Dividing fractions: how many fit
A toy store has 4 yards of ribbon and cuts it into ties one third of a yard long for its gift bags. How many ties does it get?
Where you are 6 in Dividing fractions
Why this exists
The toy store wraps every gift in a paper bag and closes the bag with a ribbon tie. Dyani has 4 yards of ribbon on a spool, and each tie takes \displaystyle \frac{1}{3} of a yard. How many ties can she cut?
That is a division, but not the sharing kind. In the last lesson, Dividing fractions: sharing into equal parts, an amount was split into a known number of equal shares, and the question was how big one share is. Here the size of each piece is known, and the question is how many pieces fit. You have met this reading with whole numbers: 12 \div 3 also asks how many groups of 3 fit in 12 (What division means). What is new is that the piece is a fraction, and then something surprising happens: the answer is bigger than the number you started with.
The lesson takes four days. Part 1 counts how many unit fractions fit in a whole number. Part 2 checks an answer by multiplying back and sets the two meanings of division side by side. Part 3 tells the idea as stories and explains why the answer comes out bigger. Part 4 divides one fraction by another by giving them the same bottom number and counting pieces, including a piece left over.
Definition
A unit fraction has 1 on top, like \displaystyle \frac{1}{3}: one of 3 equal parts of a whole (Unit fractions).
Let a and d be two amounts measured in the same unit, with d not 0. Then a \div d is the number of copies of d that fit in a. If the copies do not come out even, the part left over is counted as a fraction of one copy of d.
Put that many copies of d together and you must get a back. So the answer is the number q that makes this true:
Reading division as “how many of these fit” is the how-many-fit meaning of division. Its older names are the measurement meaning, because d is used like a ruler to measure a, and the quotitive meaning, from a Latin word for “how many”.
Intuition
Part 1: how many unit fractions fit in a whole number
Draw Dyani’s ribbon as a long tape, 4 yards long. Mark each yard. Then cut every yard into 3 equal pieces. Each piece is \displaystyle \frac{1}{3} of a yard, which is exactly one tie.
Now count the ties. Each yard holds 3 of them, and there are 4 yards, so there are 4 \times 3 = 12 ties in all:
The number line under the tape says the same thing. Start at 0 and jump \displaystyle \frac{1}{3} at a time. It takes 3 jumps to reach 1, 3 more to reach 2, and so on. Reaching 4 takes 12 jumps. A jump on the number line is one copy of \displaystyle \frac{1}{3}, so the number of jumps is the number of copies that fit (Fractions on the number line).
Nothing here depended on the numbers 4 and 3. One whole holds b copies of \displaystyle \frac{1}{b}, because \displaystyle b \times \frac{1}{b} = \frac{b}{b} = 1. So n wholes hold n groups of b copies:
Part 2: checking by multiplying back, and the two meanings
Checking. Multiplication and division undo each other (Multiplication and division undo each other). Haoran has 3 yards of ribbon and cuts ties \displaystyle \frac{1}{6} of a yard long. Each yard holds 6 sixths, so he gets 3 \times 6 = 18 ties:
To check, put the 18 ties back end to end. Eighteen pieces, each \displaystyle \frac{1}{6} yard, make \displaystyle 18 \times \frac{1}{6} = \frac{18}{6} yards (A fraction times a whole number). Eighteen sixths is 3 wholes, because 18 \div 6 = 3. That is the ribbon he started with, so 18 is right.
Two meanings of one division. With whole numbers, 12 \div 3 can be read two ways. Share 12 toys among 3 bags, and each bag gets 4. Or ask how many bags of 3 toys you can fill, and the answer is 4 bags. The answers match because both readings look for the missing number in 3 \times 4 = 12: one asks for the size of a group, the other for the number of groups.
The same is true with fractions, though the sharing reading needs a different story. Read \displaystyle 3 \div \frac{1}{6} as sharing and you would be sharing 3 yards among a sixth of a person, which makes no sense. But the sharing meaning also has a form that asks “how much is in one whole group?” Suppose 3 yards of ribbon is only \displaystyle \frac{1}{6} of what a full spool holds. Then 6 such sixths make the full spool, which is 6 \times 3 = 18 yards. Both stories look for the missing number in
and both give 18. Dividing by a unit fraction can always be read as “how many fit”. It can be read as sharing only in the “how much in one whole group” form.
