Transcript Slide 1

Grade 8 Algebra1
Review of
Proportion and
Percent
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Warm Up
1) Find each percent change from 25 to 45.
2) Find the result when 40 is increased by 40%.
3) Find the result when 70 is increased by 9%.
4) What is the final price on a $165 leather jacket that
is on sale for 30% off?
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Rates, Ratios, and Proportions
A ratio is a comparison of two quantities by division.
The ratio of a to b can be written a:b or a ,
b
where b ≠ 0. Ratios that name the same comparison
are said to be equivalent.
A statement that two ratios are equivalent,
such as 1 = 2 , is called a proportion.
12 24
Read the proportion
1 =
x
15
675
“1 is to 15 as x is to 675.”
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Using Ratios
The ratio of faculty members to students at a
college is 1:15. There are 675 students. How many
faculty members are there?
Faculty = 1
Students
15
Write a ratio comparing faculty to students.
1 = x
15
675
675 1
15
= 675
Write a proportion. Let x be the number
of faculty members.
x
675
Since x is divided by 675, multiply
both sides of the equation by 675.
x = 45
There are 45 faculty members.
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A rate is a ratio of two quantities with different
units, such as or 34mi ,
2gal
Rates are usually written as unit rates. A unit rate is a
rate with a second quantity of 1 unit, such as
34mi ,
2gal
or 17 mi/gal. You can convert any rate to a unit rate.
Conversion factor
A rate such as 12in. ,
1 ft
in which the two quantities are equal but use
different units, is called a conversion factor.
To convert a rate from one set of units
to another, multiply by a conversion factor.
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Finding Unit Rates
Garry ate 53.5 hot dogs in 12 minutes to win a
contest.
Find the unit rate. Round your answer to the nearest
hundredth.
53.5
12
=
4.46 ≈ x
x
1
Write a proportion to find an equivalent
ratio with a second quantity of 1.
Divide on the left side to find x.
The unit rate is approximately 4.46 hot dogs per minute.
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Conversion factor
The dwarf sea horse Hippocampus zosterae swims at a
rate of 52.68 feet per hour. What is this speed in
inches per minute?
Step1: Convert the speed to inches per hour.
52.68ft
1h
×
12 in
1ft
To convert the first quantity in a rate,
multiply by a conversion factor with
that unit in the second quantity.
632.16 in.
1h
The speed is 632.16 inches per hour.
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Step2: Convert this speed to inches per minute.
632.16 in
1h
×
1h
60 min
To convert the second quantity in a
rate, multiply by a conversion factor
with that unit in the first quantity.
10.536 in
1 min
The speed is 10.536 inches per minute.
Check that the answer is reasonable.
The answer is about 10 in./min.
•There are 60 min in 1 h, so 10 in./min is
60 (10) = 600 in./h.
• There are 12 in. in 1 ft, so 600 in./h is
600 = 50 ft/h. This is close to the rate
12
given in the problem, 52.68 ft/h.
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Solving Proportions
Solve the proportion.
A) 5
9
= 3
w
5
9
= 3
w
5 (w) = 9 (3)
Use cross products.
5w = 27
5w = 27
5
5
Divide both sides by 5.
w = 27
5
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A scale is a ratio between two sets of
measurements, such as 1 in : 5 mi.
A scale drawing or scale model uses a scale to
represent an object as smaller or larger than the
actual object.
A map is an example of a scale drawing.
A scale written without units, such as 32 : 1,
means that 32 units of any measure
correspond to 1 unit of that same measure.
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Scale Drawings and Scale Models
A) On the map, the distance from Houston to
Beaumont is 0.8 in. What is the actual distance?
1 in : 100 mi
Solution:
map =
actual
0.8 in
x
=
1 in
100 mi
1 in
100 mi
x · 1 = 100 (0.8)
Write the scale as a fraction.
Let x be the actual distance.
Use cross products to solve.
x = 80
The actual distance is 80 mi.
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B) The actual distance between Bryan- College Station and
Galveston is 127 mi. What is this distance on the map?
1 in : 100 mi
Solution:
map =
actual
x
127
=
1 in
100 mi
1 in
100 mi
127 = 100x
Write the scale as a fraction.
Let x be the distance on the map.
Use cross products to solve.
