Ratio

A ratio is a relationship between two numbers. It indicates how many of the first number is included in the second number. Ratios can be written in three different ways: words, fractional notation, and colon notation.

Example 1: Words Notation:
Suppose there are 5 red apples and 3 green apples in a basket. The ratio of red apples to green apples can be expressed in words as "the ratio of red apples to green apples is 5 to 3."

Example 2: Fractional Notation:
Consider a group of 12 students, where 7 are boys and 5 are girls. The ratio of boys to girls can be written in fractional notation as \( \frac{7}{5} \).

Example 3: Colon Notation:
In a recipe for pancakes, the ratio of flour to milk is 2:1. This can be expressed using colon notation as "the ratio of flour to milk is 2 to 1."

Writing Ratio as a Fraction:

A ratio compares two quantities by division and can be expressed as a fraction. The first number in the ratio represents the numerator, and the second number represents the denominator.

Example:

The ratio of boys to girls in a class is \(3:5\). This can be written as the fraction \( \frac{3}{5} \), where \(3\) represents the number of boys, and \(5\) represents the number of girls.

Simplifying Ratios:

To simplify a ratio, find the greatest common divisor (GCD) of the two numbers in the ratio and divide both numbers by it.

Example:

Simplify the ratio \(6:9\).

1. Find the GCD of \(6\) and \(9\), which is \(3\).

2. Divide both numbers by \(3\) to simplify the ratio.

\[6 \div 3 = 2\]

\[9 \div 3 = 3\]

So, the simplified ratio is \(2:3\).

Worked Examples:

Example 1: Writing Ratio as a Fraction:

The ratio of apples to oranges in a basket is \(4:7\). Write this ratio as a fraction.

Solution:

The ratio \(4:7\) can be written as the fraction \( \frac{4}{7} \). Therefore, the ratio of apples to oranges is \( \frac{4}{7} \).

Example 2: Simplifying Ratios:

Simplify the ratio \(12:18\).

Solution:

1. Find the GCD of \(12\) and \(18\), which is \(6\).

2. Divide both numbers by \(6\) to simplify the ratio.

\[12 \div 6 = 2\]

\[18 \div 6 = 3\]

So, the simplified ratio is \(2:3\).

Example 3: Applying Simplified Ratio:

In a recipe, the ratio of flour to sugar is \(3:2\). If you need \(9\) cups of sugar, how many cups of flour do you need?

Solution:

1. Given the ratio \(3:2\) is already simplified.

2. We know the ratio of flour to sugar is \(3:2\).

3. Since the ratio represents the same relationship throughout, we can set up a proportion:

\[ \frac{\text{flour}}{\text{sugar}} = \frac{3}{2} \]

4. Substitute the given value of sugar:

\[ \frac{\text{flour}}{9} = \frac{3}{2} \]

5. Cross multiply:

\[ 2 \times \text{flour} = 3 \times 9 \]

\[ 2 \times \text{flour} = 27 \]

6. Solve for flour:

\[ \text{flour} = \frac{27}{2} \]

\[ \text{flour} = 13.5 \]

Therefore, you need \(13.5\) cups of flour.