Math 208, Section 4.3 Solutions: Adding and Subtracting Fractions
2 3
+ , explain why we add fractions the way we do. In other words, what
3 4
is the logic behind this procedure? Explain your answer.
1. Using the example
I think that we can explain the procedure for adding fractions with different denominators by appealing
to the “apples-and-oranges” cliche: namely, that you can’t add apples and oranges. BUT....if you call
them both fruits, then you can add them.
[I can’t add 3 apples to 2 oranges but I can add 3 fruits to 2 fruits].
Another example: to add 3 yards and 2 ½ feet I can change them both to inches and add 108 inches
to 30 inches. But I wouldn’t add 3 to 2 ½ and expect to get anything useful.
In other words....to add quantities, the numbers involved must be expressed in terms of the same unit.
In this respect, fourths and thirds are just like feet and yards; I can’t add 2 thirds to 3 fourths directly
since they aren’t expressed in terms of the same unital fraction. But if I change them so that they are
expressed in terms of the same unital fraction, I can add the two.
So:
2 3 2 × 4 3× 3 8
9 17
+ =
+
=
+
=
3 4 3 × 4 4 × 3 12 12 12
5
2. Using the example 3 , describe the procedure for turning a mixed number into an improper
6
fraction and explain in your own words why this procedure makes sense. What is the logic
behind this procedure? Draw pictures to support your explanation.
5
into an improper fraction we start with 3 wholes and 5/6 of a whole. If we
6
divide up the 3 wholes into 6ths, then each whole will consist of six pieces. There are three wholes, so
we will have 18 sixths from the 3 wholes. But we also had 5 sixths originally, so we have a total of
18+5 = 23 sixths. This explains why we multiply the whole-number part times the denominator of the
fraction: the denominator tells us how many pieces we need to break each whole up into. So—if there
are several wholes—we multiply the number of wholes by the number of pieces in each whole, which
means multiplying by the denominator.
i. In words: to turn 3
ii. In pictures:
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one whole
three and five sixths wholes
still three and five sixths wholes but
now we can see that there are 23
sixths in the part
3
2
5. Show how to solve 5 + 1 in two different ways. In each case, express your answer as a
4
3
mixed number. Explain why both of your methods are legitimate.
3
2
3
2
3 2
17
5
5
One way: 5 + 1 = 5 + + 1 + = 5 + 1 + + = 6 +
= 6 +1+
=7
4
3
4
3
4 3
12
12
12
In this way I used the fact that ¾ + 2/3 = 17/12, which we got in problem 1.
3
2 23 5 23 × 3 5 × 4 69 20 89
5
Another way: 5 + 1 =
+ =
+
=
+
=
=7
4
3 4 3 4 × 3 3 × 4 12 12 12
12
In this way I turned the mixed numbers into improper fractions and then expressed each improper
fraction with the common denominator 12.
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2
3
6. Show how to solve 4 − 1 in two different ways. In each case, express your answer as a mixed
3
4
number. Explain why both of your methods are legitimate.
2
3
2
3
2 3
2 × 4 3× 3
8 9
1
11
One way: 4 − 1 = 4 + − 1 − = 4 − 1 + − = 3 +
−
= 3+ −
= 3−
=2
3
4
3
4
3 4
3× 4 4× 3
12 12
12
12
In this way, I did not convert to improper fractions, but I did have to get common denominators.
2
3 4 × 3 + 2 1 × 4 + 3 14 7 14 × 4 7 × 3 56 21 35
11
Another way: 4 − 1 =
−
=
− =
−
=
−
=
=2
3
4
3
4
3 4 3 × 4 4 × 3 12 12 12
12
In this way I turned the mixed numbers into improper fractions and then expressed each improper
fraction with the common denominator 12.
2 2 4
+ = and uses the picture in Figure 4.28 as evidence. Discuss what is wrong
3 3 6
with John’s reasoning. What underlying misconception does John have? Don’t just explain how
to do the problem correctly; explain why John’s reasoning is faulty.
8. John says
This is a very common mistake that children make, and—as usual—it has to do with the question:
What is the whole? John is correct in saying that the two pieces added to the two pieces make four
pieces. But he also adds the two wholes and pretends that it is a new whole, consisting of six pieces.
This means that his answer is expressed in terms of a different whole from the original one.
Remember that in adding and subtracting, we want to express both given numbers and the answer in
terms of the same whole.
So John should recognize that the small squares in his picture are each one third of the whole. So his
answer should be four thirds (of the original whole).
Actually, if we were willing to let John use a different whole for the answer, he could say that his
answer is 4/6 of (the pair of wholes). And this would be entirely correct, if not expressed in the usual
way.
15a. Write and solve a story problem for
3 2
+ .
4 3
There were two large pizzas delivered to the fraternity. Charles ate ¾ of one pizza and 2/3 of the other.
How much of a large pizza did Charles eat?
We have solved the problem ¾ + 2/3 in problem 1; the answer is 17/12.
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15b. Write and solve a story problem for
3 2
− .
4 3
There was ¾ of a large pizza left on the table. From that amount, Charles ate 2/3 of a pizza. How
much of a pizza was left after Charles finished?
Answer: 1/12 of a pizza. (9/12 – 8/12).
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