127 = 100x Since x is multiplied by 100, divide both
100
100 sides by 100 to undo the multiplication.
1.27 = x
The actual distance is 80 mi.
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Now you try!
Solve each proportion.
1) 3
z
3) x
3
= 1
8
= 1
5
2) f + 3
12
= 7
2
4) -1
5
3
2d
=
5) The ratio of the sale price of a jacket to the
original price is 5 : 7. The original price is
$105. What is the sale price?
6) Find the unit rate.
A computer’s fan rotates 2400 times in 20 seconds.
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Similar Figures
Similar figures have the same shape (but not
necessarily the same size) and the following properties:
 Corresponding sides are proportional. That is, the
ratios of the corresponding sides are equal.
 Corresponding angles are equal.
Corresponding sides of two figures are in the same
relative position, and corresponding angles are in the
same relative position. Two figures are similar if and only
if the lengths of corresponding sides are proportional and
all pairs of corresponding angles have equal measures.
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corresponding
angles
A
6
62°
55°
B
X
7.5
63°
62°
C
8
7.5
63°
55°
corresponding
sides.
10
Z
AB = BC = CA
XY
YZ
ZX
10
Y
m A=m X
m B=m Y
m C=m Z
ABC
XYZ
Make sure corresponding vertices are in the same order.
It would be incorrect to write
ABC
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Scale factor
C
D
Any two corresponding sides in two
similar figures have a common
ratio called the scale factor.
A
B
H
E
G
F
The trapezoids ABCD and EFGH shown above are similar. So,
AB = AD = CD = BC = k = scale factor.
EF
EH
HG FG
 The ratio of areas of two similar
figures is the square of the scale factor.
 The ratio of volumes of two similar
figures is the cube of the scale factor.
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Changing Dimensions
B) Every dimension of a cone with radius 4 cm and height 6 cm is
multiplied by 0.5 to form a similar cone. How is the ratio of the
volumes related to the ratio of corresponding dimensions?
3 cm
6 cm
A
Volume = 1 ×∏r2h
3
B
2cm
4 cm
Cone A
Cone B
1 ×∏(4)2 ×6 = 32∏
3
1 ×∏(2)2 ×3 = 4∏
3
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Radii = 4 = 2 =2
2
1
Heights = 6 = 2 =2
3 1
Volumes = 32∏ = 8 = 23
4∏
1
3 cm
6 cm
2cm
4 cm
 The ratio of the volumes is thrice the ratio of the
corresponding dimensions.
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Now you try!
1) Parallelogram ABCD
parallelogram EFGH. Find
the value of x.
x
D
A
12
E
B
C
15
5
H
F
G
2) Sam is 7 feet tall and casts a shadow 4 feet
long. At the same time, the pole
outside his house casts a shadow 15 feet long.
Write and solve a proportion to find
the height of the pole.
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3) Triangle ABC
triangle ADE. Find the value of x.
A
4
4
D
B
x
4
C
4
E
10
4) A rectangle has length 12 feet and width 8 feet.
Every dimension of the rectangle is multiplied by 3
4
to form a similar rectangle. What is the ratio of the
areas of the two triangles?
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Percent
 Percent is a ratio that compares a number to 100. For
example,
20% = 20
100
 To find the Fraction equivalent of a percent, write the
percent as a ratio with a denominator of 100. Then simplify.
20% = 20 = 1
100
5
To find the decimal equivalent of a percent, divide by 100.
20% = 20 = 0.2
100
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Some Common Equivalents
10%
20%
25%
33 1 %
3
40%
50%
60%
66 2 %
3
75
%
80%
100
%
1
10
1
5
1
4
1
3
2
5
1
2
3
5
2
3
3
4
4
5
1
0.1
0.2
0.25
0.3
0.4
0.5
0.6
0.6
0.75
0.8
1.0
The greatest percent shown in the table is 100%, or 1. But
percents can be greater than 100% or less than 1%.
For example,
130% = 130 = 1.3
and 0.6% = 0.6 = 0.006
100
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Finding the Percent
A) What percent of 60 is 15?.
Method 1 Use a proportion.
part
= percent
whole
100
15
60
= x
100
60x = 1500
x = 25
Use the percent proportion.
Let x represent the percent.
Find the cross products.
