Definition: A ratio gives the relationship between quantities of two different things.
Example with Apples and Oranges
Counts:
Apples: 6
Oranges: 9
Initial Ratio of Apples to Oranges
Expression: 6 to 9
Notation: Can be written as (6:9) or (6 \text{ to } 9)
Reducing Ratios
Process: Similar to reducing fractions, divide both numbers by their greatest common divisor (GCD).
GCD of 6 and 9 is 3.
Divide both by 3:
6 ÷ 3 = 2
9 ÷ 3 = 3
Reduced Ratio of Apples to Oranges: 2 to 3
Conceptual Understanding: For every 2 apples, there are 3 oranges.
Visualized as dividing total items into 3 equal groups.
Converting and Understanding Ratios in Reverse
Ratio of Oranges to Apples:
Original counts:
Oranges: 9
Apples: 6
Expression: 9 to 6
Reduction:
Divide by GCD (3):
9 ÷ 3 = 3
6 ÷ 3 = 2
Reduced Ratio of Oranges to Apples: 3 to 2
Summary
Key Concept: Ratios can be expressed in multiple ways and reduced like fractions.
Understanding both directions: It's important to clarify whether the ratio is apples to oranges or oranges to apples, as the direction affects the numbers used in the ratio.