Transcript Ratios

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
Unit rates are often used to make comparisons.
Drivers often use Miles Per Gallon(mpg) or Miles Per Hour(mph)
Example: the 8th grade parents collect $ 2450 for pizza sales and the
7th grade parents collect $ 2250 for selling Subway. If there were 102 8th graders and 90 - 7th graders, which group collect more per
student?
We have 2 rates.
We can make each a unit rate to see which
group had more dollars per student.
8th
$ 2450 = $ 24.90/ student
102
7th
$ 2250 = $ 25 per student
90
The 7th grade parents collect $ 0.10 more per student
Shop Rite has 24 oz box of Hulk Cereal for $ 3.99. Path Mark has a
1lb box for$ 2.59. Which is the better buy?
Notice in this box the units are different (oz, lbs)
Before we solve we need to make the units the same. 1 lb = 16 oz
The rates are: We find the unit rates by division.
Shop Rite
$ 3.99
24 oz
= about $ 0.17per oz
Path Mark
$ 2.59 = about $ 0.16 per oz
16 oz
Shop Rite has the better buy on Hulk Cereal
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
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
We have used a proportion to solve percent problems for P, R, and B.
We can rewrite the proportion to an equation.
Let us look at the previous
P=RxB
problems using the equation.
1. Dinner cost $75 and you wish to leave a 20% tip. How much
will the tip be?
P = R
x
P = 20% x
B
$75
P = 0.2 x
75
P = $15
Formula
Substitute
Solve: use the decimal form of the %
The tip will be $15
2. If we left a 20% tip which was $25, how much was the bill?
P = R
x
$ 25 = 20% x
25

125 = B
B
Formula
B
Substitute
0.2 = B
Solve using inverse operation
The dinner bill was $125
3. If Dinner cost $125 and we left a $35 tip what percent of the bill
was the tip?
P = R
x
$ 35 = R x
35

B
Formula
$125
125 = R
Substitute
Solve using inverse operation
Note: We are dividing by numbers not
the rate so we use the numbers.
0.28 = R
Since our answer is a decimal we
convert that decimal to get a percent
28% = R
The tip was 28% of the bill
When money is borrowed, interest is charged for the use of that money for a
certain period of time. When the money is paid back, the principal (amount of
money that was borrowed) and the interest is paid back. The amount to interest
depends on the interest rate, the amount of money borrowed (principal) and the
length of time that the money is borrowed.
The formula for finding
simple interest is:
Interest = Principal * Rate * Time
How much will the interest be if we borrow 20,000 for 2 years at 6%?
I = P x R xT
I = 20000 x 6% x 2
I = 20000 x 0.06 x 2
I = 2400
Note: we use the decimal form to
multiply and the length of time is based
on 1 year.
The interest will be $ 2400
If we put 20000 in the bank at 5.5% for 18 months, How much will
we have at the end of 18 months? There are differences in this problem
1. The interest rate has a decimal
2. The time is not if full years
3. We are asked for a total not just the interest.
I = P x R x T
I = 20000 x 0.055 x 18
12
I = 20000 x 5.5% 18 months
Note: since 1 year = 12 months we use
12 as a denominator. We could reduce
(1 1 ) or use the decimal form(1.5)
2
This is the interest. We now add
that to the principle of 20000
I = 1650
1650 + 20000 = 21650
The total at the end of the time period is $21650
Many of the things we buy, the money we earn and the places we
live come with a tax. .The tax is found by finding a percentage of
the purchase or income called the tax rate.
By finding the percentage we can calculate tax. In much the same way
we did with simple interest except we eliminate the time component of
the formula. The formula can be written:
Tax = Principle * tax rate
When a total is asked for we add the percentage to
the original amount as we did with simple interest.
We purchased a car for $28,568. If the tax rate is 6%,
how much tax did we pay?
Tax = Principle * tax rate
T = $28568 * 6%
The tax on our car is $ 1714.08
T = 28568 * 0.06
T = 1714.08
What is the total cost of the car?
This problem requires us to just add the tax and the price
of the car much the same as we did with the total for
simple interest. $28568 + $ 1714.08 = $30282.08
$30282.08 is the total cost of the car
Retailers often offer sales. They are usually in percents, We can solve
these problems in the same way using percents.
A DVD is on sale for 20% off. It originally sells for $275. How much
will we save? We can put our % equation in this form.
D = P *R
D = $275 * 20%
D = 275 * 0.2
The discount on the DVD is $ 55
D = 55
What is the sale price of the DVD?
To solve we just subtract our discount from the original price
$ 275 - $55 = $ 225
The sale price of the DVD is $ 225
Macy’s is having a 20% off sale. If you buy it
today you receive an additional 15% of the sale
price.If you buy a $45 sweater today how much
will it cost?
In this problem it looks as if you will get 35% off. But we will only
get 15% off the sale price not the original price.
D = $ 45 *20 %
D = 45 * 0.2
D= $9
$ 45 - $ 9 = $36
The sale price is $ 36. Now we take off the 15%.
D = $ 36 *15 %
D = 36 * 0.15 D = $ 5.40 $ 36 - $ 5.40 = $31.60
The final price is $ 31.60
But how much would it be if there was a 6% sales tax???
Often times when we buy things they can either increase in value
(appreciate) or decrease in value (depreciate).
The differences can looked at as Percent of Change. We refer to
these situations as either the percent of increase (appreciation) or
the percent of decrease (depreciation)
Usually, the homes we buy appreciate. When we sell our homes
we often get more than we paid for them. We also buy stocks in
the hope that they will also go up (not always the case)
On the other hand, cars often go down in value as they
get older.
Joe Smith bought his home in 1999 for $ 325,000 and sold it in
2003 for $ 545,000. What was his percent of increase?
1. Determine whether this is an increase or a decrease. ( If the new
price is higher it is an increase. If the new price is lower it is a
decrease. In this case the price is an increase).
2. Subtract the higher price
from the lower price to find the
difference.
3. Make a fraction by placing the
difference over the original price.
4. Change the fraction to a
decimal by division. Then to a
percent by moving the
decimal point.
$545,000 - $325,000
= $220,000
difference
$ 220,000
$ 325,000
Original
price
The percent of increase is 67.7%
(Answer rounded to the nearest tenth
of a percent).
Mrs. Princing bought stock in the I.O.U company worth $5500
in May. In June the stock was worth $3000 . Find the percent of
change in the stock.
Since the new price is lower we will be finding the percent of decrease.
Find the difference.
Make a fraction
$ 2500
$ 5500
$5500 - $3000 = $2500
Change to decimal
0.4545
Change to percent
45.5%
The percent of change (decrease) for Mrs. Princing’s
stock is 45.5 % ( rounded to the nearest tenth).