CONFIDENTIAL 1

Grade 8 Algebra1Grade 8 Algebra1

Review of Review of Proportion and Proportion and

PercentPercent

CONFIDENTIAL 2

Warm UpWarm 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?

1) 80% increase.

2) 56

3) 76.3

3) $115.5

CONFIDENTIAL 3

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.”

CONFIDENTIAL 4

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 = 1Students 15

1 = x15 675

1 = x 15 675675 675

x = 45

There are 45 faculty members.

Write a ratio comparing faculty to students.

Write a proportion. Let x be the numberof faculty members.

Since x is divided by 675, multiply both sides of the equation by 675.

CONFIDENTIAL 5

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.

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 unitsto another, multiply by a conversion factor.

Conversion factor

CONFIDENTIAL 6

Garry ate 53.5 hot dogs in 12 minutes to win a contest.

Find the unit rate. Round your answer to the nearest hundredth.

Finding Unit Rates

4.46 ≈ x

Write a proportion to find an equivalent ratio with a second quantity of 1.

Divide on the left side to find x.

53.5 = x 12 1

The unit rate is approximately 4.46 hot dogs per minute.

CONFIDENTIAL 7

The dwarf sea horse Hippocampus zosterae swims at a rate of 52.68 feet per hour. What is this speed in

inches per minute?

To convert the first quantity in a rate, multiply by a conversion factor with that unit in the second quantity.

52.68ft × 12 in 1h 1ft

The speed is 632.16 inches per hour.

632.16 in. 1h

Step1: Convert the speed to inches per hour.

Next page

Conversion factor

CONFIDENTIAL 8

To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity.

632.16 in × 1 h 1h 60 min

The speed is 10.536 inches per minute.

10.536 in 1 min

Step2: Convert this speed to 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.

CONFIDENTIAL 9

Solving Proportions

A) 5 = 3 9 w

Use cross products.

w = 27 5

Solve the proportion.

5w = 27

Divide both sides by 5.

5 = 39 w

5 (w) = 9 (3)

5w = 27 5 5

CONFIDENTIAL 10

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.

CONFIDENTIAL 11

Solution:

x · 1 = 100 (0.8)

Let x be the actual distance.

The actual distance is 80 mi.

A) On the map, the distance from Houston to Beaumont is 0.8 in. What is the actual distance?

1 in : 100 mi

x = 80

Scale Drawings and Scale Models

Use cross products to solve.

Write the scale as a fraction. map = 1 in actual 100 mi

0.8 in = 1 in x 100 mi

CONFIDENTIAL 12

Solution:

127 = 100x

Let x be the distance on the map.

The actual distance is 80 mi.

B) The actual distance between Bryan- College Station and Galveston is 127 mi. What is this distance on the map?

1 in : 100 mi

1.27 = x

Use cross products to solve.

Write the scale as a fraction. map = 1 in actual 100 mi

x = 1 in 127 100 mi

127 = 100x100 100

Since x is multiplied by 100, divide both sides by 100 to undo the multiplication.

CONFIDENTIAL 13

1) 3 = 1 z 8

Solve each proportion.

2) f + 3 = 7 12 2

1) z = 24 2) f = 39

Now you try!

3) x = 1 3 5

4) -1 = 3 5 2d

3) x = 0.6 4) d = -7.5

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.

5) $75

6) 200 times/sec

CONFIDENTIAL 14

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.

CONFIDENTIAL 15

ABC XYZ

62°

55°63°

A

B

C

62°

55°

63°

X

Z

Y

corresponding angles

corresponding sides.

6

7.5

7.5

8

10

10

AB = BC = CAXY YZ ZX

m A = m X

m B = m Y

m C = m Z

Make sure corresponding vertices are in the same order.

It would be incorrect to write ABC EFD.

CONFIDENTIAL 16

Scale factor

Any two corresponding sides in two similar figures have a common ratio called the scale factor. A B

CD

E F

H G

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.

