Solving Proportions Through Multiple Representations A proportion is an equation that states that two ratios are equivalent. 3 9
Numbers: = : The proportion is read, “3 is to 5 as 9 is to 15.” 5 15
a c
Algebra: = where b and d are nonzero numbers. b d
Solving proportions through multiple representations will develop student understanding of proportions and develop relational and flexible thinking. Ask students, “What is a proportion?” This is a great opportunity for a think, pair, share. Accept reasonable responses. “A proportion is the equality of ratios.” 8 c
Example 1: Solve the proportion = . Have students choral respond, “8 is to 3 as 3 18
c is to 18.” 8 c
3 = 18
8 6 c
• =
3 6 18 48 = c
18 18
∴c = 48
To solve the proportion, look for a relationship. Because 3 × 6 = 18 , multiply the fraction by the 6
equivalent form of 1= to solve 6
for c. 1
8 c
=
3 18
8
c
18i = 18i
3
18
2i3i3i2i2i2
c
= 18i
3
18
2i3i2i2i2 = c
48 = c
Multiply both sides of the proportion by the LCM of the denominators to solve. Page 1 of 7 8 c
=
3 18
3ic = 8i18
3c = 144
3c 144
=
3
3
3i48
c=
3
c = 48
Cross Products Property: a c
If = where b and d are b d
nonzero numbers, then ad = bc . MCC@WCCUSD 12/11/13 5 y
You Try #1: Solve the proportion = three different ways, as in Example 1. 2 10
4 x
Have students choral respond, “4 is to 5 as x Example 2: Solve the proportion =
. is to 15.” 5 15
4 x
4 x
4
x
=
=
=
5 15
5 15
5 15
4 3 x
4 4+4+4
i =
=
5 3 15
5 5+5+5
12 x
=
15 15
∴x = 4 + 4 + 4
∴ x = 12
x = 12
Using extended ratios, we can see that “4 is to 5 as 12 Using the rules of equality If two ratios are equal and is to 15.” and decomposition, “one 4 is their denominators are ∴
x
=
12
to 5 as 3 groups of 4 are to 3 the same, their groups of 5.” numerators must also be the same. 5
f
You Try #2: Solve the proportion =
three different ways, as in Example 2. 7 28
Look for choral response 33
opportunities when simplifying . 33
6
22
= . Example 3: Solve the proportion 22 k
33 6
=
33 6
22 k
=
3i11 6
22 k
=
2i11 k
3i11 6
If two ratios are =
3 6
2i11 k
By simplifying first, equal, then their =
2 k
proportions can be 3 6
reciprocals are =
2
k
solved m
uch e
asier. equal. 2 k
=
3 6
3 2 6
i =
2
k
2 2 k
6i = 6i
3
6
6 6
=
12
4 k
=k
3
∴k = 4
4=k
Page 2 of 7 MCC@WCCUSD 12/11/13 You Try #3: Solve the proportion Example 4: Solve the proportion 3 s
=
using bar models. 4 16
3 s
=
4 16
0 1 2 3 4 28 12
=
by simplifying first. 35 x
0 4 8 12 16 ∴ s = 12
3 s
=
4 16
0 1 2 4 8 0 3 12 4 4 16 16 ∴ s = 12
Bar models are great visuals that show the relationships between parts and whole and the equality of each ratio. You Try #4: Solve the proportion u
4
= using a bar model. 28 7
Example 5: Set up a proportion to solve the Word Problem. At a high school basketball game, the ratio of females to males in attendance is 2:3. If there are 120 female spectators in the gymnasium, what is the number of total spectators? Page 3 of 7 MCC@WCCUSD 12/11/13 Set up a proportion to find the number of boys and then add the number of boys and girls together to determine the total number of spectators. 2 120
=
3
x
2 60 120
• =
180 + 120
3 60
x
120 120
= 300
=
180
x
∴ there are 300 total spectators
∴ x = 180
So there are 180 boys.
