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### Ratios and Proportion

- A ratio is a comparison of two quantities by division.
- A proportion is an equation stating that two ratios are equal.
- Two ratios are equal if and only if their cross products are equal. To solve a proportion, use the fact that the cross products are equal. Set the cross products equal to each other and solve for the unknown value.
- A rate is a ratio of two quantities having different units of measurement.
- Solve problems involving rates by using proportions.
- A scale is a ratio that compares the size of a model to the size of the actual object. Proportions can be used to solve problems involving scales.

### Ratios and Proportion

The temperature increases from thirty to sixty degrees. What is the percent of change?

- percentage change = [(final value − original value)/original value] ×100
- original temperature = 30

final temperature = 60 - percentage ∆ = [(60 − 30)/60] ×100
- percentage ∆ = [30/60] ×100 percentage ∆ = 0.5 ×100 = 50

percentage ∆ = 50%

The temperature increases from sixty - two to seventy - five degrees. What is the percent of change?

- percentage change = [(final value − original value)/original value] ×100
- original temperature = 62 final temperature = 75
- percentage ∆ = [(75 − 62)/75] ×100
- percentage ∆ = [13/75] ×100
- percentage ∆ = [1300/75] = 17.―3

percentage ∆ ≈ 17.3%

A coat costs thirty - five dollars. The sales tax is seven percent. What is the total price?

- Price of coat + sales tax = total price
- sales tax = (.07)(35) = 2.45

35 + 2.45 = $37.45

A computer costs four hundred seventy - two dollars. The sales tax is eight percent. What is the total price?

- Price of computer + sales tax = total price
- sales tax = (.08)(472) = 37.76

472 + 37.76 = $ 509.76

A bicycle was originally priced at one hundred twelve dollars, but is discounted fifteen percent. What is the discounted price of the bicycle?

- Original price - amount of discount = final price

discount = (.15)(112) - 112 − (.15)(112) =
- 112 − 16.8 =

95.2

A dress was originally priced at sixty - nine dollars, but is discounted ten percent. How much is the dress now?

- Original price - amount of discount = final price

discount = (.10)(69) - 69 − (.10)(69) =
- 69 − 6.9 =

62.1

A suit was originally $160, but is discounted 30% . A sales tax of 7.5% is added, based on the discounted price.

What is the final price of the suit?

To the nearest percent, what is the percent of change between the original price and the final price paid, including sales tax?

What is the final price of the suit?

To the nearest percent, what is the percent of change between the original price and the final price paid, including sales tax?

- Original price - discount = discounted price

160 - (.30)(160) = discounted price - 160 − 48 = $ 112
- Discounted price + amount of sales tax = final price

112 + (.075)(112) = final price - 112 + 8.4 = $ 120.4
- percentage change = [(final value − original value)/original value] ×100
- [(120.4 − 160)/160] ×100
- [( − 39.6)/160] ×100
- − 0.2475 ×100 =

− 24.75

25% decrease

25% decrease

A camera was originally $225, but is discounted 15% . A sales tax of 7.5% is added, based on the discounted price.

What is the final price of the camera?

To the nearest percent, what is the percent of change between the original price and the final price paid, including sales tax?

What is the final price of the camera?

To the nearest percent, what is the percent of change between the original price and the final price paid, including sales tax?

- Original price - discount = discounted price

225 − (.15)(225) = discounted price - 225 − 33.75 = 191.25
- Discounted price + amount of sales tax = final price

191.25 + (.075)(191.25) = final price - 191.25 + 14.34 = $ 205.59
- percentage change = [(final value − original value)/original value] ×100
- [(205.59 − 225)/225] ×100
- [( − 19.41)/225] ×100

≈ − 8.627

9% decrease

9% decrease

A chair costs $12. The sales tax is 8.25% . What is the total price?

- price of chair + sales tax = total price
- sales tax = (.0825)(12) = 0.99

12 + .99 = $12.99

A television costs $540 but is discounted 20%. What is the discounted price of the television?

- original price - amount of discount = final price
- discount = (.20)(540) = $108

540 − 108 = $ 432

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Ratios and Proportion

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Definitions 0:07
- Ratio
- Different Representations
- Proportion
- Example
- Cross Product 1:08
- Cross Multiplication
- Example
- Rates 3:33
- Rates in Real Life
- Example 1: Form a Proportion 4:43
- Example 2: Cross Multiply 7:15
- Example 3: How Long to Drive 9:00
- Example 4: Cross Products 12:13

0 answers

Post by Venise Martinez on March 24, 2016

I think the first question in the practice questions is incorrect. Aren't you supposed to divide by original instead of final degree. ((75-62)/62)*100.

Thanks for your help!

0 answers

Post by Damien Thomas on May 5, 2013

You're beautiful, Dr. Eaton.

1 answer

Last reply by: Dr Carleen Eaton

Mon Nov 12, 2012 6:42 PM

Post by Thomas Powers on October 26, 2012

Fantastic lessons. Thank you!

1 answer

Last reply by: Dr Carleen Eaton

Thu Mar 1, 2012 2:21 PM

Post by Marcos Puglielli on February 29, 2012

Dr. you're the best - I can finally pass college algebra :D

1 answer

Last reply by: Dr Carleen Eaton

Wed May 12, 2010 12:07 AM

Post by shah Mahmoodi on April 23, 2010

Thank u Dr carleen excellent presentation and easy to understand the math formula.