CONFIDENTIAL 17

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?

Cone A Cone B

Volume = 1 ×∏r2h

3

1 ×∏(4)2 ×6 = 32∏ 3

1 ×∏(2)2 ×3 = 4∏ 3

Next page

6 cm

4 cm

3 cm

2cmAB

CONFIDENTIAL 18

Radii = 4 = 2 =2 2 1

The ratio of the volumes is thrice the ratio of the corresponding dimensions.

Heights = 6 = 2 =2 3 1

Volumes = 32∏ = 8 = 23 4∏ 1

6 cm

4 cm

3 cm

2cm

CONFIDENTIAL 19

Now you try!

1) Parallelogram ABCD parallelogram EFGH. Find the value of x.

A B

D C

E F

H G

1215

5x1) 3 units

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.

2) 8.6 feet

CONFIDENTIAL 20

A

B C

D E

x4

4

4

10

4

3) Triangle ABC triangle ADE. Find the value of x.

4) A rectangle has length 12 feet and width 8 feet. Every dimension of the rectangle is multiplied by 3

4to form a similar rectangle. What is the ratio of the

areas of the two triangles?

3) 5 units

4) 1.78 units

CONFIDENTIAL 21

Percent

Percent is a ratio that compares a number to 100. For example,

20% = 20100

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

CONFIDENTIAL 22

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

100 100

CONFIDENTIAL 23

A) What percent of 60 is 15?.

Method 1 Use a proportion.

part = percent Use the percent proportion. whole 100

15 = x Let x represent the percent. 60 100

60x = 1500 Find the cross products.

x = 25 Since x is multiplied by 60, divide both sides by 60 to undo the multiplication.

Finding the Percent

15 is 25% of 60.

CONFIDENTIAL 24

B) 60 is 0.6% of what number?

Finding the Whole

Method 2 Use an equation.

60 = 0.6% of x

60 = 0.006x

Write an equation. Let x represent the whole.

Write the percent as a decimal.

Since x is multiplied by 0.006, divide both sides by 0.006 to undo the multiplication.

60 = 0.006x0.006 0.006

10000 = x

60 is 0.6% of 10000

CONFIDENTIAL 25

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.

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 = percent whole 100

14 = x

24 100

24x = 1400

24x = 1400

24 24

x = 58.3

CONFIDENTIAL 26

1) 1 is what percent of 15?

2) 48 is 40% of what number?

3) 13 is 20% of what number?

1) 6.67%

2) 120

3) 65

Now you try!

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.

12) 68.75

CONFIDENTIAL 27

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 Interest rate peryear as a decimal

Time in years

Applications of Percent

CONFIDENTIAL 28

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.

total pay = base salary + commission

= base salary + % of total sales

= 40,000 + 2.5% of 800,000

SOLUTION:

Next page

Write the formula for total pay.

Write the formula for commission.

Substitute values given in theproblem.

Business Application

CONFIDENTIAL 29

= 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.

CONFIDENTIAL 30

Finance Application

A) Find the simple interest paid annually for 3 years on a $1200 loan at 15% per year.

I = P r t

I = (1200) (0.15) (2)

I = 540

SOLUTION:

Write the formula for simple interest.

Substitute known values. Write the interest rate as a decimal.

The amount of interest is $540.

CONFIDENTIAL 31

Estimating with Percents

A) The dinner check for Maria’s family is $67.95. Estimate a 15% tip.

Step 1: First round $67.95 to $70.

SOLUTION:

Move the decimal point one place left.

The tip should be about $10.50.

Step 2: Think: 15% = 10% + 5% 10% of $70 = $7.00

Step 3: Think: 5% = 10% ÷ 2 = $7.00 ÷ 2 = $3.50

Step 4: 15% = 10% + 5% = $7.00 + $3.50 = $10.50

CONFIDENTIAL 32

Estimating with Percents

B) The sales tax rate is 6.25%. Estimate the sales tax on a shirt that costs $29.50.

