December 12, 2014

Today:TGIF, December 12, 2014

Warm-Up Percent Proportion

Review for Test Tuesday Complete Classwork 2.7

2. You have $100 and are playing poker. On the first hand you bet 50% of your money, and lose 100% of the bet.

On the second hand, you bet 100% of the remaining money and win 100% of your bet back plus 50%. (150% of your bet)

After the first two hands, how much money do you have?

2a. What is 70% of ½?

1. What percent of 28 is 𝟐

𝟕?

5. On a test with 80 questions, you answered 68 correctly. What was your percent score?

3. Five of 34 students were absent yesterday. What percent of students were present?

Mentis Arithmetica

2. What is 45% of 80? 3. 36 is what % of 30?

Try using mental math first:

4) 300% of 9

1) 25% of 80 3) 15% of 5002) 120% of 50

5) 5% of 1500 6) What % of 35 is 7?

Test Review

1. 0.6 = 5

2. 0.14 = 50

? ?

Fill in the missing numbers:

? 4. 0.012 = 250

3. 0.016 = 125

?

Write as a decimal:

5. One-tenth of a dollar 6. One-tenth of ninety cents

7. One-tenth of two dollars and seventy cents.

Vocabulary :Compliments:

Two numbers are compliments when their sum is equal to 100%. In other words, the sum of the two parts equals the whole.Every number has it’s own compliment.

Ratio: A comparison of two numbers by division expressed

in the following ways: 𝟏

𝟐, 1 to 2, 1:2

Ratio’s Must use the same Unit of Measurement

Finding Percent Increase & Decrease

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 be reduced.

Finding Price Before Increase or Decrease

This formula is used to find the original price of an item when the price is known after increases such as taxes, tips, and markups.

The formula is also used when the original price is not known after decreases such as markdowns and discounts.

Original Price: Price after increase or decrease1 + percent or

The cost of lunch after a 15% tip was $24.15. What was the cost of the lunch alone? $24.15

1.15 1 +15%

= $21.00

Common percent changes are discounts and

markups.A discount is an amount by which an original price is reduced.

discount = % of original price

final price = original price – discount

A markup is an amount by which a wholesale price is increased.

final price = wholesale cost markup+

markup wholesale cost= % of

Applying Percent Changes

Class Notes & Practice Problems:

Find each percent change. Tell whether it is a

percent increase or decrease.From 8 to 10

= 0.25

= 25% Write the answer as a percent.

Simplify the fraction.

8 to 10 is an increase, so a change from 8 to 10 is a 25%

increase.

Ex. 1A: Percent Increase and Decrease

Change to a decimal.

From 75 to 30

= 0.6


Simplify the numerator.


75 to 30 is a decrease, so a change from 75 to 30 is a

60% decrease.

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

Write as a decimal

Ex. 1B: Finding Percent Increase and Decrease

1. From 200 to 110

= 0.45




200 to 110 is an decrease, so a

change from 200 to 110 is a 45%

decrease.

Find the percent change. Tell whether it is a percent

increase or decrease.

Practice 1: Percent Increase and Decrease

Write as a decimal

2. From 25 to 30

= 0.20




25 to 30 is an increase, so a change from 25 to

30 is a 20% increase.

Find each percent change. Tell whether it is a

percent increase or decrease.

Practice 2: Percent Increase and Decrease

Write as a decimal

A. Find the result when 12 is increased by 50%.

0.50(12) = 6 Find 50% of 12. This is the amount of increase.

12 + 6 =18It is a percent increase, so add 6

to the

original amount. 12 increased by 50% is 18.

B. Find the result when 55 is decreased by

60%.0.60(55) = 33 Find 60% of 55. This is the amount of decrease.

55 – 33 = 22 It is a percent decrease so subtract 33 from the original amount.

55 decreased by 60% is 22.

Example 1: Percent Increase and Decrease

A. Find the result when 72 is increased by 25%.

0.25(72) = 18 Find 25% of 72. This is the amount of increase.

72 + 18 =90 It is a percent increase, so add 18

to the original amount.

72 increased by 25% is 90.

B. Find the result when 10 is decreased

by 40%.0.40(10) = 4 Find 40% of 10. This is the amount of

decrease.10 – 4 = 6 It is a percent decrease so subtract 4

from the original amount.

10 decreased by 40% is 6.

Example 2: Percent Increase and Decrease

Common percent changes are discounts and

markups.A discount is an amount by which an original price is reduced.

discount = % of original price

final price = original price – discount

A markup is an amount by which a wholesale price is increased.

final price = wholesale cost markup+

markup wholesale cost= % of

Applying Percent Changes

The entrance fee at an amusement park is $35.

