Math Workshop II

Unit 6

Ratios and Proportions

Ratios

Proportions Direct, Inverse, and Joint Variation

Percents Discount, Tax, and Interest

Name ___________________ Pd ________

6-3 Ratios and Proportions

• What is a ratio? A comparison of two numbers

• Example: If there are five girls and four boys in a class, what is the ratio of women to men?

The ratio of women to men is 5 to 4

• Ratios can be written in symbols two ways:

1.) With a colon: 5:4 (read “5 to 4”)

2.) As a fraction: 45 (read “5 to 4”)

• Ratios as fractions:

-Always reduced to lowest terms.

-A ratio written as an improper fraction does not need to be changed to a mixed

number.

A whole number ratio is always written with a denominator of 1.

Example: Reduce the ratio 10 to 4 to lowest terms.

ADDITIONAL EXAMPLES: Express each ratio in lowest terms.

1. 497 2.

945 3.

4411 4.

80150

5. 9:15 6. 15:20 7. 100:28 8. 50:25 9. 12 to 8 10. 3 to 9 11. 16 to 4 12. 5 to 35

• What is a proportion? A proportion is made up of two equal ratios.

• Proportions can be written in symbols in two ways:

1.) With a colon: 3:4 = 9:12 (read “Three is to four as nine is to twelve”)

2.) As a fraction: 43 =

129 (read “Three is to four as nine is to twelve”)

- Example: Two is to six as eight is to sixteen.

• Determine if two proportions are equal 1.) Determine the cross product 2.) Write or= ≠

- Example:. 1. 54 ___

1512 2.

3012 ___

2510

• In a proportion, the cross-products are equal. To find the cross-products,

cross multiply.

- Example: 43 =

129

• To find a missing term in a proportion:

1.) Write the proportion as two equal fractions.

2.) Represent the missing term by a variable.

3.) Use cross multiplication to write an equation for the unknown and solve.

Example 1: Find the missing term in the proportion x:8 = 4:16.

Example 2: Find the missing term in the proportion 2030 =

y90 .

ADDITIONAL EXAMPLES:

DIRECTIONS: Find the missing term in each proportion.

1. _ __:4 = 6:8 2. 1: = 4:20

3. 9:12 = 15: 4. ?5 =

610

5. 6

2 x+ = 9

12

6-3 Ratios and Proportions

Solve each proportion. Show work!

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6-3 Ratios and Proportions

Use a proportion to solve each problem.

1. To make a model of the Guadeloupe River bed, Henry used 1 inch of clay for the 5 miles of the river’s actual length. His model was 50 inches long. How long is the Guadeloupe River?

2. Josh finished 24 math problems in 1 hour. At this rate, how many hours will it take him to complete 72 problems.

3. Mike’s boat used 5 gallons of gasoline in 4 hours. At this rate, how many gallons of gasoline will the boat use in 10 hours?

4. Shauna paints a room that has 400 square feet of wall space in 2.5 hours. At this rate, how long will it take her to paint a room that has 720 square feet of wall space?

5. Walker is planning a summer vacation. He wants to visit Petrified National Forest and Meteor Crater , Arizona, the 50,000 year old impact site of a large meteor. On a map with a scale where 2 inches equals 75 miles, the two areas are about 1.5 inches apart. What is the distance between petrified National Forest and Meteor Crater?

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6-3 Ratios and Proportions

Use cross products to determine if each pair of ratios forms a proportion. Write yes or no.

7.

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6-3 Ratios and Proportions

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Write a proportion to solve each problem. Show work!

6-4 Variation

Direct Variation – y varies directly as x if there is some constant k such that y = kx is called the constant of variation. Formula: y = kx Proportion : x1 = x2

y1 y2 Examples :

1. If y varies directly as x and y = 16 when x = 4, find x when y = 20

2. Given that m varies directly as n and m = 75 when n = 25, find the constant of variation, k

Inverse Variation – y varies inversely as x if there is some constant k such that xy = k or

y = xk .