Step 1: First round 6.25% to 6% and $29.50 to $30

SOLUTION:

Move the decimal point two places to the left.

The sales tax is about $1.80.

Step 2: Think: 6% = 6 (1%) 1% of $30 = $0.30

Step 3: 6% = 6 (1%) = 6 ($0.30) = $1.80

CONFIDENTIAL 33

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%.

Total number of valid votes cast = 100 ×1600 = 20,000 8

Total number of valid votes cast is 20,000

8% of the total valid votes cast = 1600.

i.e. 8 of the total votes = 1600. 100

CONFIDENTIAL 34

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%.

CONFIDENTIAL 35

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.

1) $2824.5

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.

2) $42575

3) Find the total amount owed after 6 months on a loan of $900 at an annual interest rate of 8.5%.

3) $229.5

4) Estimate the tip on a $42.65 check using a tip rate of 15%.

4) $6.45

CONFIDENTIAL 36

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

CONFIDENTIAL 37

A) from 25 to 56

= 0.96

SOLUTION:

Simplify the numerator.

Write the answer as a percent.

Finding Percent Increase or Decrease

Find each percent change. Tell whether it is a percent increase or decrease.

percent change = amount of increase original amount

= 56 - 25 25

= 31 25

= 96%

CONFIDENTIAL 38

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

20 increased by 40% is 28.

Find 40% of 20. This is the amount of the increase.

It is a percent increase, so add 8 to the original amount.

B) Find the result when 75 is decreased by 60%.

0.60 (75) = 45

75 - 45 = 30

75 decreased by 60% is 30.

Find 60% of 75. This is the amount of the decrease.

It is a percent decrease, so subtract 45 from the original amount.

CONFIDENTIAL 39

Common applications of percent change are discounts and markups.

A discount is an amount by which an original priceis reduced.

discount = % of original price

final price = original price - discount

markup = % of wholesale cost

final price = wholesale cost + markup

A markup is an amountby which a wholesale costis increased.

CONFIDENTIAL 40

Markups

48 + 36 = 84Add to 48. This is the selling price.

0.75 (48) = 36 Find 75% of 48. This is the amount of the markup.

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?

Method 1: A markup is a percent increase. So find $48 increased by 75%.

Next Page

Solution:

CONFIDENTIAL 41

B) Lars purchased a daily planner for $32. The wholesale cost was $25. What was the percent markup?

The markup was 28%.

32 - 25 = 7

7 = x (25)

Find the amount of the markup.

7 is what percent of 25? Let x represent the percent.

0.28 = x

Since x is multiplied by 25, divide both sides by 25 toundo the multiplication.

7 = x . (25)25 25

Write the answer as a percent.28% = x

Solution:

CONFIDENTIAL 42

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

Hence, the trader gains 5%.

Therefore, net selling price = Marked price – Discount = $(140 – 35) = $105

Therefore, gain = 5%

CONFIDENTIAL 43

A) from 25 to 56

= 0.96

SOLUTION:

Simplify the numerator.

Write the answer as a percent.

Finding Percent Increase or Decrease

Find each percent change. Tell whether it is a percent increase or decrease.

percent change = amount of increase original amount

= 56 - 25 25

= 31 25

= 96%

CONFIDENTIAL 44

B) from 25 to 17

= 0.32

SOLUTION:

Simplify the numerator.

Write the answer as a percent.

percent change = amount of decrease original amount

= 25 - 17 25

= 8 25

= 32%

CONFIDENTIAL 45

Now you try!

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.

3)Gain = 16%

Find each percent change. Tell whether it is a percent increase or decrease

1) 35 to 45

2) 11 to 8

1) 28.57% increase

2) 27.27% decrease

4) What is the final price on a $175 leather jacket that is on sale for 40% off?

4) $105

CONFIDENTIAL 46

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