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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 รท 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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