1. Ratios and Proportion
UMI: July 9, 2015

2. 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

3. 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

4. 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

5. 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

6. 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.

7. "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

8. 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

9. 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

10. 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.

11. 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.

12. 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

13. 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.

14. 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?

15. 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?

16. 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?

17. 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?

18. Thank You References: A problem solving approach to mathematics
for elementary school teachers by Billstein,
Libeskind, Lott
Math Playground:
Math is Fun :