is that it is a division. An order names the amounts, never the operation.

This lesson is about that decision. Reach for it whenever a situation hands you two amounts and asks for a third: decide whether to multiply or divide, which amount goes first, and what the answer means once you have it. Day 1 is the decision itself, between the two meanings of how many fit and sharing into equal parts. Day 2 is problems with more than one step, answers that do not come out whole, and checking that an answer makes sense.

The procedure

Day 1: how many groups, or how much in one group

Every one of these problems is a multiplication with a gap in it. Flour at the mill comes in equal groups (bags, bins, batches), and three numbers describe it:

(\text{number of groups}) \times (\text{amount in one group}) = \text{total}

A word problem gives two of the three and asks for the third. Before computing anything, write this sentence with a ? in the gap. Which gap it is decides everything.

The total is missing. Multiply. Five bags of \displaystyle \frac{3}{4} kg hold \displaystyle 5 \times \frac{3}{4} kg.
The number of groups is missing. This is the “how many fit” question. \displaystyle ? \times \frac{2}{3} = 2 asks how many \displaystyle \frac{2}{3} kg bags 2 kg fills, and the division that undoes the multiplication is \displaystyle 2 \div \frac{2}{3}.
The amount in one group is missing. This is the sharing question. \displaystyle \frac{2}{3} \times ? = 2 says that 2 kg is \displaystyle \frac{2}{3} of one group, so one whole group is \displaystyle 2 \div \frac{2}{3}.

In both divisions the total goes first. That is not a habit but a consequence: division undoes the multiplication, and the total is its product. The number of groups need not be whole. A bin can be \displaystyle \frac{2}{3} full, which is two thirds of one group; Finding the whole when only a fraction of it is known treats that case on its own.

Figure 1Both pictures are \displaystyle 2 \div \frac{2}{3} = 3. On top the 3 counts bags; underneath it is the 3 kg a full bin holds.

The two questions in the figure have the same numbers and the same quotient, \displaystyle 2 \div \frac{2}{3} = 3, but different answers: 3 bags, and 3 kg. The units come from the gap. A “how many groups” answer counts groups (bags, batches, tins); a “how much in one group” answer is an amount of flour, in kg.

Words are a clue, not a rule. “Each”, “per” and “how many” turn up in multiplication problems as readily as in division ones. “Each bag holds \displaystyle \frac{3}{4} kg; how much do 5 bags hold?” has “each” and “how much” and is a multiplication. Only the sentence with the gap settles it.

Estimate before computing. A group of less than 1 kg fits into a total more times than the total has kilograms; a group of more than 1 kg fits fewer times. A rough number now catches a division written the wrong way round, which gives the reciprocal of the right answer.

Day 2: several steps, part groups, and sense

Find the total first. Many orders need a step before the division: two sacks are tipped together, some flour is kept back, a share is taken out. Each step is an addition, subtraction or multiplication you already know; the division comes once the sentence with the gap is ready. Flour pours together, so a combined total is real: what is left in one sack can top up a bag from the next.

A “how many groups” quotient that is not whole has two parts. The whole number counts full groups. The fraction is part of one more group, not a number of kilograms, because the quotient counts groups. To turn it into flour, multiply it by the size of a group, or subtract the full groups from the total; the two ways must agree. Then the question decides which whole number answers it, as with remainders in whole-number division: “how many full bags” rounds down, “how many tins to hold all of it” rounds up, and “how many batches” may keep the fraction, since a recipe can be scaled down.

Check that the answer makes sense, three ways. Multiply back: the answer put in the gap must make the sentence true. Compare with the estimate. Read the units: a count of bags that comes out as a small fraction, or one baker’s share larger than the whole amount shared, means the division was written backwards.

Worked examples

Example 1

Day 1: four questions, two numbers

Each question uses 8 and \displaystyle \frac{4}{5}. Write the sentence with the gap and answer it.

(a) Catalina packs 8 kg of rye flour into bags holding \displaystyle \frac{4}{5} kg each. How many bags does she fill?

(b) Femi tips 8 kg of rye flour into an empty bin and it is \displaystyle \frac{4}{5} full. How much does the full bin hold?

(d) Malia shares \displaystyle \frac{4}{5} kg of rye flour equally among 8 trays of rolls. How much goes on each tray?

Solution

(a) The bags are the groups, and their number is missing: \displaystyle ? \times \frac{4}{5} = 8. Undo it: \displaystyle 8 \div \frac{4}{5} = 8 \times \frac{5}{4} = 10. She fills 10 bags. Check: \displaystyle 10 \times \frac{4}{5} = 8.

