# Dividing Fractions

**Class Activity 0C: "How Many in One Group?" Frac-
tion Division Problems
Class Activity 0D: Are These Division Problems?
Exercises for Section 0.1 on Dividing Fractions
**

1. Write a "how many groups?" story problem for Use the story

problem and a diagram to help you solve the problem .

2. Write a "how many in one group?" story problem for Use the

situation of the story problem to help you explain why the answer is

3. Annie wants to solve the division problem by using the following

story problem:

I need cup of chocolate chips to make a batch of cookies.

How many batches of cookies can I make with of a cup of

chocolate chips?

Annie draws a diagram like the one in Figure 3. Explain why it would

be easy for Annie to misinterpret her diagram as showing that

How should Annie interpret her diagram so as to conclude that

4. Which of the following are solved by the division problem For

those that are, which interpretation of division is used? For those that

are not, determine how to solve the problem, if it can be solved.

(a) of a bag of jelly worms make a cup. How many cups of jelly

worms are in one bag?

1/2 cup makes one batch |
1/4 cup left |

Figure 3: How Batches of Cookies Can We Make With
of a Cup of Chocolate

Chips if 1 Batch Requires
Cup of Chocolate
Chips?

(b)
of a bag of jelly
worms make
a cup. How many bags
of jelly

worms does it take to make one cup?

(c) You have of a bag
of jelly worms and a recipe that calls for
of

a cup of jelly worms. How many batches of your recipe can you

make?

(d) You have of a cup
of jelly worms and a recipe that calls for
of

a cup of jelly worms. How many batches of your recipe can you

make?

(e) If of a pound of
candy costs of a
dollar, then how many pounds

of candy should you be able to buy for 1 dollar?

(f) If you have of a
pound of candy and you divide the candy in
,

then how much candy will you have in each portion?

(g) If of a pound of
candy costs $1, then how many dollars should

you expect to pay for
of a pound of candy?

5. Frank, John, and David earned $14 together. They want to divide it

equally, except that David should only get a half share, since he did half

as much work as either Frank or John did (and Frank and John worked

equal amounts). Write a division problem to find out how much Frank

should get. Which interpretation of division does this story problem

use?

6. Bill leaves a tip of $4.50 for a meal. If the tip is 15% of the cost of

the meal, then how much did the meal cost? Write a division problem

to solve this. Which interpretation of division does this story problem

use?

7. Compare the arithmetic needed to solve the following
problems.

(a) What fraction of a
cup measure is filled when we pour in cup

of water?

(b) What is one quarter of
cup?

(c) How much more is
cup than cup?

(d) If cup of water fills
of a plastic
container, then how much

water will the full container hold?

8. Use the meanings of multiplication and division to solve the following

problems.

(a) Suppose you drive 4500 miles every half year in your car. At the

end of years, how many miles will you have
driven?

(b) Mo used 128 ounces of liquid laundry detergent in
weeks. If

Mo continues to use laundry detergent at this rate, how much will

he use in a year?

(c) Suppose you have a 32 ounce bottle of weed killer concentrate.

The directions say to mix two and a half ounces of weed killer

concentrate with enough water to make a gallon. How many gallons

of weed killer will you be able to make from this bottle?

9. The line segment below is
of a unit long. Show
a line segment that

is of a unit long. Explain how this problem
is related to fraction

division.

unit

**Answers To Exercises For Section 0.1 on Dividing Frac-
tions**

1. A simple "how many groups?" story problem for is "how many

of a cup of water are in 1 cup of water?" Figure 4 shows 1 cup of water

and shows of a cup of water shaded. The shaded portion is divided

into 5 equal parts and the full cup is 7 of those parts. So the full cup

is of the shaded part. Thus there are of of a cup of water in 1

cup of water, so

1 cup | of a cup |
each piece is of the shaded portion |

Figure 4: Showing Why
by Considering How Many
of a Cup of

Water are in 1 Cup of Water

2. A "how many in one group?" story problem for
is "if 1 ton of

dirt fills a truck
full, then how many tons of dirt will be needed to

fill the truck completely full?". We can see that this is a "how many

in one group?" type of problem because the 1 ton of dirt fills
of a

group (the truck) and we want to know the amount of dirt in 1 whole

group. Figure 5 shows a truck bed divided into 4 equal parts with 3

of those parts filled with dirt. Since the 3 parts are filled with 1 ton

of dirt, each of the 3 parts must contain
of a ton of dirt. To
ll the

truck completely, 4 parts, each containing
of a ton of dirt are
needed.