Part 3: stories, and why the answer is bigger
A how-many-fit story has three parts: an amount, the size of one piece in the same unit, and the question “how many pieces?” Jumoke has 6 yards of ribbon. Look at what happens to the number of ties as the ties get shorter:
| Length of one tie | Division | Number of ties |
|---|---|---|
| 2 yards | 6 \div 2 | 3 |
| 1 yard | 6 \div 1 | 6 |
| \displaystyle \frac{1}{2} yard | \displaystyle 6 \div \frac{1}{2} | 12 |
When a tie is longer than 1 yard, fewer ties than yards fit, so the answer is less than 6. When a tie is exactly 1 yard, one tie fits in each yard, so the answer is 6. When a tie is shorter than 1 yard, more than one tie fits in every yard, so the answer is more than 6.
So “dividing makes a number smaller” is true only when the piece is bigger than 1. Dividing a number that is not 0 by a unit fraction \displaystyle \frac{1}{b} with b more than 1 always gives a bigger answer, because each whole holds b pieces, and n \times b is more than n. The smaller the piece, the more pieces fit.
Part 4: dividing fractions by counting pieces
Now the amount is a fraction too. Jumoke has \displaystyle \frac{3}{4} of a yard of ribbon, and a small tie for a doll’s bag takes \displaystyle \frac{1}{8} of a yard. How many fit?
Fourths and eighths are different sizes, so first write both amounts in the same size of piece. Every fourth is 2 eighths, so \displaystyle \frac{3}{4} = \frac{6}{8} (Equivalent fractions). Renaming fractions so their pieces match is Common denominators and the least common denominator: multiply the top and the bottom by the same number. Now the ribbon is 6 eighths and each tie is 1 eighth.
Counting the shaded pieces gives the answer:
Check: \displaystyle 6 \times \frac{1}{8} = \frac{6}{8} = \frac{3}{4}.
Once both fractions have the same bottom number, the bottom number only names the size of the piece. The question becomes “how many groups of these pieces fit in those pieces?”, which is a division of the top numbers:
The top numbers may not divide evenly. Then some pieces are left over, too few to make another whole copy. Those left-over pieces are written as a fraction of one copy, and a \div b is the fraction \displaystyle \frac{a}{b} (A fraction as division). Example 8 works one through.
Results
A whole number divided by a unit fraction. For counting numbers n and b,
Fractions with the same bottom number. For counting numbers a, b and c,
Fractions with different bottom numbers are first renamed to a common one.
Checking by multiplying back. If a \div d = q, then q \times d = a.
Size of the answer. For an amount a that is not 0, the answer a \div d is bigger than a when d is less than 1, equal to a when d = 1, and smaller than a when d is more than 1.
Why it is true
The unit fraction rule. One whole is b copies of \displaystyle \frac{1}{b}, since \displaystyle b \times \frac{1}{b} = \frac{b}{b} = 1. Then n wholes are n groups of b copies, which is n \times b copies. It fits the definition, because putting the copies back together gives
The last step divides the top and the bottom by b, which gives an equal fraction. No other number works: one more copy would be more than n, one fewer would be less.
The common denominator rule. The amount \displaystyle \frac{a}{c} is a pieces of size \displaystyle \frac{1}{c}, and \displaystyle \frac{b}{c} is b pieces of the same size. Asking how many copies of \displaystyle \frac{b}{c} fit in \displaystyle \frac{a}{c} is asking how many groups of b pieces fit in a pieces, and that is a \div b. Check with the rule for multiplying fractions (A fraction times a fraction):
dividing the top and the bottom by b. So \displaystyle \frac{a}{b} copies of \displaystyle \frac{b}{c} make exactly \displaystyle \frac{a}{c}. Renaming a fraction to a different bottom number does not change the amount, so the rule works for fractions with different bottoms once they are renamed.
The size of the answer. The copies of d, put together, make a. If d is less than 1, then a copies of d make less than a, so more than a copies are needed. If d is more than 1, then a copies would make more than a, so fewer are needed.
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.