(b) The bin is the group, and \displaystyle \frac{4}{5} of it holds 8 kg: \displaystyle \frac{4}{5} \times ? = 8. So \displaystyle ? = 8 \div \frac{4}{5} = 10, and the full bin holds 10 kg. The quotient is the same as in (a), but now it is kilograms, not bags. Check: \displaystyle \frac{4}{5} of 10 kg is 8 kg.

(c) The number of groups and the amount in each are both known; the total is missing: \displaystyle 8 \times \frac{4}{5} = ?. Multiply: \displaystyle \frac{32}{5} = 6\frac{2}{5} kg. There is no division at all, though “each” is in the question.

(d) The eight trays are the groups, and each tray’s amount is missing: \displaystyle 8 \times ? = \frac{4}{5}. The total, \displaystyle \frac{4}{5}, goes first: \displaystyle \frac{4}{5} \div 8 = \frac{4}{5} \times \frac{1}{8} = \frac{1}{10} kg a tray. Check: \displaystyle 8 \times \frac{1}{10} = \frac{4}{5}. Dividing the other way, \displaystyle 8 \div \frac{4}{5} = 10, would put 10 kg on each tray, more than the whole amount shared.

Example 2

Day 1: batches from a bag

Jakob’s bakery uses \displaystyle 1\frac{1}{4} kg of flour in each batch of bread. He buys a \displaystyle 7\frac{1}{2} kg bag from the mill. How many batches can he bake?

Solution

The batches are the groups, each uses \displaystyle 1\frac{1}{4} kg, and their number is missing:

? \times 1\frac{1}{4} = 7\frac{1}{2}

so the number of batches is \displaystyle 7\frac{1}{2} \div 1\frac{1}{4}. Estimate: a batch is a little over 1 kg, so there are a little fewer batches than kilograms, perhaps 6. Renaming as in Dividing mixed numbers,

7\frac{1}{2} \div 1\frac{1}{4} = \frac{15}{2} \times \frac{4}{5} = \frac{60}{10} = 6

Jakob can bake 6 batches. Check: \displaystyle 6 \times 1\frac{1}{4} = 6 + \frac{6}{4} = 7\frac{1}{2}.

Example 3

Day 1: less than one group

Malia has \displaystyle \frac{5}{6} kg of spelt flour. A full batch of her spelt loaves needs \displaystyle 2\frac{1}{2} kg of flour. How many batches can she make?

Solution

The batch is the group, and the number of batches is missing: \displaystyle ? \times 2\frac{1}{2} = \frac{5}{6}. The total, \displaystyle \frac{5}{6}, goes first even though it is the smaller number:

\frac{5}{6} \div 2\frac{1}{2} = \frac{5}{6} \times \frac{2}{5} = \frac{10}{30} = \frac{1}{3}

She can make \displaystyle \frac{1}{3} of a batch, using a third of every ingredient. That is sensible: she has much less flour than one batch needs. Check: \displaystyle \frac{1}{3} \times 2\frac{1}{2} = \frac{1}{3} \times \frac{5}{2} = \frac{5}{6}.

The reverse division, \displaystyle 2\frac{1}{2} \div \frac{5}{6} = 3, answers a different question: how many of her \displaystyle \frac{5}{6} kg amounts make one full batch. She would need 3 times the flour she has.

Example 4

Day 2: two sacks, then bags

Femi has two sacks of wholemeal flour, one with \displaystyle 4\frac{1}{2} kg and one with 3 kg. He tips both into the hopper and fills bags of \displaystyle \frac{5}{8} kg. How many bags does he fill?

Solution

First the total: \displaystyle 4\frac{1}{2} + 3 = 7\frac{1}{2} kg. Now the sentence is \displaystyle ? \times \frac{5}{8} = 7\frac{1}{2}, so the number of bags is

7\frac{1}{2} \div \frac{5}{8} = \frac{15}{2} \times \frac{8}{5} = \frac{120}{10} = 12

Estimate: a bag is between half a kilogram and a whole one, so the number of bags is between \displaystyle 7\frac{1}{2} and 15. 12 fits. Check: \displaystyle 12 \times \frac{5}{8} = \frac{60}{8} = 7\frac{1}{2}.

Adding first relies on the flour mixing. Bagging each sack on its own gives \displaystyle 4\frac{1}{2} \div \frac{5}{8} = 7\frac{1}{5} and \displaystyle 3 \div \frac{5}{8} = 4\frac{4}{5}, only 7 + 4 = 11 full bags, with two part bags that together make the twelfth.

Example 5

Day 2: full bags or enough tins

Jakob has 5 kg of rye flour.