Therefore the truck takes tons of dirt to
fill it completely, and

so

the 1 ton of dirt is divided equally among 3 parts |
truck bed |
4 parts are needed to fill the truck; each part is 1/3 of a ton, so 4/3 tons of dirt are needed to fill the truck |

Figure 5: Showing Why
by Considering How Many Tons of Dirt

it Takes to Fill a Truck if 1 Ton Fills it
Full

3. Annie's diagram shows that she can make 1 full batch of cookies from

her
of a cup of chocolate
chips and that cup of
chocolate chips will

be left over. Because
cups of chocolate chips are left over, it would

be easy for Annie to misinterpret her picture as showing

But the answer to the problem is supposed to be the number of batches

Annie can make. In terms of batches , the remaining
cup of chocolate

chips makes of a batch
of cookies. We can see this because 2 quarter-cup

sections make a full batch, so each quarter-cup section makes
of

a batch of cookies. So by interpreting the remaining
cup of chocolate

chips in terms of batches , we see that Annie can make
batches of

chocolate chips, thereby showing that not

4. (a) This problem can be rephrased as "if
of a cup of jelly
worms

fill of a bag, then
how many cups fill a whole bag?", therefore

this is a "how many in one group?" division problem illustrating

, not
Since there are
of a cup of

jelly worms in a whole bag.

(b) This problem is solved by ,according to
the "how many in

each group?" interpretation. A group is a cup and each object is

a bag of jelly worms.

(c) This problem can't be solved because you don't know how many

cups of jelly worms are in
of a bag.

(d) This problem is solved by , according to
the "how many

groups?" interpretation. Each group consists of
of a cup of jelly

worms.

(e) This problem is solved by , according to
the "how many in

one group?" interpretation. This is because you can think of the

problem as saying that
of a pound of candy fills
of a group

and you want to know how many pounds fills 1 whole group.

(f) This problem is solved by , not
. It is dividing in half,

not dividing by .

(g) This problem is solved by , according to
the "how many

groups?" interpretation because you want to know how many

pounds are in of a
pound. Each group consists of
of a pound

of candy.

5. If we consider Frank and John as each representing one group, and

David as representing half of a group, then the $14 should be dis-

tributed equally among
groups. Therefore, this is a "how many in

one group" division problem. Each group should get

dollars. Therefore Frank and John should each get $5.60
and David

should get half of that, which is $2.80.

6. According to the "how many in one group?" interpretation, the problem

is solved by $4.50 ÷ 0.15 because $4.50 fills 0.15 of a group and we

want to know how much is in 1 whole group. So the meal cost

7. Each problem, except for the first and last, requires
different arithmetic

to solve it.

(a) This is asking:
equals what times ? We
solve this by calculating

, which is
. We can also think
of this as a division problem

with the "how many groups?" interpretation because we want to

know how many of a
cup are in of a cup.
According to the

meaning of division, this is .

(b) This is asking: what is
of
? We solve this by
calculating

(c) This is asking: what is The answer is
which happens to

be the same answer as in part (b), but the arithmetic to solve it

is different.

(d) Since cup of water
fills of a plastic
container, the full container

will hold 3 times as much water, or of a
cup. We can

also think of this as a division problem with the "how many in one

group?" interpretation.
cup of water is put into
of a group.

We want to know how much is in one group. According to the

meaning of division it's , which again is
equal to .

8. (a) The number of
years in years is
There will be that

many groups of 4500 miles driven. So after
years you will have

driven

miles.

(b) Since one year is 52 weeks there are
groups of weeks

in a year. Mo will use 128 ounces for each of those groups, so Mo

will use

ounces of detergent in a year.

(c) There are groups of
ounces in 32 ounces. Each of those

groups makes 1 gallon. So the bottle makes
gallons

of weed killer.

9. One way to solve the problem is to determine how many
units are in

units. This will tell us how many of the
unit long segments to
lay end

to end in order to get the unit long
segment. Since

there are segments of length
units in a segment of
length units.

So you need to form a line segment that is 3 times as long as the one

pictured, plus another
as long:

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