People over the age of 65 receive a 20% discount.

What is the amount of the discount? How much do

people over 65 pay?Method 1: A discount is a percent decrease. So find

$35 decreased by 20%.

0.20(35) = 7 Find 20% of 35. This is the

amount of the discount.

35 – 7 = 28 Subtract 7 from 35. This is the

entrance fee for people over

the age of 65.

Practice 1: Percent Discounts

Method 2: Subtract the percent discount from

100%.

100% – 20% = 80% People over the age of 65 pay 80% of

the regular price, $35.

0.80(35) = 28 Find 80% of 35. This is the entrance

fee for people over the age of 65.

35 – 28 = 7 Subtract 28 from 35. This is the

amount of the discount.

By either method, the discount is $7. People over the

age of 65 pay $28.00.


A $220 bicycle was on sale for 60% off. Find the sale

price. Use Method 2:

100% – 60% = 40% The bicycle was 60% off of 100% .

0.40(220) = 88 Find 40% of 220.

By this method, the sale price is

$88.


The wholesale cost of a DVD is $7. The markup is

85%. What is the amount of the markup? What is the

selling price?Method 1

A markup is a percent increase. So find $7 increased by 85%.

0.85(7) = 5.95

7 + 5.95 = 12.95

Find 85% of 7. This is the amount of the

markup.

Add to 7. This is the selling price.

Subtract from 12.95. This is the

amount of the markup.12.95 ÷ 7 = 5.95

By either method, the amount of the markup is

$5.95. The selling price is $12.95.

Method 2

Add percent markup to 100%

The selling price is 185% of the

wholesale price, 7.100% + 85% = 185%

Find 185% of 7. This is the selling price.1.85(7) = 12.95

Practice 1: Percent Markups

A video game has a 70% markup. The wholesale cost

is $9. What is the selling price?

Method 1

A markup is a percent increase. So find $9 increased

by 70%.

0.70(9) = 6.30 Find 70% of 9. This is the amount of the markup.

9 + 6.30 = 15.30 Add to 9. This is the selling price.

The amount of the markup is $6.30. The selling price is

$15.30.

Practice 2: Percent Markups

1. from 20 to 28.

2. from 80 to 62.

3. from 500 to 100.

4. find the result when 120 is increased by 40%.

5. find the result when 70 is decreased by 20%.

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

40% increase

22.5% decrease

80% decrease

168

56

Lesson Quiz: Part I

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

6. A movie ticket costs $9. On Mondays, tickets are 20% off. What is the amount of discount? How much would a ticket cost on a Monday?

7. A bike helmet cost $24. The wholesale cost was $15. What was the percent of markup?

$1.80; $7.20

60%

Lesson Quiz: Part II

Example 2: Measurement Application

A flagpole casts a shadow that is 75 ft long at the

same time a 6-foot-tall man casts a shadow that is 9 ft

long. Write and solve a proportion to find the height

of the flag pole.

The flagpole is 50 feet tall.

Since h is multiplied by 9, divide both sides

by 9 to undo the multiplication.

Percents

Warm Up What is 70% of ½?

Change each percent to a decimal.

1. 73% 2. 112%

3. 0.6% 4. 1%

Change each fraction or mixed number to a decimal.

5. 6. 7. 8.

Solve each proportion.

9. 10.

0.73

0.006

0.8

1.12

1.2

0.01

0.5

12 4.2

0.3

Percents

Example 3B: Finding the Whole

20 is 0.4% of what number?

Method 2 Use an equation.

20 = 0.4% of x

20 = 0.004 • x

5000 = x

Write an equation. Let x represent the whole.

Write the percent as a decimal.

Since x is multiplied by 0.004, divide both sides by 0.004 to undo the multiplication.

20 is 0.4% of 5000.

Percents

90 is 120% of what number?

Method 1 Use a proportion.

Use the percent proportion.

120x = 9000

x = 75

120% of 75 is 90.

Let x represent the whole.

Find the cross products.

Since x is multiplied by 120, divide both sides by 120 to undo the multiplication.

Percents

Lesson Quiz: Part 1

1. Find 20% of 80.

2. What percent of 160 is 20?

3. 35% of what number is 40?

4. 120 is what percent of 80?

5. Find 320% of 8.

6. 65 is 0.5% of what number?

Find each value. Round to the nearest tenth if necessary.

16

12.5%

114.3

150%

25.6

13,000

1. Order from least to greatest: 2/8, 2.8%, 8/2, .28

3. 20 is 40% of what number?