(a) How many full \displaystyle \frac{3}{8} kg bags can he fill, and how much flour is left?

(b) A customer wants all 5 kg delivered in tins holding \displaystyle \frac{3}{8} kg each. How many tins are needed, and how full is the last?

Solution

Both are \displaystyle ? \times \frac{3}{8} = 5, so both start from the same quotient:

5 \div \frac{3}{8} = 5 \times \frac{8}{3} = \frac{40}{3} = 13\frac{1}{3}

(a) Thirteen bags fill completely, and the \displaystyle \frac{1}{3} is one third of a fourteenth bag, which is not full, so the answer is 13 bags. The leftover is \displaystyle \frac{1}{3} of a bag’s worth: \displaystyle \frac{1}{3} \times \frac{3}{8} = \frac{1}{8} kg. Subtracting agrees: thirteen bags use \displaystyle \frac{39}{8} = 4\frac{7}{8} kg, and \displaystyle 5 - 4\frac{7}{8} = \frac{1}{8}.

(b) Thirteen tins leave \displaystyle \frac{1}{8} kg undelivered, so the answer rounds up to 14 tins. The last holds \displaystyle \frac{1}{8} kg, so it is \displaystyle \frac{1}{3} full, the same fraction the quotient showed.

Example 6

Day 2: bags shared among bakeries

Tenzin packs 14 kg of oat flour into \displaystyle \frac{2}{3} kg bags and shares the bags equally among 4 bakeries, whole bags only. How many bags, and how much flour, does each bakery get? Check that the answer makes sense.

Solution

Bags first: \displaystyle ? \times \frac{2}{3} = 14, so \displaystyle 14 \div \frac{2}{3} = 14 \times \frac{3}{2} = 21 bags.

Now share 21 bags among 4 bakeries. This is whole-number division: 21 = 4 \times 5 + 1, so each bakery gets 5 bags and 1 bag stays at the mill, since a sealed bag is not split. Each bakery’s flour is \displaystyle 5 \times \frac{2}{3} = \frac{10}{3} = 3\frac{1}{3} kg.

Check by building the 14 kg back up: \displaystyle 4 \times 3\frac{1}{3} = 13\frac{1}{3} kg is delivered, and the bag at the mill holds \displaystyle \frac{2}{3} kg; \displaystyle 13\frac{1}{3} + \frac{2}{3} = 14. Also, an equal share of 14 kg among 4 is \displaystyle 3\frac{1}{2} kg, and each bakery gets a little less, as it must when some is held back.

Sharing the flour first gives the same picture: \displaystyle 14 \div 4 = 3\frac{1}{2} kg each, and \displaystyle 3\frac{1}{2} \div \frac{2}{3} = 5\frac{1}{4} bags each. The four quarter bags that cannot be handed out are the 1 whole bag left at the mill.

Boundaries

Division needs the total. If the total is the missing number, the problem is a multiplication, whatever words it uses.

The divisor is never 0. A bag holding 0 kg, or 0 groups to share among, makes the question meaningless, and the division has no answer.

The order is fixed. Dividing the two numbers the other way gives the reciprocal, which answers a different question (Example 3) or no sensible one (Example 1(d)).

Whether a part group counts depends on the thing. Part of a batch can be baked by scaling; part of a bag is not a full bag; part of a bakery does not exist. The quotient is the same; reading it is the situation’s job.

Combining amounts first needs amounts that combine. Flour pours together; sealed bags do not, which is why Example 6 kept its last bag whole.

This lesson stays with positive fractions, mixed numbers and whole numbers: amounts of flour and counts of bags. Comparing two amounts by division, and rates such as kilograms ground in an hour, come later.

Where this goes next

The “how much in one group” question with a fractional group is Finding the whole when only a fraction of it is known. When the group is an hour or a kilometer rather than a bag, the same division gives a rate, which Unit rates when the amounts are fractions takes up. And the sentence with a gap, \displaystyle ? \times \frac{4}{5} = 8, is already an equation with an unknown: later work writes the ? as a letter and solves it the same way, by undoing the multiplication.

Exercises

Day 1: which question is it?

For each, say whether the number of groups, the amount in one group or the total is missing, write the multiplication sentence with a ?, and answer.

Femi has 4 kg of flour and fills bags that hold \displaystyle \frac{2}{3} kg each. How many bags?
Jakob shares \displaystyle \frac{3}{4} kg of rye flour equally among 3 mixing bowls. How much goes in each?
Malia fills 5 bags, each holding \displaystyle \frac{3}{8} kg. How much flour does she use?
Tenzin pours 3 kg of flour into an empty tub and fills \displaystyle \frac{3}{5} of it. How much does the full tub hold?
Hugo has \displaystyle \frac{1}{2} kg of rye flour and makes sample packets of \displaystyle \frac{1}{10} kg. How many packets?
Catalina has \displaystyle 5\frac{1}{3} kg of flour, and each batch of rolls needs \displaystyle \frac{2}{3} kg. How many batches?

