Pre Algebra
Section 2.2
Unit 2 - Fractions
Section 2 – Multiplying and Dividing
Multiplying Fractions
ଵ
ଵ
Consider ଶ ∙ ଷ. In words this is read “ half of a third”. So lets think of a shape that is broke into thirds
Now lets cut it in half.
Notice the resulting size. It now takes 6 pieces to make a whole.
1 1 1
∙ =
2 3 6
Multiplication of fractions
௔
௕
௖
௔∙௖
∙ ௗ = ௕∙ௗ ,
݊‫ ≠ ܾ ݁ݐ݋‬0 ܽ݊݀ ݀ ≠ 0
Example 1)
૛
૜
ࡹ࢛࢒࢚࢏࢖࢒࢟ − ૞ ∙ ૞
ଶ ଷ
ଶ∙ଷ
Recall: A negative times a
positive will give you a
negative answer
଺
− ହ ∙ ହ = − ହ∙ହ = − ଶହ
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Pre Algebra
Section 2.2
Example 2)
૝
ࡹ࢛࢒࢚࢏࢖࢒࢟ − ∙ ቀ−
૞
ସ
− ହ ∙ ቀ−
=
ଵ଴
ଷ
૚૙
૜
ቁ
ቁ
ିସሺିଵ଴ሻ
ହ∙ଷ
ସ଴
= ଵହ
We need to reduce to lowest terms.
=
ଶ∙ଶ∙ଶ∙ହ
=
ଶ∙ଶ∙ଶ
Here we use the prime factorization to help us.
ଷ∙ହ
Cancel common factors.
ଷ
଼
=ଷ
Example 2) Revisted with cross canceling.
Another way to get to the answer in lowest terms is to cross cancel – This keeps the numbers smaller
and easier to work with!
ସ
ଵ଴
ସ
ଶ∙ହ
ସ
ଶ
− ହ ∙ ቀ−
− ହ ∙ ቀ−
ଷ
ቁ
ଷ
− ଵ ∙ ቀ− ଷቁ
ିସ∙ିଶ
ଵ∙ଷ
=
ቁ
Notice that both fractions are in lowest terms.
There is a factor of 5 on the top and on the bottom – they cancel.
Notice there is a 1 left behind when we canceled the 5.
଼
ଷ
Canceling common factors can shorten the work – I showed every step however the “work” can be
much more condensed.
We did get the same answer. Notice we left the answer as an “improper” fraction in both cases. We
ଶ
could have also given the mixed number answer 2 ଷ.
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Pre Algebra
Section 2.2
Multiplying with Mixed Numbers
When multiplying with mixed numbers we begin by making them improper fractions so we can multiply
the same as in the previous examples.
Example 3)
૝
૛
ࡹ࢛࢒࢚࢏࢖࢒࢟ − ૛ ∙ ૜ ૞
૜
ଶ
ସ
଼ ଵଽ
−2 ଷ ∙ 3 ହ = − ଷ ∙
=−
଼∙ଵଽ
ଷ∙ହ
=−
We must turn mixed numbers in to improper fractions.
ହ
ଵହଶ
Multiplying straight across.
ଵହ
ଶ
= −10 ଵହ
Example 4)
૛
૜
ࡹ࢛࢒࢚࢏࢖࢒࢟ − ૛ ૜ ∙ ቀ−૜ ૝ቁ
ଷ
ଶ
−2 ଷ ∙ ቀ−3 ସቁ
଼
= −ଷ∙ −
=−
ଶ∙ସ
ଷ
ଵହ
We must make mixed numbers improper fractions.
ସ
∙−
ଷ∙ହ
We can cross cancel common factors.
ସ
ହ
ଶ
= − ∙ −ଵ
ଵ
= −2 ∙ −5
= 10
Example 5)
૞
ࡹ࢛࢒࢚࢏࢖࢒࢟ ૟ ∙ ૛૚
ହ
6 ∙ ଶଵ
଺
ହ
= ଵ ∙ ଶଵ
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We begin by writing 6 as a fraction.
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Pre Algebra
=
ଶ∙ଷ
Section 2.2
ହ
∙ ଷ∙଻
ଵ
Then cross cancel any common factors.
ହ
ଶ
=ଵ∙଻
ଶ∙ହ
= ଵ∙଻
=
Multiply straight across.
ଵ଴
଻
Example 6)
૛
ࡹ࢛࢒࢚࢏࢖࢒࢟
࢞
૜
∙࢞
Even though there are variables we still follow the same rules - we still multiply across the top for the
numerator and across the bottom for the new denominator.
ଶ
௫
ଷ
∙௫
ଶ∙ଷ
= ௫∙௫
Multiplying straight across.
଺
= ௫మ
Dividing Fractions
Division means to multiply by the reciprocal
ܽ ܿ ܽ d ܽ∙d
÷ = ∙ =
,
ܾ ݀ b c ܾ∙ܿ
݊‫ ≠ ܾ ݁ݐ݋‬0 , ݀ ≠ 0 ܽ݊݀ ܿ ≠ 0
Example 7)
ࡰ࢏࢜࢏ࢊࢋ
ଷ
ସ
૜
૝
ૢ
÷ ૜૛
ଽ
÷ ଷଶ
ଷ
=ସ∙
ଷଶ
ଽ
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To divide we multiply by the reciprocal.