Two yards in fifths
Dyani cuts 2 yards of ribbon into ties that are each \displaystyle \frac{1}{5} of a yard long. How many ties does she get? Use a number line.
SolutionThe amount is 2 yards and each piece is \displaystyle \frac{1}{5} yard, so the question is \displaystyle 2 \div \frac{1}{5}.
Draw a number line from 0 to 2 and mark every fifth. From 0 to 1 there are 5 jumps of \displaystyle \frac{1}{5}, and from 1 to 2 there are 5 more. That is 2 \times 5 = 10 jumps:
Dyani gets 10 ties, and no ribbon is left over, because the last jump lands exactly on 2.
Seven yards without drawing every piece
Haoran has 7 yards of ribbon. Each tie is \displaystyle \frac{1}{4} yard. How many ties can he cut?
SolutionThe question is \displaystyle 7 \div \frac{1}{4}. Drawing 28 little pieces would work, but the picture is not needed. Each yard holds 4 fourths, and there are 7 yards, so
Haoran can cut 28 ties. The rule \displaystyle n \div \frac{1}{b} = n \times b is just the count from the picture, done by multiplying instead of counting one by one.
Count the fits, then multiply back
Find \displaystyle 5 \div \frac{1}{8}, and check the answer by multiplying.
SolutionEach whole holds 8 eighths, so 5 wholes hold 5 \times 8 = 40 of them:
Check: 40 copies of \displaystyle \frac{1}{8} make \displaystyle 40 \times \frac{1}{8} = \frac{40}{8}, and \displaystyle \frac{40}{8} = 5 because 40 \div 8 = 5. Putting the pieces back together gives the 5 we started with, so the answer is right.
One division, two stories
Answer both questions, and explain why they give the same number.
(a) Jumoke cuts 6 yards of ribbon into ties \displaystyle \frac{1}{4} yard long. How many ties does she get?
(b) Jumoke’s 6 yards of ribbon is \displaystyle \frac{1}{4} of what a full spool holds. How many yards does a full spool hold?
Solution(a) This asks how many quarter yards fit in 6 yards. Each yard holds 4 of them, so \displaystyle 6 \div \frac{1}{4} = 6 \times 4 = 24 ties.
(b) This asks how much is in one whole spool when 6 yards fills \displaystyle \frac{1}{4} of it. Four fourths make the whole spool, and each fourth holds 6 yards, so the spool holds 4 \times 6 = 24 yards. This is also \displaystyle 6 \div \frac{1}{4}, read the sharing way.
Both answers are 24 because both stories look for the missing number in
In (a) the 24 is a number of ties and in (b) it is a number of yards, but the division is the same. Notice that 4 \times 6 and 6 \times 4 are equal by the commutative property.
Is there enough ribbon?
The toy store needs 50 ties for a party order. Each tie is \displaystyle \frac{1}{6} yard, and Dyani has 8 yards of ribbon. Is it enough? If not, how much more ribbon is needed?
SolutionThe number of ties in 8 yards is
That is 2 ties short of 50. Two more ties take \displaystyle 2 \times \frac{1}{6} = \frac{2}{6} = \frac{1}{3} yard of ribbon. So 8 yards is not enough, and Dyani needs \displaystyle \frac{1}{3} yard more.
Notice that 48 is much bigger than 8. The ties are shorter than 1 yard, so 6 of them fit in every yard.
Testing an answer by its size
Haoran says \displaystyle 6 \div \frac{1}{3} = 2. Without working it out first, explain why his answer cannot be right. Then find the right answer.
SolutionA third of a yard is shorter than 1 yard, so more than one third fits in every yard of 6 yards. The answer has to be more than 6, and 2 is less. Multiplying back says the same: \displaystyle 2 \times \frac{1}{3} = \frac{2}{3}, not 6.
Haoran found 6 \div 3, the number of 3-yard pieces in 6 yards. The question asks for \displaystyle \frac{1}{3}-yard pieces. Each yard holds 3 of them, so
Check: \displaystyle 18 \times \frac{1}{3} = \frac{18}{3} = 6.
Same bottom number, a tie that is not a unit fraction
The store’s large bags use a tie \displaystyle \frac{2}{9} yard long. Jumoke has \displaystyle \frac{8}{9} yard of ribbon. How many large ties can she cut?