Day 1: order and meaning

Write two flour questions answered by \displaystyle 3 \div \frac{1}{4}: one asking how many groups, one asking how much in one group. Answer both, with units.
Femi has \displaystyle \frac{3}{8} kg of flour, and a batch of scones needs \displaystyle \frac{3}{4} kg. How many batches can he make?
Jakob shares \displaystyle \frac{5}{6} kg of flour equally among 5 bakers and writes \displaystyle 5 \div \frac{5}{6} = 6. What went wrong, and how much does each baker get?
Malia’s scoop holds \displaystyle \frac{3}{8} kg. How many scoops are in \displaystyle 2\frac{1}{4} kg?
Tenzin shares \displaystyle 1\frac{7}{8} kg of flour equally among 3 bakeries. How much does each get?

Day 2: more than one step

Hugo has 3 bags of semolina, each holding \displaystyle 1\frac{1}{2} kg. He tips them together and repacks the semolina in \displaystyle \frac{3}{8} kg bags. How many bags does he fill?
Catalina has 6 kg of flour. She keeps \displaystyle 2\frac{1}{4} kg for her own bread and packs the rest in \displaystyle \frac{5}{8} kg bags. How many bags?
A sack holds 12 kg. Tenzin sells \displaystyle \frac{2}{3} of it to one bakery and packs what remains in \displaystyle \frac{1}{2} kg bags. How many bags?
Jakob shares \displaystyle 6\frac{1}{4} kg of flour equally among 2 bakeries, and each bakery stores its share in \displaystyle \frac{5}{8} kg tins. How many tins does each bakery fill?

Day 2: parts of a group

Malia has 4 kg of flour and \displaystyle \frac{3}{4} kg bags. How many full bags can she fill, and how much flour is left?
Tenzin must store 7 kg of flour in tins holding \displaystyle 1\frac{1}{4} kg each. How many tins does he need, and how full is the last one?
Hugo has \displaystyle 2\frac{1}{2} kg of flour, and a batch of bagels needs \displaystyle \frac{2}{3} kg. How many full batches can he make, what fraction of another batch is left, and how much flour is that?
Catalina packs 10 kg of flour into \displaystyle \frac{2}{3} kg bags and shares the bags equally among 4 bakeries, whole bags only. (a) How many bags does each bakery get, and how many are left? (b) How much more flour would the mill need for every bakery to get 4 bags?

Day 2: does it make sense?

Femi works out how many \displaystyle \frac{3}{4} kg bags \displaystyle 4\frac{1}{2} kg fills and gets \displaystyle 3\frac{3}{8}. Show with an estimate that this cannot be right, find the right answer, and say what he computed instead.
Malia packs \displaystyle 3\frac{3}{4} kg of flour into \displaystyle \frac{5}{8} kg bags and says she fills 6 bags with \displaystyle \frac{1}{4} kg left over. Check her answer by multiplying back.

Reasoning

Write two flour questions that both use the word “each”: one answered by multiplying and one by dividing. Explain what makes the difference.
Bags of \displaystyle \frac{5}{8} kg are filled from a sack, and the division gives \displaystyle 5\frac{2}{5}. What do the 5 and the \displaystyle \frac{2}{5} each mean? How much flour is left over, and how much was in the sack?
Explain why, in a “how many groups” problem, a group of less than 1 kg always gives more groups than there are kilograms in the total.