Since x is multiplied by 60, divide both
sides by 60 to undo the multiplication.
15 is 25% of 60.
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Finding the Whole
B) 60 is 0.6% of what number?
Method 2 Use an equation.
60 = 0.6% of x
Write an equation. Let x
represent the whole.
60 = 0.006x
Write the percent as a
decimal.
60
= 0.006x
0.006
0.006
Since x is multiplied by 0.006,
divide both sides by 0.006 to
undo the multiplication.
10000 = x
60 is 0.6% of 10000
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A) Jewelers use the karat system to determine the amount of
pure gold in jewelry. Pure gold is 24 karat, meaning the item
is 100% gold. A 14-karat gold ring contains 14 parts gold and
10 parts other metal. What percent of the ring is gold? Round
your answer to the nearest percent.
part
whole
14
24
= percent
100
= x
100
24x = 1400
24x = 1400
24
24
x = 58.3
Use the percent proportion.
Let x represent the percent.
Find the cross products.
Since x is multiplied by 24, divide
both sides by 24 to undo the
multiplication.
A 14-karat gold ring is approximately 58% gold.
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Now you try!
1) 1 is what percent of 15?
2) 48 is 40% of what number?
3) 13 is 20% of what number?
4) On average, sloths spend 16.5 hours per day sleeping.
What percent of the day do sloths spend sleeping? Round
your answer to the nearest percent.
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Applications of Percent
A commission is money paid to a person or a
company for making a sale. Usually the commission
is a percent of the sale amount.
Interest is the amount of money charged for borrowing
money, or the amount of money earned when saving or
investing money.
Principal is the amount borrowed or invested. Simple
interest is interest paid only on the principal.
Formula for Simple Interest Paid Annually
Simple Interest, I = P × r × t
Principal
Time in years
Interest rate per
year as a decimal
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Business Application
A) Daniela earns a base salary of $40,000 plus a 2.5%
commission on sales. Her total sales one year were
$800,000. Find her total pay for the year.
SOLUTION:
total pay = base salary + commission
Write the formula for total pay.
= base salary + % of total sales
Write the formula for commission.
= 40,000 + 2.5% of 800,000
Substitute values given in the
problem.
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= 40,000 + 2.5 of 800,000
100
= 40,000 + (0.025)(800,000)
Write the percent as a decimal.
= 40,000 + 20,000
Add.
= 60,000
Daniela total pay was $60,000.
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Finance Application
A) Find the simple interest paid annually for 3 years on
a $1200 loan at 15% per year.
SOLUTION:
I=Prt
Write the formula for simple interest.
I = (1200) (0.15) (2)
Substitute known values. Write the
interest rate as a decimal.
I = 540
The amount of interest is $540.
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Estimating with Percents
A) The dinner check for Maria’s family is $67.95.
Estimate a 15% tip.
SOLUTION:
Step 1: First round $67.95 to $70.
Step 2: Think: 15% = 10% + 5%
10% of $70 = $7.00
Move the decimal point one place left.
Step 3: Think: 5% = 10% ÷ 2
= $7.00 ÷ 2 = $3.50
Step 4: 15% = 10% + 5%
= $7.00 + $3.50 = $10.50
The tip should be about $10.50.
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Estimating with Percents
B) The sales tax rate is 6.25%. Estimate the sales tax
on a shirt that costs $29.50.
SOLUTION:
Step 1: First round 6.25% to 6% and $29.50 to $30
Step 2: Think: 6% = 6 (1%)
1% of $30 = $0.30
Move the decimal point two places to
the left.
Step 3: 6% = 6 (1%)
= 6 ($0.30) = $1.80
The sales tax is about $1.80.
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Some more examples
A) Two candidates A and B contest an election. A gets
46% of the valid votes and is defeated by 1600 votes.
Find the total number of valid votes cast in the election?
SOLUTION:
A gets 46% of the valid votes.
B gets (100 – 46)% = 54% of the valid votes.
Therefore, % difference if the votes by which A is
defeated is = 54% - 46% = 8%.
8% of the total valid votes cast = 1600.
i.e. 8 of the total votes = 1600.
100
Total number of valid votes cast = 100 ×1600 = 20,000
8
Total number of valid votes cast is 20,000
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B) My income was increased by 10% and later decreased by
10%. What is the total change in the percent in my income?
