Download Lesson 6-1

Chapter 6: Lessons 6-1 to 6-4, 6-7
STUDY GUIDE
This study guide provides you with an overview of the types of problems and skills that we
will learn or have learned in the lessons listed above. In addition to this guide, we will
complete additional review problems in class, as we get closer to the test. Review your class
notes too!
Lesson 6-1: Ratios and Rates
Example 1 Write a Ratio in Simplest Form
MARBLES A collection of marbles has 8 blue marbles and 6 green marbles. Write the
ratio in simplest form that compares the number of blue marbles to the number of green
marbles. Then explain its meaning.
÷2
blue marbles →
green marbles →
8
4
6 = 3
The GCF of 8
and 6 is 2.
÷2
4
The ratio of blue marbles to green marbles is 3, 4 to 3, or 4:3. This means that for every
4 blue marbles there are 3 green marbles.
Example 2 Use Ratios to Compare Parts to a Whole
BALLS A large bin of balls in the equipment room of the gym contains 20 balls,
5 of which are soccer balls. Write the ratio that compares the number of soccer balls to the
total number of balls.
÷5
soccer balls →
total balls →
5
1
=
20
4
The GCF of 5
and 20 is 5.
÷5
1
The ratio of soccer balls to the total number of balls is 4, 1 to 4, or 1:4.
One out of every 4 balls is a soccer ball.
Example 3 Find a Unit Rate
CEREAL A 16-ounce box of cereal is priced at $3.68. Find the cost per ounce.
÷16
Write the rate that compares the price to the number
$3.68
of ounces. Then divide to find the unit rate.
$0.23
16 ounces = 1 ounce
So, the cost of the cereal is about $0.23 per ounce.
÷16
Lesson 6-2: Ratio Tables
Example 1 Equivalent Ratios of Larger Quantities
TEA To make iced tea, Sahara adds 5 scoops of sugar to 1 gallon of tea. Use the ratio
table to find how many scoops of sugar she would use for 4 gallons of iced tea.
1
5
Gallons of Tea
Scoops of Sugar
METHOD 1
4
■
Find a pattern and extend it.
For 2 gallons of iced tea, Sahara would need a total of 5 + 5 or 10 scoops.
+1
1
5
Gallons of Tea
Scoops of Sugar
+1
2
10
+5
METHOD 2
3
15
+5
+1
4
20
Continue this pattern
until you reach 4 gallons.
+5
Multiply each quantity by the same number.
×4
1
5
Gallons of Tea
Scoops of Sugar
4
20
Since 1 × 4 = 4, multiply
each quantity by 4.
×4
So, Sahara would need 20 scoops of sugar for 4 gallons of iced tea.
Example 2 Equivalent Ratios of Smaller Quantities
VACATION On vacation, Miguel’s family
traveled 260 miles in 4 hours. If they traveled at a
constant rate without stopping, use the ratio table to
determine about how many miles they traveled
every hour.
÷2
Miles
Time (h)
260
4
÷2
130
2
÷2
Miles
Time (h)
65
1
Divide each quantity by one or
more common factors until you
reach a quantity of 1 hour.
÷2
So, Miguel’s family traveled about 65 miles every hour.
260
4
■
1
Example 3 Use Scaling
YOGURT Containers of yogurt are on sale
at 12 for $8. Use the ratio table to find the
cost of 18 containers of yogurt.
Containers
of Yogurt
Cost in
Dollars
12
18
8
■
There is no whole number by which you can multiply 12 to get 18. So, scale back to 6 and then
scale forward to 18.
÷2
Containers
of Yogurt
Cost in
Dollars
×3
12
6
18
Divide each quantity by a common
factor, 2.
8
4
12
Then, since 6 × 3 = 18, multiply each
quantity by 3.
×3
÷2
So, 18 containers of yogurt would cost $12.
Real-World Example 4 Use a Ratio Table
ARCADE Lashonda exchanged $10 for 45 tokens to play arcade games. Use a ratio table
to find how many tokens she would receive for $16.
Set up a ratio table.
Tokens
Dollars
■
16
45
10
Label the rows with the two
quantities being compared. Then
fill in what is given.
Use scaling to find the desired quantity.
÷5
Tokens
Dollars
45
10
×8
9
2
÷5
72
16
×8
Divide each quantity by a common
factor, 5.
Then, since 2 × 8 = 16, multiply each
quantity by 8.
Lashonda would receive 72 tokens for $16.
Lesson 6-3: Proportions
Example 1 Use Cross Products
Determine if the quantities in the pair of rates are proportional. Explain your reasoning
and express each proportional relationship as a proportion.
12 laps in 4 minutes; 14 laps in 7 minutes
84
56
𝟏𝟐 𝟏𝟒
=
𝟒
𝟕
The rates are not proportional because the cross products are not equal.
Example 2 Use Cross Products
Determine if the quantities in the pair of rates are proportional. Explain your reasoning
and express each proportional relationship as a proportion.
2 hats for $18; 4 hats for $36
72
72
𝟐
𝟒
=
𝟏𝟖 𝟑𝟔
The rates are proportional because the cross products are equal.
Example 3 Use Cross Products
Determine if the quantities in the pair of ratios or rates are proportional. Explain your
reasoning and express each proportional relationship as a proportion.
