Transcript Ratios: a comparison of two numbers using division

Ratios: a comparison of two
numbers using division
To see if a ratio is equivalent
1. Compare the fractions by finding
the LCD
2. Simplify the ratios
3. Compare by using cross products
We use ratios to make comparisons between two things.
We are comparing rectangles
to triangles.
Ratios can be written 3 ways.
1. As a fraction 3
5
2. Using the word to
3 to 5
3. Using a colon 3:5
equivalent ratios
Ratios that name the same comparisons
To see if to ratios are equivalent
1. Change each to a decimal and
compare the decimals.
2. Reduce both ratios and compare.
3. Use cross products.
Rate: is a ratio of 2 measurements with different units
Example: It rained 4 inches in 30 days Here we are comparing days to
inches
The rate is 4
We can reduce to 2
30
15
A rate that has 1 unit as its second term (denominator)
If a car travels 325 miles and uses 11 gallons
of gas what is the mile per gallon?
This is an example of a unit rate. How many miles per 1 gallon?
Create a ratio Miles
Gallons
325
11
Since every fraction is a division
problem we divide 325/ 11
Our Unit Rate is 29. 54 miles per gallon
An equation that shows that two ratios are equal
We can write proportions in 2 forms.
a:b = c:d
If 2 ratios are equal then their cross product will be equal. a * d = b * c
A car travels 125 miles in 5 hours. How many
miles will the car travel in 8 hours?
Solve using
proportions.
Proportion 125 = m
5
8
Set an equation using cross products
125 * 8 = 5 * m
Simplify
1000 = 5m
Solve by inverse operation
1000 /5 = m
(The opposite of multiplication is
division )
200 = m
In 8 hours a car can travel 200 miles
On a map 1.5 inches is equal to 5 miles. If the distance in real life
is 22 mile how big will it be on the map?
Proportions can help us with this problem. We know 1 ratio is
1.5 in: 5 m. We know 1 part of the second ratio is 22 m.
Proportion
1.5 in = X m
5m
22 m
Cross Products
1.5 x 22 = 5 x X
Simplify
Inverse Operation
33
33/5
6.6
= 5X
= X
= X
Notice we lined
up m to m and in
to in
22 miles is equal to 6.6
inches on the map
Another form of writing a piece of a number is by using percents.
Percent means "out of 100.” With percents we are using a comparison of
decimals and fractions to 100 pieces.
10 out of 100 =10% = 0.1 = 1
10
5 out of 100 = 5% = 0.05 = 1
20
A decimal can be written as a percent, by moving the decimal
point two places to the right like this:
Follow the same procedure for decimals larger than 1
2.35 = 2.35 = 235%
A percent can always be written as a decimal by moving the decimal
point two places to the left like this:
68% = 68. = 0.68
Follow the same procedure for percents larger than 100
345% = 345. = 3.45
To convert the fraction 3 to a percent
5
To convert 138% to a
fraction
Change the fraction to a decimal
0.6
5 3
Place the percent over 100
Then move the decimal 2 places
right
0.6
= 60%
138
100
138
100
Then simplify
=
38
1 100 =
1
19
50
Dinner cost $75 and you wish to leave a 20% tip. How much
will the tip be? We can use the percent proportion to solve.
P is the percentage ( a value that is a number for the
percent
P =R
B is the base or the original amount
B
R is the rate(the percent number over 100)
In this problem the Base is $75, the Rate is 20 over 100 and we are
solving for the Percentage ( how much money is equal to 20%)
P
$75
= 20
100
Cross products 75 x 20 = P x 100
Simplify
1500 = 100P
Inverse operation 1500/ 100 = P
$15 = P
The tip will be $15
We have seen 1 type of percent problem. Let’s look at 2 others.
If we left a 20% tip which was $25, how much was the bill?
We know R is 20% and P is $25. We need to find B.
Proportion
25 = 20
B
100
Cross products
Simplify
Inverse Operation
25 x 100 = 20 x B
2500 = 20B
2500/ 20 = B
$125 = B
The dinner bill was
$125
If Dinner cost $125 and we left a $35 tip what percent of the
bill was the tip?
P is $35, B is $125. We are trying to find R.
Proportion
35
125
=
R
100
Cross Products
35 x 100
=
Simplify
3500
= 125 R
Inverse Operation
3500/ 125
= R
28%
125 x R
= R
The tip was 28% of
the bill