Download 3.1-3.2 Ratios Proportions Percents

RATIOS
A ratio is a comparison of two quantities. You can write a ratio in a variety of ways, including as
a fraction, decimal, or percent.
Writing Ratios
Two common ways to write a ratio are by using a colon or the word "to" between the two
quantities being compared.
The ratio of squares to circles is 3:2 or 3 to 2.
The ratio of circles to squares is 2:3 or 2 to 3.
A A A A A A B B B
The ratio of A’s to B’s is 5:3 or 5 to 3.
The ratio of B’s to A’s is 3:5 or 3 to 5.
Writing Ratios as Fractions
Ratios can be written as fractions.
You can buy 5 apples for 2 dollars
5 apples
2 dollars
You can get 3 rides for each ticket
3 rides
1 ticket
Writing Ratios as Decimals
Because ratios can be written as fractions, they can also be written as decimals. Imagine that a
jar holds 3 red marbles and 4 marbles total.
3 red marbles
4 total marbles
The ratio of red marbles to total marbles is ¾ or 0.75.
(Remember that fractions can be changed to decimals by dividing the numerator by the
denominator. In this case, 3 divided by 4 = 0.75)
Writing Ratios as Percents
Percent mean "of one hundred." A ratio of 80 to 100 is 80%. Here is how to change a ratio to a
percent.
Step 1:
Write the ratio as a decimal.
85 = 0.85
100
Step 2:
2 = 0.40
5
1 = 0.10
10
Multiply by 100 the easy way – by moving the decimal point two
places to the right. Write the number with a percent sign.
0.85 = 85%
0.40 = 40%
0.10 = 10%
Equivalent Ratios
Ratios are equivalent when they represent the same relationship between quantities.
Example:
If a punch is made by using 2 cups of orange juice and 1 cup of Sprite,
it uses ratio of 2:1 (2 cups of orange juice to 1 cup of Sprite).
A larger quantity of punch could be made by combining 2 gallons of
orange juice and 1 gallon of Sprite. This is an equivalent ratio.
Another batch of punch could be made using 4 cups of orange juice
and 2 cups of Sprite. This ratio, 4:2, is equivalent to the ratio 2:1, and
the punch will taste the same.
Finding Equivalent Ratios
Equivalent ratios can be found mathematically. Because ratios can be written as fractions, finding
equivalent fractions is one way to find equivalent ratios. Equivalent ratios (when written as
fractions) can be found by multiplying or dividing by a fraction equal to one. (A fraction is equal to
one when it has the same numerator and denominator.)
Example: The ratio 3
5
is equivalent to
both
3
2
X
5
2
=
6
10
6 and 9
10
15
3
3
9
=
X
5
3
15
PROPORTIONS
A pair of equivalent fractions is a proportion. This proportion shows the ratio of boys to children
in two groups is the same.
Boys
Children
3
7
12
25
One group has 3 boys for every 7 children. The other has 12 boys out of 28 children. Both
fractions show the same relationship, so this is a proportion.
You can check to see if fractions are equivalent by crossing multiplying. When cross multiplying,
you multiply the numerator of one fraction by the denominator of the other. Then you multiply the
opposite numerator by the opposite denominator. If the two products are equal, the fractions are
equivalent and you have a proportion.
Example:
3
7
12
28
3 x 28 = 84
12 x 7 = 84
So, these fractions are equivalent and a proportion exists.
Cross multiplying can also be used to figure out a missing number in a proportion.
Example:
3
7
n
28
Because this is a proportion, 3 x 28 = n x 7.
84 = n x 7
So
And
12 = n
RATES
A rate is a special type of ratio. It compares two quantities that are measured in different units.
140 heart beats
2 minutes
This ratio shows that a heart beats 140 times in 2
minutes.
3 miles walked
1 hour
This ratio shows that 3 miles were walked in each
hour.
Unit Rates
A rate is called a unit rate when the denominator is 1. This tells the rate for 1 unit. To change a
rate to a unit rate, set up a proportion with an unknown.
Example: A heart beats 140 times in 2 minutes.
Write this as a unit rate.
140 heart beats
2 minutes
n heart beats
1 minute
Use cross multiplication:
140 x 1 = n x 2
140 = n x 2
70 = n
140 hearts beats in 2 minutes can be written as the unit rate:
70 heart beats
1 minute
PERCENT
Percent means "of 100"
25/100
=
25%
16/100
=
16%
3/10
=
30/100
= 30%
2/5
=
40/100
= 40%
Picturing Percents
You can talk about what part of a whole unit is shaded by using percent. Percent tells what part
of a hundred.
1
4
=
25
100
= 25%
1
2
=
50
100
=
50%
3
4
=
75
100
=
75%
3 or 75% of this rectangle is yellow.
4
1 or 50% of this triangle is blue.
2
Calculating a Percent of a Number
When calculating percent, remember that the word "of" means multiply.
Example:
10% of 50 means 10% x 50
15% of 20 means 15% x 20
Change the percent into a decimal or a fraction to complete the problem.
Example:
10% of 50 means 10% x 50
10% can be written as .10, so 10% x 50 = .10 x 50 = 5.0 or 5
15% of 20 means 15% x 20
15% can be written as .15, so 15% x 20 = .15 x 20 = 3.0 or 3
Using Percent Bars
Here is how to use percent bars to find out how many of the 30 students in your class equal
70%.
1. Draw two bars of the same height.
Draw a line across the bottom of
them.
100%
30
2. The left bar always shows the
percent. Label the bar Percent and
mark 0% at the bottom and 100% at
the top.
70%
21
Percent
Students
3. The right bar shows what you are
estimating. Label the bar and put 0
at the bottom. Put the total number,
in this example 30, at the top.
4. Estimate 70% on the left bar. Draw a
line across both bars at that point.
0%
0
5. Estimate the number of students on
the right bar. The number is 21.
0%
0