4. 36 is what percent of 30?

5. What is the total cost of a $21.00 lunch and 15% tip?

6. Which fraction must have more than two decimal

places?

A.) ¼ B.) 2/5 C.) 12/50 D.) 5/6 E.)

None

Warm-Up:

* A certain item used to sell for seventy-five cents a pound, you see that it's

been marked up to eighty-one cents a pound. What is the percent increase?

First, I have to find the increase: 81 – 75 = 6

The price has gone up six cents. Now I can find the percentage increase over the

original price.

Note this language, "increase/decrease over the original", and use it to your

advantage: it will remind you to put the increase or decrease over the original value,

and then divide.

This percentage increase is the relative change:

6/75 = 0.08 or an 8% increase in price per pound.

An important category of percentage exercises is markup and markdown problems.

For these, you calculate the markup or markdown in absolute terms (you find by how

much the quantity changed), and then you calculate the percent change relative to

the original value. So they're really just another form of "increase - decrease"

exercises.

* A computer software retailer used a markup rate of 40%. Find the selling

price of a computer game that cost the retailer $25.

The markup is 40% of the $25 cost, so the markup is: (0.40)(25) = 10

A golf shop pays its wholesaler $40 for a certain club, and then sells it to a golfer for $75. What is the markup rate? First, I'll calculate the markup in absolute terms: 75 – 40 = 35

Then I'll find the relative markup over the original price, or the markup rate: ($35) is (some percent) of ($40), or: 35 = (x)(40) so the relative markup over the original price is: 35 ˜ 40 = x =0.875

Since x stands for a percentage, I need to remember to convert this decimal

value to the corresponding percentage. The markup rate is 87.5%.

* A shoe store uses a 40% markup on cost. Find the cost of a pair of shoes that

sells for $63.

This problem is somewhat backwards. We have the selling price, which is cost

plus markup, and they gave me the markup rate, but they didn't tell me the actual

cost or markup. So I have to be clever to solve this.

I will let "x" be the cost. Then the markup, being 40% of the cost, is 0.40x. And

the selling price of $63 is the sum of the cost and markup, so:

63 = x + 0.40x 63 = 1x + 0.40x 63 = 1.40x 63 ˜ 1.40 = x= 45

The shoes cost the store $45.

* An item originally priced at $55 is marked 25% off. What is the sale price?

First, find the markdown. The markdown is 25% of the price of $55, so: x =

(0.25)(55) = 13.75

By subtracting this markdown from the original price, find the sale price:

55 – 13.75 = 41.25 The sale price is $41.25.

* An item that regularly sells for $425 is marked down to $318.75. What is the

discount rate?

First, I'll find the amount of the markdown: 425 – 318.75 = 106.25

Then I'll calculate "the markdown over the original price", or the markdown

rate: ($106.25) is (some percent) of ($425), so: 106.25 = (x)(425)

...and the relative markdown over the original price is: x = 106.25 ˜ 425 =

0.25

Since the "x" stands for a percentage, I need to remember to convert this

decimal to percentage form. The markdown rate is 25%.

* An item is marked down 15%; the sale price is $127.46. What was the original

price?

This problem is backwards. They gave me the sale price ($127.46) and the

markdown rate (15%), but neither the markdown amount nor the original price. I

will let "x" stand for the original price. Then the markdown, being 15% of this

price, was 0.15x. And the sale price is the original price, less the markdown, so I

get: x – 0.15x = 127.46 1x – 0.15x = 127.46

0.85x = 127.46 x = 127.46 ˜ 0.85 = 149.952941176...

This problem didn't state how to round the final answer, but dollars-and-cents

is always written with two decimal places, so: The original price was $149.95.

Note in this last problem that I ended up, in the third line of calculations, with an

equation that said "eighty-five percent of the original price is $127.46". You can

save yourself some time if you think of discounts in this way: if the price is 15%

off, then you're only actually paying 85%. Similarly, if the price is 25% off, then

you're paying 75%; if the price is 30% off, then you're paying 70%; and so on.

While the values below do not refer to money, the procedures used to solve these

problems are otherwise identical to the markup - markdown examples on the

previous page.

* Your friend diets and goes from 125 pounds to 110 pounds. What was her

percentage weight loss? First, I'll find the absolute weight loss: 125 – 110 = 15

This fifteen-pound decrease is some percentage of the original, since the rate

of change is always with respect to the original value. So the percentage is

"change over original", or: 15 = (x)(125)

15 ˜ 125 = x (The change, 15, is over the original, 125.) 15 ˜ 125 = 0.12

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December 12, 2014