SOLUTION:
Let my income be $100.
10% increase means that my income becomes $110.
Decreased income = $110 - $ 10 × 110
100
= $110 - $11 = $ 99.
% change in income = Change in income ×100%
original income
= $100 - $99 ×100% = 1%
$100
The percent in my income is 1%.
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Now you try!
1) A sales representative earns a 3.5%
commission on sales. Find the commission earned
when the total sales are $80,700.
2) Karen earns a salary of $32,500 per year plus
a 3.25% commission on sales. Find her total
salary for a year when the sales are $310,000.
3) Find the total amount owed after 6 months on a
loan of $900 at an annual interest rate of 8.5%.
4) Estimate the tip on a $42.65 check using a tip
rate of 15%.
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Applications of Percent
A percent change is an increase or decrease given as a
percent of the original amount.
Percent increase describes an amount that has grown
and percent decrease describes an amount that has been
reduced.
Percent Change
percent change = amount of increase or decrease
original amount
, expressed as a percent
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Finding Percent Increase or Decrease
Find each percent change. Tell whether it is a percent
increase or decrease.
A) from 25 to 56
SOLUTION:
percent change = amount of increase
original amount
= 56 - 25
25
= 31
25
Simplify the numerator.
= 0.96
= 96%
Write the answer as a percent.
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Finding the Result of a Percent Increase or Decrease
A) Find the result when 20 is increased by 40%.
0.40 (20) = 8
20 + 8 = 28
Find 40% of 20. This is the
amount of the increase.
It is a percent increase, so
add 8 to the original amount.
20 increased by 40% is 28.
B) Find the result when 75 is decreased by 60%.
0.60 (75) = 45
Find 60% of 75. This is the
amount of the decrease.
75 - 45 = 30
It is a percent decrease, so subtract
45 from the original amount.
75 decreased by 60% is 30.
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Common applications of percent change are
discounts and markups.
A discount is an amount
by which an original price
is reduced.
discount = % of original price
A markup is an amount
by which a wholesale cost
is increased.
markup = % of wholesale cost
final price
= original price - discount
final price
= wholesale cost + markup
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Markups
A) Kale buys necklaces at a wholesale price of $48
each. He then marks up the price by 75% and sells
the necklaces. What is the amount of the markup?
What is the selling price?
Solution:
Method 1: A markup is a percent increase. So
find $48 increased by 75%.
0.75 (48) = 36
48 + 36 = 84
Find 75% of 48. This is the
amount of the markup.
Add to 48. This is the
selling price.
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B) Lars purchased a daily planner for $32. The
wholesale cost was $25. What was the percent markup?
Solution:
Find the amount of the markup.
32 - 25 = 7
7 = x (25)
7 is what percent of 25? Let
x represent the percent.
7 = x . (25)
25
25
Since x is multiplied by 25,
divide both sides by 25 to
undo the multiplication.
0.28 = x
28% = x
Write the answer as a percent.
The markup was 28%.
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Some more examples
A) A trader marks his goods 40% above cost price and allows
a discount of 25% .What gain percent does he make?
SOLUTION:
Let the cost price be $100.
Then marked price = $140.
Discount = 25% of marked price = $140 × 25 = $35
100
Therefore, net selling price = Marked price – Discount
= $(140 – 35) = $105
Therefore, gain = 5%
Hence, the trader gains 5%.
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Finding Percent Increase or Decrease
Find each percent change. Tell whether it is a percent
increase or decrease.
A) from 25 to 56
SOLUTION:
percent change = amount of increase
original amount
= 56 - 25
25
= 31
25
Simplify the numerator.
= 0.96
= 96%
Write the answer as a percent.
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B) from 25 to 17
SOLUTION:
percent change = amount of decrease
original amount
= 25 - 17
25
=8
25
Simplify the numerator.
= 0.32
= 32%
Write the answer as a percent.
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Now you try!
Find each percent change. Tell whether it is a percent
increase or decrease
1) 35 to 45
2) 11 to 8
3) A dealer marked his goods 45% above the cost
price and allows a discount of 20% on the marked
price. Find his gain or loss percent.
4) What is the final price on a $175 leather jacket that
is on sale for 40% off?
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You did a great job
today!
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