Ratios and Proportion UMI: July 9, 2015

Ratios A ratio compares values: A ratio says how much of one thing there is compared to another thing 3:5 There are 3 diamonds to 5 hearts

Ratios can be shown in different ways: Using the ":" to separate the values: 3 : 5 Instead of the ":" we can use the word "to": 3 to 5 Or write it like a fraction: 𝟑 𝟓

3:5 A ratio can be scaled up: Here the ratio is also 3 diamonds to 5 hearts even though there are more diamonds and hearts

Using Ratios The trick with ratios is to always multiply or divide the numbers by the same value. Examples: 3 : 5 is same as 𝟑×𝟐:𝟓×𝟐=𝟔:𝟏𝟎 𝟔:𝟏𝟎 is same as 𝟔÷𝟐:𝟏𝟎÷𝟐=𝟑:𝟓

Example: A Recipe for pancakes uses 3 cups of flour and 2 cups of milk. So the ratio of flour to milk is 3 : 2 To make pancakes for a LOT of people we might need 4 times the quantity, so we multiply the numbers by 4: 3×4 : 2×4 = 12 : 8 In other words, 12 cups of flour and 8 cups of milk. The ratio is still the same, so the pancakes should be just as yummy.

"Part-to-Part" and "Part-to-Whole" Ratios: Example: There are 20 students, 5 are girls, and 15 are boys Part-to-Part: The ratio of boys to girls is 15:5=3:1 or 3/1 The ratio of girls to boys is 5:15=1:3 or 1/3

Part-to-Whole: The ratio of boys to all students is 15:20=3:4 or 3/4 The ratio of girls to all students is 5:20=1:4 or 1/4

Answer the following regarding the English alphabet: Try It Yourself Answer the following regarding the English alphabet: Determine the ratio vowels to constants. What is ratio of constants to vowels? What is ratio of constants to letters in the English alphabet? Write a word that has a ratio of 2:3 of vowels to constants. Part-to-Part: The ratio of boys to girls is 15:5=3:1 or 3/1 The ratio of girls to boys is 5:15=1:3 or 1/3

Proportions Proportion says that two ratios (or fractions) are equal. Example: So 1-out-of-3 is equal to 2-out-of-6 The ratios are the same, so they are in proportion.

Example: Rope A ropes length and weight are in proportion. When 20m of rope weighs 1kg, then: 40m of that rope weighs 2kg 200m of that rope weighs 10kg etc. So: 𝟐𝟎 𝟏 = 𝟒𝟎 𝟐 = 𝟐𝟎𝟎 𝟏𝟎 The ratios are the same, so they are in proportion.

𝒂 𝒃 = 𝒄 𝒅 is a proportion if, and only if, Theorem: If 𝒂, 𝒃, 𝒄, 𝒂𝒏𝒅 𝒅 are real numbers and 𝒃≠𝟎, 𝒂𝒏𝒅 𝒅≠𝟎, then 𝒂 𝒃 = 𝒄 𝒅 is a proportion if, and only if, 𝒂×𝒅=𝒃×𝒄

𝟓𝒊𝒏 𝟏𝟎𝒔𝒆𝒄 = 𝒙𝒊𝒏 𝟓𝟎 𝒔𝒆𝒄 Example: If a turtle travels 5 in. every 10 sec, how many feet does it travel in 50 sec? Solve: A proportion is set up with unit 𝟓𝒊𝒏 𝟏𝟎𝒔𝒆𝒄 = 𝒙𝒊𝒏 𝟓𝟎 𝒔𝒆𝒄 This implies that 𝒙=𝟐𝟓 𝒊𝒏. Consequently, since 12 in.=1 ft, the turtle travels 𝟐𝟓 𝟏𝟐 =𝟐 𝟏 𝟏𝟐 , or 2 ft 1 in.

Practice: The ratio of the weight of Meg’s cat to the weight of Anne’s cat is 5:7. Meg’s cat weighs 20 kg. How much more does Anne’s cat weigh?

Practice: Pat the Painter mixed 3 pints of yellow paint with 4 pints of green paint to make a nifty new color. He used 27 pints of yellow paint. How many pints of green paint will he need?

Practice: The ratio of boys to girls at the basketball game is 8:5. There are 30 girls. How many more boys are there than girls?

Practice: Ben and Matt received votes in the ratio 2:3. The total number of votes cast was 60. How many votes did Ben get?

Thank You References: A problem solving approach to mathematics for elementary school teachers by Billstein, Libeskind, Lott Math Playground: http://www.mathplayground.com/ Math is Fun : https://www.mathsisfun.com/index.htm