SolutionThe question is \displaystyle \frac{8}{9} \div \frac{2}{9}. Both amounts are already in ninths. The ribbon is 8 ninths, and each tie is 2 ninths. Groups of 2 pieces in 8 pieces:
Jumoke can cut 4 ties. Check: \displaystyle 4 \times \frac{2}{9} = \frac{8}{9}.
A piece left over
Dyani has \displaystyle \frac{2}{3} yard of ribbon left, and a tie is \displaystyle \frac{1}{4} yard. How many ties fit? What part of a tie is left over?
SolutionThe question is \displaystyle \frac{2}{3} \div \frac{1}{4}. Thirds and fourths are different sizes, so rename both in twelfths, since 12 is a multiple of both 3 and 4:
Now the ribbon is 8 twelfths and a tie is 3 twelfths. Groups of 3 pieces in 8 pieces:
Two groups of 3 use 6 pieces, and 2 pieces are left. A whole tie needs 3 pieces, so the leftover is 2 of the 3 pieces a tie needs, which is \displaystyle \frac{2}{3} of a tie:
Dyani can cut 2 whole ties, with \displaystyle \frac{2}{3} of a tie left over. As ribbon, the leftover is \displaystyle \frac{2}{12} = \frac{1}{6} yard. Both are true: \displaystyle \frac{1}{6} of a yard is \displaystyle \frac{2}{3} of a tie. The division counts ties, so it gives \displaystyle \frac{2}{3}.
Check: \displaystyle 2\frac{2}{3} = \frac{8}{3}, and \displaystyle \frac{8}{3} \times \frac{1}{4} = \frac{8}{12} = \frac{2}{3}.
Boundaries
The piece cannot be 0. No number of copies of 0 adds up to an amount that is not 0, so a \div 0 names no number. An amount of 0 is allowed: \displaystyle 0 \div \frac{1}{3} = 0, since no ties fit in no ribbon.
Both amounts must be measured in the same unit. Yards of ribbon divided by yards per tie counts ties. If the ribbon were in yards and the tie in feet, rename one first.
The answer counts copies of the piece, so a leftover is a fraction of the piece, not of the whole or of a yard. In Example 8 the leftover is \displaystyle \frac{2}{3} of a tie and \displaystyle \frac{1}{6} of a yard. Both are true, of different wholes, and the division gives the first.
In a story, a fraction of a copy may not be useful. A store cannot close a bag with \displaystyle \frac{2}{3} of a tie, so the story answer is 2 ties, even though \displaystyle \frac{2}{3} \div \frac{1}{4} is \displaystyle 2\frac{2}{3} (Remainders and what they mean).
The answer is bigger than the amount only when the piece is less than 1. Dividing by a piece of exactly 1 changes nothing, and dividing by a piece bigger than 1 gives a smaller answer.
Where this goes next
Sharing and fitting are the two meanings of dividing with fractions. Deciding which one a story asks for, and writing the division the right way round, is Word problems that divide fractions.
Counting pieces with a common bottom number always works, but renaming every time is slow. A shortcut that skips the renaming, multiplying by the second fraction turned upside down, is Dividing by multiplying by the reciprocal, and Why flipping the second fraction works shows it gives the same answer as counting pieces. Dividing amounts such as \displaystyle 2\frac{1}{2} yards is Dividing mixed numbers.
Exercises
Exercises 1 to 6 go with Part 1, 7 to 12 with Part 2, 13 to 18 with Part 3, 19 to 23 with Part 4, and the reasoning group draws on all four parts.
Part 1: how many unit fractions fit
- \displaystyle 3 \div \frac{1}{5}
- \displaystyle 2 \div \frac{1}{7}
- \displaystyle 5 \div \frac{1}{6}
- Use a number line from 0 to 2 to find \displaystyle 2 \div \frac{1}{3}. Say where the tick marks go.
- Haoran has 9 yards of ribbon and cuts ties \displaystyle \frac{1}{2} yard long. How many ties does he get?
- Describe a tape diagram for the question “how many thirds of a yard are in 3 yards?”, and answer it.