Formula: xy = k or y = xk . Proportion : x1 = x2

y2 y1 Examples :

1. If x varies inversely directly as y and x = 8 y = 12, find x when y = 4

2. Given that a varies inveresly as b and m = 3 when n = 4, find the constant of variation, k.

Joint Variation – y varies jointly as x and z if there is some constant k such that y = kxz where x 0 and z 0.. ≠ ≠ Formula: y = kxz Proportion : y1 = y2 x1z1 x2z2 Examples :

1. If y varies jointly as x and z and y = 1 when x = 2 and z = 4, find y when x = 4 and z = 3.

3. Given that y varies jointly as x and z and y = 200 when x = 25 and z = 4, find the constant of variation, k

Word Problems Examples

1. The amount of interest earned on a savings is directly proportional to the amount of money saved, If 426 is earned on $325, how much interest will be earned on $900 in the same period of time?

2. A fulcrum and a lever are inversely proportional. If a 12 g mass is 60 cm from the fulcrum of a lever, how far from the fulcrum is a 45 g mass that balances the 12 g mass?

3. The area A of a trapezoid varies jointly as its height and the sum of its bases. If the area is 240 square meters when the height is 10 meters and the sum of the bases is 12 meters, what is the area of a trapezoid when its height is 12 meters and the sum of the bases is 35 meters?

6-4 Variation

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6-4 Variation

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6-4 Variation

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6-4 Variation

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Solve each problem. Show work! 1. An employee’s wages are directly proportional to the time worked. If an

employee earns $120 for 8 hours, how much will the employ earn for 20 hours?

2. A certain car used 21 gallons of gasoline in 7 hours. If the rate of gasoline used is a direct variation, then how much gasoline will the care use on a 6 hour trip?

3. The numbers of words typed is directly proportional to the time spent typing. If a typist can type 325 word in 5 minutes, how long will it take to type a 1040 word report?

4. The volume V of gas varies inversely as its pressure P. If V = 80 cubic centimeters when P = 2000 millimeters of mercury, find V when P = 320 millimeters of mercury.

5. The area A of a trapezoid varies jointly as its height and the sum of its bases. If the area is 480 square meters when the height is 20 meters and the sum of the bases is 48 meters, what is the area of a trapezoid when its height is 8 meters and the sum of the bases is 25 meters?

6-5 Percents

Percents

Formula: ofIs =

100%

Examples :

1. 24 is 20% of what number?

2. What percent of 60 is 45? 3. What is 7% of 480? Percent of change –When an increase or decrease in an amount is expressed as a percent, the percent is called the percent of change. If the new number is greater than the original number , the percent of change is a percent increase. If the new number is less than the original number, then the percent of change is the percent decrease.

Formula: Amount of Change x 100 Original Amount Examples :

1. Find the percent of increase original: 48 new: 60

2. Find the percent of decrease original: 30 new: 22

Discount – Discounted prices are applications of percent of change. Discount is the amount by which the regular price of an item is reduced. Thus, the discounted price is an example of percent of decrease...

Formula: Original Price – Discount = Sale Price Steps:

1. ) Change the percent to a decimal by moving the decimal two places to the left.

2. ) Multiply the decimal by the original amount and round to the nearest cent to get the amount of the discount. .

3. ) Subtract the amount of the discount from the original price to get the sale price.

Examples :

1. A coat is on sale for 25% off the original price. If the original price of the coat is $75, what is the sale price?

Sales Tax – Sales tax prices are applications of percent of change. Sales tax is the amount that is added to the original price. Thus, the price including tax is an example of percent of increase.

Formula: Original Price + Tax = Final Price Steps:

1.) Change the percent to a decimal by moving the decimal two places to the left. 2.) Multiply the decimal by the original amount and round to the nearest cent to get the amount of the discount. . 3.) Add the amount of the tax to the original price to get the final price.

Examples :

1. A pair of jeans cost $35 The sales tax is 7% What is the final cost of the jeans including tax?

Interest

Formula: I = PRT I = $ Interest P = $ Principal Amount R = % rate of Interest T = yrs Time Formula: Total Amount = Principal Amount + Interest Steps: 1. Change the rate of interest to a decimal by moving the decimal two places to

the left. 2. Change the time to years (÷12) or months (x 12) if needed. 3. Substitute the values into the formula 4. Solve

Examples :

