Ratios and Proportion

UMI: July 9, 2015

RatiosA ratio compares values: A ratio says how much of one thing there is compared to another thing

There are 3 diamonds to 5 hearts

Ratios can be shown in different ways:

1. Using the ":" to separate the values: 3 : 5

2. Instead of the ":" we can use the word "to": 3 to 5

3. Or write it like a fraction:

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 is same as

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 3/1The 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

Try It Yourself

Answer the following regarding the English alphabet:

1. Determine the ratio vowels to constants.

2. What is ratio of constants to vowels?3. What is ratio of constants to letters in

the English alphabet?4. 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 3/1The ratio of girls to boys is 5:15=1:3 or 1/3

1. Determine the ratio vowels to constants.

Solve:

2. What is ratio of constants to vowels?Solve:

3. What is ratio of constants to letters in the English alphabet?

Solve:

4. Write a word that has a ratio of 2:3 of vowels to constants.

Solve:

Part-to-Part:The ratio of boys to girls is 15:5=3:1 or 3/1The ratio of girls to boys is 5:15=1:3 or 1/3

ProportionsProportion 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: RopeA 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:

The ratios are the same, so they are in proportion.

Theorem:If are real numbers and then

is a proportion if, and only if,

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

This implies that . Consequently, since 12 in.=1 ft, the turtle travels , or 2 ft 1 in.

Practice:

1. 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?

Solve:

Answer: Anne’s cat weighs 8 kg more than Meg’s cat.

Practice:

2. 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?

Solve:

Answer: He will need 36 pints of green paint .

Practice:

3. 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?

Solve:

Answer: There are 18 more boys than girls.

Practice:

4. Ben and Matt received votes in the ratio 2:3. The total number of votes cast was 60.

How many votes did Ben get?

Solve:

Answer: Ben gets 24 votes.

Thank You

References:1. A problem solving approach to mathematics for elementary school teachers by Billstein, Libeskind, Lott2. Math Playground: http://www.mathplayground.com/3. Math is Fun : https://www.mathsisfun.com/index.htm