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7th Grade Pre-algebra
Chapter 6 Notes
6.1 Ratios and Rates
Vocabulary
Ratio: a comparison of two numbers by division.
Rate: a ratio of two measurements having different kinds of units.
Unit Rate: when a rate is simplified so that it has a denominator of 1. Ex. mph
Writing Ratios as Fractions
Express the ratio 9 goldfish out of 12 goldfish as a fraction in simplest form.
Express the ratio of girls to boys in this class as a fraction in simplest form
Express the ratio of boys to girls in this class as a fraction in simplest form.
12
9
boys
girls
18
15
girls
boys
15
18
Write Ratios as Fractions
Units of measure must be the same in order to simplify!!!
Express 3 feet to 16 inches as a fraction in simplest form.
4
9
16
36
16
3
in
in
in
ft
Find Unit Rates
Since we want the cost of 1 soda, put the soda on the
bottom of our fraction
Hint: Money will almost ALWAYS be the numerator of the fraction!!
canper
cans
129.0$
1
29.
6
6
6
74.1$
Converting Rates
To convert a rate such as miles per hour to feet per second, you can use dimensional analysis. Carry the units throughout the computation.
A Grizzly bear can run 30 miles in 1 hour. How many feet is this per second?
ond
feet
hr
mi
sec?
?
1
30
ond
feet
hr
mi
sec600,3
400,158
1
30
There are 5280 ft in a mile. 30 x 5280 = 158,400
There are 3600 second in an hour (60min x 60sec)
ond
feet
sec600,3
400,158
sec1
44 ft
Simplify to get 1 in the denominator by dividing both the numerator and denominator by 3600
You Try…
You Try…
Problem Solving
6.2 Using Proportions
Vocabulary
Proportion: a statement of equality of two ratios
Cross Products: when you multiply the numerator of the first ratio by the denominator of the second ratios and compare that product to the product of the denominator of the first ratios and the numerator of the second ratio.
168
24
84
12 )24(84)168(12 20162016
Use Cross Products to identify Proportions
Pairs of ratios are only proportional if their cross products are equal.
2x20 = 408x5 = 40
= equal
3x24 = 724x18 = 72
= equal
2.5x6 = 15.02x7.5 = 15.0
= equal
Solving Proportions
We can solve for the unknown value in a proportion by using cross multiplication.
35x = 385
1135
385
35
35
x
x
Solving Proportions
1m = 216 m = 216 2x = 135
x = 67.5
Problem Solving
18
362122
3
x
x
x
6.3 Scale Drawings and Models
Vocabulary
Scale Drawing: a drawing used to represent an object that is too large or too small to be drawn at actual size.
Scale Model: a model used to represent an object that is too large or too small to be built at actual size.
Scale: gives the relationship between the measurements on the drawing or model and the measurements of the real object.
Scale Factor: The ratio of a length on a scale drawing or model to the corresponding length on the real object.
Using Scales
.5/10=.45/x
.5/10=1/x
.5/10=.75/x
6.4 Fractions, Decimals, and Percents
Vocabulary
Percent: a ratio that compares a number to 100
Writing Percents as Decimals
To write a percent as a decimal, divide by 100 and remove the percent symbol
Divide by 100
Remove % sign
62.10062%62
Writing Decimals as Percents
To write a decimal as a percent, multiply by 100 and add the percent symbol
multiply by 100
Add % sign
%4210042.42.
Writing Percents as Fractions
To write a percent as a fraction, express the ratio as a fraction with a denominator of 100
100
68%68
25
17
100
68
Put % over 100
Simplify
Writing Fractions as percentsMethod #1
To write a Fraction as a percent, write an equivalent fraction with a denominator of 100. Write the new numerator as a percent
multiply by 20 to get the denominator
to be 100
Write the numerator
as a percent
%40100
40
205
202
5
2
Writing Fractions as PercentsMethod #2
To write a fraction as a percent, first change the fraction to a decimal, then change the decimal to a percent
change fractions to a decimal by dividing the
numerator by the denominator
Add % sign
%5.87100875.0875.08
7
change decimal to a percent by
multiplying by 100
Practice
= 90%
10
9
100
90
= 135%
20
27
100
135
= 62%
50
31
100
62
Practice
=50%=.50 =75%
=.75=125%=1.25
Problem Solving
5
3
100
60
6.5 Using the percent Proportion
Vocabulary
Percent Proportion: a proportion in which a ratio is being compared to a percent
Part: the number being compared to the whole (always the numerator)
Base: The whole quantity (the denominator)
Find the Percent
Five is what percent of 8?
Step 1: set up a proportion
1008
5 p
5 is the part
8 is the base
percent is the unknown
always 100
Find the PercentFive is what percent of 8?
Step 2: Find the cross products
Step 3: Solve for the variable (p)
1008
5 p 500 = 8p
500 = 8p
8
8
8
500 p
62.5 = p
Divide each side by 8
62.5%62.5%
Add % sign since we were looking for the
percent
Find the Part
What number is 5.5% of 650?
Step 1: set up a proportion
100
5.5
650
athe part is unknown
650 is the base
5.5 is the percent
always 100
Find the PercentFive is what percent of 8?
Step 2: Find the cross products
Step 3: Solve for the variable (p)
100
5.5
650
a100a = 3575
100a = 3575
100
3575
100
100
a
a = 35.75
Divide each side by 100
35.7535.75
Practice
10050
14 x
1400 = 50x
x = 25
100
300
12
x
100x = 3600
x = 36
Apply the Percent Proportion
20.19$
20.19$8.1232$
8.12
1280100100
40
32
x
x
x
20.19$
60.32 or
Since the jeans are 40% off, the remaining price is 60% of the original price