Ratios Tutorial

Introduction to Ratios:

A ratio is a comparison of two quantities that have the same unit. It represents how many times one quantity is contained in another. Ratios are often written in the form of "a : b" or "a/b" to show the relationship between the two quantities.

Ratios are used in various real-world scenarios, such as in cooking recipes, map scales, and financial calculations. They help us understand the relative sizes of different quantities and make comparisons.

Representing Ratios:

To represent ratios, we use appropriate language to describe the relationship between the two quantities. For example:

"The ratio of apples to oranges is 3 to 2."
"The ratio of boys to girls in a class is 2 to 5."
"The ratio of blue marbles to red marbles in a bag is 4 : 7."

Making Tables of Equivalent Ratios:

In a ratio, equivalent ratios are those that represent the same comparison but have different values. To make tables of equivalent ratios, you can multiply or divide both quantities in the ratio by the same number.

Ratio     Equivalent Ratio
2 : 3     4 : 6
2 : 3     6 : 9
2 : 3     8 : 12
    

Finding Missing Values in Tables of Ratios:

Tables of equivalent ratios can be used to find missing values. Given one ratio, you can find the missing value in another ratio. For example:

4 : 6 is equivalent to 2 : 3
Find missing value in 2 : 4
2 : 4 is equivalent to 1 : 2
    

Solving Proportion Problems:

Proportions are statements that two ratios are equal. Solve for unknowns by cross-multiplying:

3 : 6 = x : 12
3 * 12 = 6 * x
36 = 6x
x = 6
    

Describing Rates and Unit Rates:

A rate is a special type of ratio comparing two quantities with different units, e.g., "miles per hour" or "dollars per pound".

A unit rate is a rate in which the second quantity is 1 unit. For example:

Car travels 60 miles in 2 hours
Rate = 60 / 2 = 30 miles per hour
Unit rate = 30 miles per hour
    

First Ratios Problem Example:

Problem: In a bag of marbles, the ratio of red marbles to blue marbles is 2 : 5. If there are 15 blue marbles, how many red marbles are there?

Solution:

2 : 5 = x : 15
2 * 15 = 5 * x
30 = 5x
x = 6
There are 6 red marbles
    

Second Ratios Problem Example:

Problem: In a recipe, the ratio of flour to sugar is 3 : 2. If we use 6 cups of flour, how many cups of sugar do we need?

Solution:

3 : 2 = 6 : x
3 * x = 2 * 6
3x = 12
x = 4
We need 4 cups of sugar