9 books for $45; 3 books for $15
135
135
𝟗
𝟑
=
𝟒𝟓 𝟏𝟓
The rates are proportional because the cross products are equal.
Lesson 6-4: Algebra: Solving Proportions
Example 1 Solve Using Equivalent Fractions
Solve the proportion.
2 b
=
5 30
Find a value for b so the fractions are equivalent.
×6
2
b
5 = 30
Since 5 × 6 = 30, multiply the numerator and denominator by 6.
×6
2
12
=
5
30
Since 2 × 6 = 12, b = 12.
Example 2 Solve Using Equivalent Fractions
Solve the proportion.
21 3
=
56 x
÷7
21
3
=
56
x
Since 21 ÷ 7 = 3, divide the numerator and denominator by 7.
÷7
21
3
56 = 8
Since 56 ÷ 7 = 8, x = 8.
Example 3 Make Predictions in Proportional Situations
READING Scott read 225 pages of a novel in 5 days. At this rate, how many pages can he
read in 10 days?
Step 1
Set up the proportion. Let p represent the number of pages Scott can read in
10 days.
𝒑𝒂𝒈𝒆𝒔
𝟐𝟐𝟓 𝒑𝒂𝒈𝒆𝒔
𝒅𝒂𝒚𝒔
𝟓 𝒅𝒂𝒚𝒔
Step 2
Solve.
=
𝒑 𝐩𝐚𝐠𝐞𝐬
𝟏𝟎 𝒅𝒂𝒚𝒔
×2
𝒑𝒂𝒈𝒆𝒔
𝟐𝟐𝟓 𝒑𝒂𝒈𝒆𝒔
𝒅𝒂𝒚𝒔
𝟓 𝒅𝒂𝒚𝒔
=
𝟒𝟓𝟎 𝐩𝐚𝐠𝐞𝐬
𝟏𝟎 𝒅𝒂𝒚𝒔
×2
So, the value of p is 450. At the given rate, Scott can read 450 pages in 10 days.
Example 3 Make Predictions in Proportional Situations
LUNCH If 21 of the 28 students in a gym class consider pizza their favorite lunch food,
use this ratio to predict how many consider pizza their favorite lunch food in a school of
200 students.
Write and solve a proportion. Let p represent the number of students who can be expected to
consider pizza their favorite lunch food.
𝒑𝒊𝒛𝒛𝒂
𝟐𝟏 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔
𝒕𝒐𝒕𝒂𝒍
𝟐𝟖 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔
=
=
𝒑 𝐬𝐭𝐮𝐝𝐞𝐧𝐭𝐬
𝟐𝟎𝟎 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔
The denominators 28 and 200 are not easily related by multiplication, so simplify the ratio 21 out
of 28. Then solve using equivalent fractions.
×50
÷7
𝒑𝒊𝒛𝒛𝒂
𝟐𝟏 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔
𝒕𝒐𝒕𝒂𝒍
𝟐𝟖 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔
=
÷7
𝟑
=
𝟒
𝟏𝟓𝟎 𝐬𝐭𝐮𝐝𝐞𝐧𝐭𝐬
𝟐𝟎𝟎 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔
×50
So, about 150 out of 220 students in the school can be expected to consider pizza their favorite
lunch food.
Lesson 6-7: Proportions and Equations
Example 1 Write an Equation for a Function
Write an equation to represent the function
displayed in the table.
Input, x
Output, y
1
6
2
12
3
18
4
24
5
30
Examine how the value of each input and output changes.
Input, x
1
2
3
4
5
Multiply by 6
1×6
2×6
3×6
4×6
5×6
Output, y
6
12
18
24
30
Each output y is equal to 6 times the input x. So, the equation that represents the function is
y = 6x.
Example 2 Real-World Example
POOLS The cost of admission at a public pool
is shown in the table. Write an equation to
describe the data. Then find the total cost of
admission for a group of 7 friends.
Number of
People, p
1
2
3
4
Total
Cost ($), t
8
16
24
32
The cost of admission is $8 for each person.
The total cost t is $8 times the number of people p. Therefore, t = 8p. Use this equation to find
the total cost t of admission for 7 people.
t = 8p
t = 8 ∙ 7 or 56
Write the equation.
Replace p with 7. Multiply.
The total cost of admission for 7 people is $56.
Example 3 Real-World Example
a.) MONEY Aidan mows lawns each summer. He charges $15 for each lawn he mows.
Make a table to show the relationship between the number of lawns mowed ℓ and the total
amount earned t.
The total earned (output) is equal to $15 times the number of lawns mowed (input).
Lawns
Mowed, ℓ
1
2
3
4
Multiply
by 15
1 × 15
2 × 15
3 × 15
4 × 15
Total
Earned ($), t
15
30
45
60
b.) Write an equation to find the total amount earned t for mowing ℓ lawns.
Study the table from Example 2. The total earned equals $15 times the number of lawns mowed.
Words
Total earned equals $15 times the number of lawns mowed.
Variable
Let t represent the total earned and
ℓ represent the number of lawns mowed.
t
Equation
=
15
∙
So, the equation is t = 15ℓ.
c.) How much will Aidan earn if he mows 20 lawns?
t = 15ℓ
t = 15 ∙ 20
t = 300
Write the equation.
Replace ℓ with 20.
Multiply.
Aidan will earn $300 for mowing 20 lawns.
ℓ