OR 0 0 180 + 120
1 60 ∴ there are 180 boys = 300
2 ∴ there are 300 total spectators
120 180 3 You Try #5: Set up a proportion to solve the Word Problem. A survey of sixth graders found that the ratio of students who prefer rock to hip-­‐hop is 2 to 7. The total number of sixth graders surveyed was 72. How many of the students surveyed prefer rock? Extra Problems: Solve each proportion three different ways. 12 d
48 24
4. =
=
1. 16 40
30
x
66 11
c 20
= 5. 2. =
x 15
9 45
4
x
3. =
7 42
1
Page 4 of 7 MCC@WCCUSD 12/11/13 Answers to Extra Problems: 1. x = 15 2. c = 4 3. x = 24 4. d = 30 5. x = 90 Worked-­Out You Tries: Explanation of Cross Products Property: a c
=
b d
! a$
! a$
bd # & = bd # &
" b%
" d%
abd cbd
=
b
d
ad = cb
You Try #1: Solve the proportion 5 y
=
2 10
5 5 y
• =
2 5 10 25 y
=
10
10
∴ 25 = y
To solve the proportion, look for a relationship. Because 2 × 5 = 10 , multiply the fraction by the 5
equivalent form of 1 = to 5
solve for y. 1
5 y
= three different ways. 2 10
5 y
=
2 10
5
y
10i = 10i
2
10
2i5i5 10y
=
2
10
5i5 = y
25 = y
Multiply both sides of the proportion by the LCM of the denominators to solve. Page 5 of 7 b and d must be nonzero numbers. 5 y
=
2 10
2iy = 5i10
2y = 50
2y 50
=
2
2
y = 25
Cross Product Property: a c
If = where b and d are b d
nonzero numbers, then ad = bc . MCC@WCCUSD 12/11/13 You Try#2: Solve the proportion 5
f = 7 28
5 5 + 5 + 5 + 5
= 7 7+7+7+7
∴ f = 5 + 5 + 5 + 5
f = 20
of equality and Using the rules decomposition, one 5 is to 7 as 4 groups of 5 a re to 4 groups of 7. 5
f
=
7 28
5
f
=
7 28
5 4
f
i =
7 4 28
20
f
=
28 28
∴ f = 20
Using extended ratios, we can see that 5 is to 7 as 20 is to 28. You Try #3: Solve the proportion 28 12
=
35 x
4 i 7 = 12
5 i 7 x
4 12
=
5 x
4 3 12
i =
5 3 x
12 = 12
15 x
∴ x = 15
By simplifying first, proportions can be solved much easier. 5
f
=
three different ways. 7 28
If two ratios are equal and their denominators are the same, their numerators must also be the same. 28 12
= by simplifying first. 35 x
Page 6 of 7 28 12
=
35 x
4 i 7 12
=
5i7 x
4 12
=
5 x
5 x
=
4 12
5
x
12 i = 12 i
4
12
60 12x
=
4
12
15 = x
If two ratios are equal, then their reciprocals are equal. MCC@WCCUSD 12/11/13 You Try #4: Solve the proportion u
4
= using a bar model. 28 7
0 0 1 4 2 8 12 3 ∴u = 16 16 4 20 5 6 24 7 28 You Try #5: Set up a proportion to solve the Word Problem. A survey of sixth graders found that the ratio of students who prefer rock to hip-­‐hop is 2 to 7. The total number of sixth graders surveyed was 72. How many of the students surveyed prefer rock. 2
x
=
9 72
The ratio of students who prefer rock to 2 8
x
hip-­‐hop is 2 to 7 which means that 2 out of i =
9 8 72 every nine prefer rock. That’s the ratio 16
x
that must be used in the proportion. =
72 72
x = 16
∴16 students
surveyed prefer rock. Answers to Warm-­‐Up Problems: CST/CAHSEE: D 2
Current: f = 24 Review: x = 32 Other: 3
Page 7 of 7 MCC@WCCUSD 12/11/13