1. Find the amount of interest charged and the total amount if $120 is at a 2% per month for 4 months.

2. Find the amount of interest charged and the total amount if $345.24 is at 12.5% per year for 5 years

3. Find the amount of interest charged and the total amount if $400 is at 5% per year for 18 months

4. Find the amount of interest charged and the total amount if $1234.75 is at .8% per month for 2 years.

6-5 Percents

Solve each problem. 1. 25% of what is 80? 2. What percent of 72 is 18? 3. 60% of what is 45? 4. What percent of 12 is 6? 5. What is 60% of 12? 6. 75% of what is 48? 7. What is 28% of 650? 8. What percent of 150 is 90? 9. What percent of 90 is 63? 10. What is 37% of 60? 11. 22.5% of what is 43 12. 45% of what is 99? 13. What percent of 210 is 10.5? 14. 160% of what is 124? 15. What is 8.25% of 160? 16. What is 250% of 14? Write an equation to model each question and solve. 17. Pablo has a goal to lose 25 pounds. He has lost 16 pounds. What percent of his goal has he reached? 18. You spent 16% of your vacation money on food. If you spent $48 on food, how much money did you spend on your vacation? 19. A writer earns $3400 a month. Last month she spent $204 on food. What percent of her income was spent on food? 20. Suppose 62.5% of freshman graduate from college. If there are 2650 freshman, how many will graduate from college?

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6-5 Percents

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6-5 Percents

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6-5 Percents

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6-5 Percents

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6-5 Percents

Find the final price of each item

1. television: $375 2. DVD player: $269 discount: 25% discount: 20% tax: 6% tax: 7%

3. printer: $255 4. Class ring: $189 discount: 30% discount: 17% tax: 5.5% tax: 5% 5. Software: $44 6. Video recorder: $110.95 discount: 21% discount: 30% tax: 6% tax: 5%

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6-5 Percents

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6-5 Percents

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6-5 Percents

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Solve each problem. Show all work! 1. The original price of a new sports video was $65. Due to the demand the price

was increased to $87.75. What was the percent of increase over the original price?

2. A high school paper increased its sales by 75% when it ran an issue featuring a contest to win a class party. Before the contest issue, 10% of the school’s 800 students bought the paper. How many students bought the contest paper?

3. Baseball tickets cost $15 for general admission or $20 for box seats. The sales tax on each ticket is 8%, and the municipal tax on each ticket is an additional 10% of the base price. What is the final cost of each type of ticket?

4. The price per share of an internet related stock decreased from $90 per share to $36 per share early 2009. By what percent did the price of the stock decrease?

5. Customers of a utility company received notices in their monthly bills that heating costs for the average customer had increased 125% over the last year because of an unusually severe winter. In January of last year, the Garcia’s paid $120 for heating. What should they expect to pay this January if their bill increased by 125%?

Name____________________________________ Mrs. Mesko Date___________ Math Workshop II Period_______ 20 Unit 6 - Percents For #1-3, determine the desired quantity. 1. What is 30% of 70? 2. 90% of what number is 72? 3. What percent of 45 is 18? For #4-10, answer the question from the problem situation. 4. The price of a pair of shoes is reduced from $80 to $68. What is the percent of discount? 5. Joe King got a 5% raise. His previous salary was $7.50/hr. What is his new pay rate? 6. Anita Jaub bought a new shirt that was originally marked $23. It is on a clearance rack marked “30%

percent off”. What will the actual price be? 7. Robbie ran 25 miles in 4 hours. At that rate how long did it take him to run an additional 20 miles? 8. Your monthly rent is $725. If your rent is 18% of your monthly income, then what is monthly income? - 18 -

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9. The amount of interest earned on savings is directly proportional to the amount of money saved. If $32 interest is earned on $410, how much interest will be earned on $1200 in the same period of time?

10. If y varies directly as x and y = 8 when x = 1.5, then find y when x = 5 11. If y varies inversely as x and y = -18 when x = 9, then find y when x = 4 12. If y varies jointly as x and z and y = 10 when z = 4 and x = 5, then find y when x = 4 and z = 2. 13. Mandy is shopping at Giant Eagle. She buys a pack of gum for $.88, toothpaste for $2.39, two DVDs for

$9.99 each and printer ink cartridge for $27.99. Mandy has a coupon for 15% off a single item. She decides she should apply it to the most expensive item in order to save the most money.

Answer each of the following and explain your work.

a.) What is the total price before tax including the discounted item on Mandy’s purchases?

b.) Using a 7% sales tax rate and assuming that she has to pay tax on all items, then how much sales tax will Mandy have to pay?

c.) What is the total price that Mandy has to pay?