Answers

Groups missing: \displaystyle ? \times \frac{2}{3} = 4, and \displaystyle 4 \div \frac{2}{3} = 4 \times \frac{3}{2} = 6 bags.
One group missing: \displaystyle 3 \times ? = \frac{3}{4}, and \displaystyle \frac{3}{4} \div 3 = \frac{1}{4} kg.
Total missing, so multiply: \displaystyle 5 \times \frac{3}{8} = \frac{15}{8} = 1\frac{7}{8} kg.
One group, the tub, missing: \displaystyle \frac{3}{5} \times ? = 3, and \displaystyle 3 \div \frac{3}{5} = 3 \times \frac{5}{3} = 5 kg.
Groups missing: \displaystyle ? \times \frac{1}{10} = \frac{1}{2}, and \displaystyle \frac{1}{2} \times 10 = 5 packets.
Groups missing: \displaystyle ? \times \frac{2}{3} = 5\frac{1}{3}, and \displaystyle \frac{16}{3} \times \frac{3}{2} = 8 batches.
Both answers are 12. For example: how many \displaystyle \frac{1}{4} kg bags does 3 kg fill? 12 bags. And: 3 kg fills \displaystyle \frac{1}{4} of a bin; how much does the full bin hold? 12 kg.
\displaystyle \frac{1}{2} batch. \displaystyle ? \times \frac{3}{4} = \frac{3}{8}, so \displaystyle \frac{3}{8} \times \frac{4}{3} = \frac{1}{2}.
\displaystyle \frac{1}{6} kg each. The total, \displaystyle \frac{5}{6}, must go first: \displaystyle \frac{5}{6} \times \frac{1}{5} = \frac{1}{6}. His 6 kg each is far more than the \displaystyle \frac{5}{6} kg shared.
6 scoops. \displaystyle \frac{9}{4} \times \frac{8}{3} = 6.
\displaystyle \frac{5}{8} kg. \displaystyle \frac{15}{8} \times \frac{1}{3} = \frac{5}{8}.
12 bags. The total is \displaystyle 3 \times 1\frac{1}{2} = 4\frac{1}{2} kg, and \displaystyle \frac{9}{2} \times \frac{8}{3} = 12.
6 bags. \displaystyle 6 - 2\frac{1}{4} = 3\frac{3}{4} kg, and \displaystyle \frac{15}{4} \times \frac{8}{5} = 6.
8 bags. \displaystyle \frac{1}{3} of the sack remains, 4 kg, and \displaystyle 4 \div \frac{1}{2} = 8.
5 tins. Each share is \displaystyle 6\frac{1}{4} \div 2 = 3\frac{1}{8} kg, and \displaystyle \frac{25}{8} \times \frac{8}{5} = 5.
5 bags, \displaystyle \frac{1}{4} kg left. \displaystyle 4 \times \frac{4}{3} = 5\frac{1}{3}; five bags use \displaystyle 3\frac{3}{4} kg.
6 tins, the last \displaystyle \frac{3}{5} full. \displaystyle 7 \times \frac{4}{5} = 5\frac{3}{5}; five tins hold \displaystyle 6\frac{1}{4} kg, so the sixth holds \displaystyle \frac{3}{4} kg, which is \displaystyle \frac{3}{5} of \displaystyle 1\frac{1}{4}.
3 batches, \displaystyle \frac{3}{4} of a batch left, which is \displaystyle \frac{1}{2} kg. \displaystyle \frac{5}{2} \times \frac{3}{2} = 3\frac{3}{4}, and \displaystyle \frac{3}{4} \times \frac{2}{3} = \frac{1}{2}.
(a) 3 bags each, 3 left. \displaystyle 10 \div \frac{2}{3} = 15 bags, and 15 = 4 \times 3 + 3. (b) \displaystyle \frac{2}{3} kg. Four bags each is 16 bags, one bag more than 15.
The answer is 6. A bag is under 1 kg, so there are more bags than kilograms, more than \displaystyle 4\frac{1}{2}. Exactly, \displaystyle \frac{9}{2} \times \frac{4}{3} = 6. He multiplied: \displaystyle 4\frac{1}{2} \times \frac{3}{4} = 3\frac{3}{8}.
6 bags, nothing left. \displaystyle \frac{15}{4} \times \frac{8}{5} = 6, and \displaystyle 6 \times \frac{5}{8} = \frac{30}{8} = 3\frac{3}{4} is all the flour, so her \displaystyle \frac{1}{4} kg leftover is wrong.
For example: “Each scoop holds \displaystyle \frac{3}{8} kg; how much do 8 scoops hold?” is \displaystyle 8 \times \frac{3}{8} = 3 kg. “Each scoop holds \displaystyle \frac{3}{8} kg; how many scoops does 3 kg fill?” is \displaystyle 3 \div \frac{3}{8} = 8 scoops. The difference is which number is missing, the total (multiply) or the number of groups (divide), not the word “each”.
5 full bags, and \displaystyle \frac{2}{5} of a sixth bag. The leftover is \displaystyle \frac{2}{5} \times \frac{5}{8} = \frac{1}{4} kg. The sack held \displaystyle 5\frac{2}{5} \times \frac{5}{8} = \frac{27}{8} = 3\frac{3}{8} kg; check: \displaystyle 5 \times \frac{5}{8} + \frac{1}{4} = 3\frac{3}{8}.
One group is less than a kilogram, so each kilogram holds more than one group, and the whole total holds more groups than it has kilograms. In symbols, dividing by a number less than 1 is multiplying by its reciprocal, which is more than 1.