Part 2: multiplying back, and two meanings
- Find \displaystyle 4 \div \frac{1}{5} and check it by multiplying.
- Find \displaystyle 7 \div \frac{1}{3} and check it by multiplying.
- Haoran cut a ribbon into ties \displaystyle \frac{1}{8} yard long and got 24 ties with nothing left over. How long was the ribbon?
- A whole number divided by \displaystyle \frac{1}{6} gives 42. What is the number?
- Jumoke says \displaystyle 2 \div \frac{1}{4} = \frac{1}{8}. Test her answer by multiplying back, and find the right answer.
- Dyani’s 5 yards of ribbon is \displaystyle \frac{1}{3} of a full spool. How many yards does a full spool hold? Write the division, and write a how-many-fit story for the same division.
Part 3: stories, and the size of the answer
- Dyani has 6 yards of ribbon, and each gift bag needs a tie \displaystyle \frac{1}{5} yard long. How many bags can she close?
- Haoran needs 25 ties, each \displaystyle \frac{1}{8} yard. He has 4 yards of ribbon. Is it enough? How many ties are spare or missing?
- Write a toy store story for \displaystyle 2 \div \frac{1}{10}, and answer it.
- Put 10 \div 5, 10 \div 1 and \displaystyle 10 \div \frac{1}{5} in order from smallest to largest, first by reasoning about the size of the piece, then by working each one out.
- Which is larger, \displaystyle 2 \div \frac{1}{6} or \displaystyle 2 \div \frac{1}{8}? By how much? Explain why the smaller piece gives the larger answer.
- A roll of 12 yards of ribbon is cut into bows \displaystyle \frac{1}{3} yard long. The store uses 30 bows. How many bows are left, and how many yards of ribbon is that?
Part 4: dividing fractions by counting pieces
- \displaystyle \frac{4}{5} \div \frac{1}{5}
- \displaystyle \frac{9}{10} \div \frac{3}{10}
- \displaystyle \frac{1}{2} \div \frac{1}{8}
- \displaystyle \frac{3}{4} \div \frac{1}{6}. Say how many whole copies fit and what is left over.
- Jumoke has \displaystyle \frac{7}{8} yard of ribbon, and a large tie takes \displaystyle \frac{3}{8} yard. How many ties fit? What fraction of a tie is left over, and how much ribbon is that?
Reasoning
- Explain, with a tape diagram in words, why \displaystyle n \div \frac{1}{b} = n \times b for any counting numbers n and b.
- Haoran says “dividing always makes a number smaller.” Explain when he is right and when he is wrong, with one example of each, and say what happens when you divide by 1.
Answers
- 15. Each whole holds 5 fifths: 3 \times 5 = 15.
- 14. Each whole holds 7 sevenths: 2 \times 7 = 14.
- 30. Each whole holds 6 sixths: 5 \times 6 = 30.
- 6. Put tall ticks at 0, 1 and 2, and short ticks at every third between them. Jumping \displaystyle \frac{1}{3} at a time from 0 takes 3 jumps to reach 1 and 3 more to reach 2, so 2 \times 3 = 6 jumps.
- 18 ties. \displaystyle 9 \div \frac{1}{2} = 9 \times 2 = 18.
- 9. Draw a tape 3 yards long, mark the yards, and cut each yard into 3 equal pieces. There are 3 \times 3 = 9 pieces, so \displaystyle 3 \div \frac{1}{3} = 9.
- 20. Check: \displaystyle 20 \times \frac{1}{5} = \frac{20}{5} = 4.
- 21. Check: \displaystyle 21 \times \frac{1}{3} = \frac{21}{3} = 7.
- 3 yards. Put the ties back together: \displaystyle 24 \times \frac{1}{8} = \frac{24}{8} = 3. Check: \displaystyle 3 \div \frac{1}{8} = 3 \times 8 = 24.
- 7. The number times 6 is 42, and 7 \times 6 = 42. Check: \displaystyle 42 \times \frac{1}{6} = \frac{42}{6} = 7.
- She is wrong: \displaystyle \frac{1}{8} \times \frac{1}{4} = \frac{1}{32}, not 2. The right answer is 8, since each whole holds 4 fourths and 2 \times 4 = 8. Check: \displaystyle 8 \times \frac{1}{4} = \frac{8}{4} = 2.