Notice it is the second fraction that flips.
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Pre Algebra
Section 2.2
ଷ∙ଵ ସ∙଼
= ସ∙ଵ ∙ ଷ∙ଷ
ଵ
Look for common factors to cross cancel.
଼
=ଵ∙ଷ
Multiply straight across.
଼
=ଷ
Example 8)
૚
ࡰ࢏࢜࢏ࢊࢋ ૛ ૛ ÷ ૜
ଵ
2ଶ ÷ 3
ହ
=ଶ÷
ହ
ଷ
We need to turn all numbers into improper fractions.
ଵ
ଵ
= ∙ଷ
ଶ
=
ହ
଺
Applications
A triangle can be formed by cutting a rectangle(or parallelogram) in half – so it has exactly half the area
of a rectangle.
h
‫=ܣ‬
ଵ
ܾ∙ℎ
‫=ܣ‬
ଵ
ܾ∙ℎ
ଶ
b
h
ଶ
Notice the b and h have to meet at a 90 degree angle.
b
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Pre Algebra
Section 2.2
Example 9)
Find the Area of a triangle whose base is 10m and whose height is 3m.
‫=ܣ‬
ଵ
ܾ∙ℎ
‫=ܣ‬
ଵ
∙ 10 ∙ 3
ଶ
ଶ
‫ =ܣ‬5∙3
‫ = ܣ‬15
The area is 15݉ ଶ
Example 10)
How big of a serving would 5 people get if they split a 7 oz bag of potato chips?
This is a division problem because we need to divide up the chips, so
7÷5
଻
=ଵ÷
଻
ହ
ଵ
ଵ
=ଵ∙ହ
଻
=ହ
଻
Each person gets ହoz.
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Pre Algebra
Section 2.2
Exercise 2.2
NAME:___________________________________
Multiply
ଶ
1.
ଷ
ଵ
ଶ
∙ହ
ସ
ଵ
4. − ହ ∙ − ଶ
7.
ସ
ହ
10.
∙−
ଵ଴
ଵଶ
ଶ଴
ଷହ
∙
ଵ
2. − ହ ∙ ଷ
5.
ଶ
∙ −ଷ
ହ
6.
ଵ
8. − ଶ ∙ ଵ଴
ଶ
11. −
ଷ
ଵ
ଽ
ଵଵ
3.
ଶ
13. −3 ∙ −
ଶ
ଷ
ଵ
16. 1 ଶ ∙ −4
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ଷଶ
଻
ଶଵ
∙ ଶ଴
ସ
ଵ
14. −1 ହ ∙ −3 ଷ
ଶ
17. −2 ∙ − ହ
65
଻ ସ
ଽ
∙ହ
଻
ଷ
∙ ଵସ
ଵଶ
9. −
ଵହ
∙ −ହ
12. −
ଵ଴
∙ଷ
ଶ
ଷ
ଶ
ଵ
ଶ
ଵ
15. 2 ହ ∙ −1 ସ
ଶ
18. −5 ∙ 3 ଻
Milano
Pre Algebra
Section 2.2
Divide
ଶ
19.
ହ
ଵ
ଷ
÷ଷ
ସ
ଵ
22. − ହ ÷ ቀ− ଻ቁ
25. −
ଵହ
ଶ
÷
ଵ
ଶ
20. − ଻ ÷ ଷ
ଵ଴
23.
ଵ
28. −2 ହ ÷ −3 ଷ
ଶ
ଽ
÷ ቀ− ଷቁ
ଵଵ
26. −
ଷ
21.
ଵ଼
ହ
଻
ଵଵ
଻
24.
଼ଵ
÷ ଶ଴
ଷ
27. −
ଷ
ଶ
÷ହ
ଷହ
÷ ଵ଼
ଵ଴
ଷ
ଵ
÷ଷ
ଶ
29. 1ହ ÷ −2
30. −5 ÷ 3 ଻
Multiply
31.
௫
௫
∙
ଷ ଺
32.
௫మ
ସ
଺
∙௫
33.
ଶ
௫
ଷ
∙௫
ଵ
The area of a triangle can be found by the formula ‫ = ܣ‬ଶ ܾℎ.
Find the area of the shape with the given base (b) and height (h)
34. b =10in and h = 4in
h
b
35. b = 3ft and h = 5ft
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Pre Algebra
Section 2.2
Complete the following by using fractions and leaving your answer as a mixed number.
ଵ
36. A bottle of Ketch up contains 40 oz of Ketchup. How many ଶ oz servings can you get out of one
bottle?
37. A recipe that serves 6 people needs to be doubled to serve a dinner party of 12. The original recipe
ଶ
calls for 1 ଷ cups of flour. How much flour should you use to double the recipe?
38. A bag of pretzels contains 16 oz. If you want to give 20 people equal servings how many ounces
should you give each person?
ଵ
39. A recipe for 15 servings calls for 2 ଷ ܿ‫ ݏ݌ݑ‬of sugar. If you are cutting the recipe in half, how much
sugar should you use?
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Pre Algebra
Section 2.2
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