Ratios, Proportions, and Percentages in Mathematics
Learn about ratios, proportions, and percentages in mathematics, along with practical examples and solutions. Find out how to write ratios, solve proportions, and make comparisons in real-world scenarios, illustrated with classroom examples. Discover the concept of cross products and understand when proportions are true or false.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
2.6 Ratio, Proportion, and Percent
Write ratios. A ratio is a comparison of two quantities using a quotient. Ratio The ratio of the number a to the number b (b 0) is written , atob a b . : , a b or The last way of writing a ratio is most common in algebra. Slide 2.6-4
CLASSROOM EXAMPLE 1 Writing Word Phrases as Ratios Write a ratio for each word phrase. 3 days to 2 weeks Solution: 3 days days 3days weeks 3 = = weeks = days 2 7 14 days 14 14 12 hr to 4 days 12 96 hours hours 1 8 hours days = = days hours = hours 4 24 96 4 Slide 2.6-5
CLASSROOM EXAMPLE 2 Finding Price per Unit A supermarket charges the following prices for pancake syrup. Which size is the best buy? What is the unit cost for that size? Solution: $3.89 36 = $0.108 $2.79 24 $1.89 12 = $0.116 = $0.158 The 36 oz. size is the best buy. The unit price is $0.108 per oz. Slide 2.6-6
Solve proportions. A ratio is used to compare two numbers or amounts. A proportion says that two ratios are equal, so it is a special type of equation. For example, 3 4 15 20 = 3 4 15 20 is a proportion which says that the ratios and are equal. In the proportion a b c d ( ) = , 0 , b d a, b, c, and dare the terms of the proportion. The terms aandd are called the extremes, and the terms bandc are called the means. We read the a c b d proportions as ais to bas c is to d. = Slide 2.6-8
Solve proportions. (contd) Beginning with this proportion and multiplying each side by the common denominator, bd, gives a b c d a b c d ( ) ( ) = = = b d bd . ad bc We can also find the products ad and bc by multiplying diagonally. bc a b c d = ad For this reason, ad andbcare called cross products. Slide 2.6-9
Solve proportions. (contd) Cross Products a c = If then the cross products ad and bc are equal that is, the product , b d of the extremes equals the product of the means. a b c d ( ) Also, if then ad bc = , = where , 0 . b d a b = If then ad = cb, or ad = bc. This means that the two proportions , c d are equivalent, and the proportion can also be written as a c b d a c b d ( ) = = , 0 . c d Sometimes one form is more convenient to work with than the other. Slide 2.6-10
CLASSROOM EXAMPLE 3 Deciding Whether Proportions Are True Decide whether the proportion is true or false. 15 62 = 930 Solution: False 21 15 62 45 = 21 45 = 945 17 91 1547 = Solution: True 13 17 91 119 = 13 119 1547 = Slide 2.6-11
CLASSROOM EXAMPLE 4 Finding an Unknown in a Proportion 35. 42 x= Solve the proportion 6 Solution: x 4 6 35 = 210 x= 42 2 42 42 5 x = The solution set is {5}. The cross-product method cannot be used directly if there is more than one term on either side of the equals symbol. Slide 2.6-12
CLASSROOM EXAMPLE 5 Solving an Equation by Using Cross Products x+ 6 2. 5 = Solve 2 ( + ) 6 5 30 Solution: x+ = 30 5 5 2 2 4 = = 5 3 6 0 x 2 5 26 5 x x = 26 5 . The solution set is When you set cross products equal to each other, you are really multiplying each ratio in the proportion by a common denominator. Slide 2.6-13
Objective 3 Solve applied problems by using proportions. Slide 2.6-14
CLASSROOM EXAMPLE 6 Applying Proportions Twelve gallons of diesel fuel costs $37.68. How much would 16.5 gal of the same fuel cost? Solution: Let x = the price of 16.5 gal of fuel. $37.68 12 gal 12 x = 16.5 621. 12 51.81 gal 72 x= 12 x = 16.5 gal of diesel fuel costs $51.81. Slide 2.6-15
Objective 4 Find percents and percentages. Slide 2.6-16
Write ratios. A percent is a ratio where the second number is always 100. Since the word percent means per 100, one percent means one per one hundred. = 1 = 1% 0.01, 1% or 100 Slide 2.6-17
CLASSROOM EXAMPLE 7 Converting Between Decimals and Percents Convert. 310% to a decimal Solution: 3.1 8% to a decimal .08 0.685 to a percent 68.5% Slide 2.6-18
CLASSROOM EXAMPLE 8 Solving Percent Equations Solve each problem. What is 6% of 80? Solution: x = x = .06 80 4.8 16% of what number is 12? 1200 16 x = x = x = .16 1200 75 What percent of 75 is 90? 9000 75 = x = = 9000 75 x 1.2 or 120% x Slide 2.6-19
CLASSROOM EXAMPLE 9 Solving Applied Percent Problems Mark scored 34 points on a test, which was 85% of the possible points. How many possible points were on the test? Solution: Let x = the number of possible points on the test. 34 x 8 00 1 5 = 34 0 0 85 8 40 x = 85 5 x = There were 40 possible points on the test. Slide 2.6-20