- 15 yards. The division is \displaystyle 5 \div \frac{1}{3}: three thirds make the spool, each holding 5 yards, so 3 \times 5 = 15. One how-many-fit story: Dyani cuts 5 yards into ties \displaystyle \frac{1}{3} yard long; she gets 5 \times 3 = 15 ties.
- 30 bags. \displaystyle 6 \div \frac{1}{5} = 6 \times 5 = 30.
- Enough, with 7 spare. \displaystyle 4 \div \frac{1}{8} = 4 \times 8 = 32 ties, and 32 - 25 = 7.
- One story: the store has 2 yards of ribbon and cuts ties \displaystyle \frac{1}{10} yard long for small bags; how many ties? \displaystyle 2 \div \frac{1}{10} = 2 \times 10 = 20 ties. Any story works if it asks how many pieces of \displaystyle \frac{1}{10} fit in 2 of the same unit.
- 10 \div 5, then 10 \div 1, then \displaystyle 10 \div \frac{1}{5}. A piece of 5 is bigger than 1, so fewer than 10 fit; a piece of 1 fits exactly 10 times; a piece of \displaystyle \frac{1}{5} is less than 1, so more than 10 fit. The values are 2, 10 and 50.
- \displaystyle 2 \div \frac{1}{8} is larger, by 4. The answers are 2 \times 6 = 12 and 2 \times 8 = 16, and 16 - 12 = 4. An eighth is a smaller piece than a sixth, so more eighths fit in each whole.
- 6 bows, which is 2 yards. The roll makes \displaystyle 12 \div \frac{1}{3} = 12 \times 3 = 36 bows, and 36 - 30 = 6. The leftover ribbon is \displaystyle 6 \times \frac{1}{3} = \frac{6}{3} = 2 yards.
- 4. Four fifths is 4 pieces of \displaystyle \frac{1}{5}: 4 \div 1 = 4.
- 3. Both in tenths: 9 \div 3 = 3. Check: \displaystyle 3 \times \frac{3}{10} = \frac{9}{10}.
- 4. \displaystyle \frac{1}{2} = \frac{4}{8}, so \displaystyle \frac{4}{8} \div \frac{1}{8} = 4 \div 1 = 4.
- \displaystyle 4\frac{1}{2}: 4 whole copies and half a copy left over. In twelfths, \displaystyle \frac{3}{4} = \frac{9}{12} and \displaystyle \frac{1}{6} = \frac{2}{12}, so the answer is \displaystyle 9 \div 2 = \frac{9}{2} = 4\frac{1}{2}. Four copies use 8 pieces, and the 1 piece left is half of the 2 a copy needs. Check: \displaystyle \frac{9}{2} \times \frac{1}{6} = \frac{9}{12} = \frac{3}{4}.
- 2 ties, with \displaystyle \frac{1}{3} of a tie left over, which is \displaystyle \frac{1}{8} yard of ribbon. Both in eighths: \displaystyle 7 \div 3 = 2\frac{1}{3}. Two ties use 6 eighths, and the 1 eighth left is 1 of the 3 eighths a tie needs. Check: \displaystyle \frac{7}{3} \times \frac{3}{8} = \frac{21}{24} = \frac{7}{8}.
- Draw a tape n wholes long and mark each whole. Cut each whole into b equal pieces; each piece is \displaystyle \frac{1}{b}, because b of them make 1. Every whole holds b pieces and there are n wholes, so there are n groups of b pieces, which is n \times b pieces. That count is the number of copies of \displaystyle \frac{1}{b} that fit in n. Multiplying back agrees: \displaystyle (n \times b) \times \frac{1}{b} = \frac{n \times b}{b} = n.
- He is right when the piece is bigger than 1: 12 \div 4 = 3, smaller than 12, because each 4-yard piece uses more than one yard. He is wrong when the piece is less than 1: \displaystyle 5 \div \frac{1}{2} = 10, bigger than 5, because two half-yard pieces fit in every yard. Dividing by 1 changes nothing, since exactly one piece fits in each whole: a \div 1 = a. (Any examples of these kinds work, as long as the amount is not 0.)