Ratios and Proportional Relationships 1

Objective: You will be able to… Explain what a ratio is in your own words Explain what a proportion is in your own words Explain what a rate is in your own words Use ratio language to describe a ratio relationship between two quantities Use proportion language to describe a ratio relationship Use rate language to describe a ratio relationship Use ratio, proportion, and rate reasoning to solve problems 2 © 2013 Meredith S. Moody

Vocabulary Ratio: A relationship between two quantities Proportion: Two or more related quantities whose ratios can be simplified to the same fraction Rate: A ratio that compares two quantities with different units 3 © 2013 Meredith S. Moody

Example 1: Ratio A small group of students in a math class has 1 girl and 3 boys The ratio of girls to boys can be expressed four different ways: ◦ Using the “:” sign  1:3 (read as 1 to 3) ◦ Using fractions  1/3 (read as 1 to 3) ◦ As a decimal .33 ◦ As a percent  33% 4 © 2013 Meredith S. Moody

Example 2: Proportion The small group of students from the previous slide is proportional to the math class as a whole This means that for every 1 girl there are 3 boys If the class has 20 students, how many girls are there? How many boys? The ratio 1:3 is proportional to the whole class You can build the proportion until your total = 20 1:3  2:6  3:9  4:12  5:15 There are 5 girls and 15 boys in the entire class We use proportions to find values of larger or smaller quantities 5 © 2013 Meredith S. Moody

Example 2: Rate The most common example of a rate is speed The number of miles a car can travel per hour is a ratio between two numbers with different units  miles vs. hours If a car travels 30 miles every hour, the rate can be expressed as 30:1  30miles:1hour (read as 30 miles per hour) 6 © 2013 Meredith S. Moody

Guided practice 1: Expressing ratios The math class from example 1 has 20 students. 5 are girls and 15 are boys Instead of expressing the ratio in terms of girls:boys, we can express the ratio in terms of girls:total students OR boys:total students What is the ratio of girls to total students? ◦ 5:20 OR 5/20 ( ¼ ) OR.25 OR 25% What is the ratio of boys to total students? ◦ 15:20 OR 15/20 ( ¾ ) OR.75 OR 75% 7 © 2013 Meredith S. Moody

Guided practice 2: Proportions The math class ratio of girls:total students is proportional to the whole school’s ratio of girls:total students. The whole school has 1000 students How many girls are there in the school? Class ratio = 5:20 How many times would I have to add 20 to get to the total (1000) number of students in the school? ◦ 50 What is repeated addition? ◦ Multiplication Treat the ratio like a fraction and create an equivalent fraction There are 250 girls in the school 8 © 2013 Meredith S. Moody

Guided practice 3: Rates City A and City B are 120 miles apart Susan drove from City A to City B in 2 hours City C and City D are 200 miles apart Joe drove from City C to City D in 3 hours Who drove at a faster rate? What is Susan’s rate? 120 miles:2 hours  120/2  60/1  60mph What is Joe’s rate? 200 miles:3 hours  200/3  66.7/1  66.7mph Joe’s rate is greater, so Joe drove faster 9 © 2013 Meredith S. Moody

Check! Write down, in your own words, what a ratio is ◦ Share your explanation with a neighbor Write down, in your own words, what a proportion is ◦ Share your explanation with a neighbor Write down, in your own words, what a rate is ◦ Share your explanation with a neighbor 10 © 2013 Meredith S. Moody

You try! Express your ratios all 4 ways What is the ratio of strawberries to oranges? ◦ 3:5 OR 3/5 OR.6 OR 60% What is the ratio of oranges to strawberries? ◦ 5:3 OR 5/3 OR 1.67 OR 167% What is the ratio of strawberries to total fruits? ◦ 3:8 OR 3/8 OR.375 OR 37.5% What is the ratio of oranges to total fruits? ◦ 5:8 OR 5/8 OR.625 OR 62.5% 11 © 2013 Meredith S. Moody

You try! Proportions The fruit ratio in the previous scenario is proportional to the fruit ratio at the entire market There are 120 total strawberries and oranges at the market How many strawberries are there? ◦ How many oranges are there? ◦ 12 © 2013 Meredith S. Moody

You try! Rates A basket of 30 strawberries costs $1.50 A crate of 50 oranges costs $5.00 Which fruit is more expensive? (hint:) Find the rate (cost) of each fruit Strawberries: 30(strawberries)/$1.50  1(strawberry) / $0.05 ◦ The strawberries cost $ 0.05 each ($0.05 per strawberry) Oranges: 50(oranges)/$5.00  1(orange) / $0.10 ◦ The oranges cost $0.10 each ($0.10 per orange) The oranges are more expensive 13 © 2013 Meredith S. Moody