2.2 Conversion to and from percentages - Fractions One of the most important concepts in business is that of percentage. To express one number as a percentage of another is to give its relative value expressed as the numerator of the equivalent fraction in which 100 is the denominator. The symbol for percentage is $$ \% $$. To convert a fraction to a percentage, multiply the fraction by 100 and write the percentage sign after the answer.

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### 2.2 Conversion to and from Percentages - Fractions

One of the most important concepts in business is that of percentage. Percentages allow us to express one number as a part of another, making it easier to compare and understand relative values. The symbol for percentage is **$$ \% $$**.

#### **Understanding Percentages**

To express one number as a percentage of another is to give its relative value expressed as the numerator of the equivalent fraction in which 100 is the denominator. Essentially, a percentage represents a fraction out of 100.

**Formula:**
$$ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100\% $$

#### **Converting Fractions to Percentages**

To convert a fraction to a percentage, follow these steps:

1. **Divide the numerator by the denominator** to get a decimal.
2. **Multiply the decimal by 100**.
3. **Add the percentage sign (%)** to the result.

**Formula:**
$$ \text{Percentage} = \left( \frac{\text{Numerator}}{\text{Denominator}} \right) \times 100\% $$

**Example 1:**
Convert the fraction $$ \frac{3}{4} $$ to a percentage.

**Solution:**
1. Divide numerator by denominator: $$ \frac{3}{4} = 0.75 $$
2. Multiply by 100: $$ 0.75 \times 100 = 75 $$
3. Add the percentage sign: **75%**

**Example 2:**
Convert the fraction $$ \frac{5}{8} $$ to a percentage.

**Solution:**
1. $$ \frac{5}{8} = 0.625 $$
2. $$ 0.625 \times 100 = 62.5 $$
3. **62.5%**

#### **Converting Percentages to Fractions**

To convert a percentage back to a fraction, reverse the process:

1. **Remove the percentage sign (%)**.
2. **Write the number as a fraction with 100 as the denominator**.
3. **Simplify the fraction** if possible.

**Example 3:**
Convert **45%** to a fraction.

**Solution:**
1. Remove the percentage sign: 45
2. Write as a fraction: $$ \frac{45}{100} $$
3. Simplify: $$ \frac{9}{20} $$

**Example 4:**
Convert **12.5%** to a fraction.

**Solution:**
1. Remove the percentage sign: 12.5
2. Write as a fraction: $$ \frac{12.5}{100} $$
3. Simplify: $$ \frac{125}{1000} = \frac{1}{8} $$

#### **Practice Problems**

1. **Convert the following fractions to percentages:**
   - $$ \frac{2}{5} $$
   - $$ \frac{7}{10} $$
   - $$ \frac{9}{16} $$

2. **Convert the following percentages to fractions and simplify:**
   - 80%
   - 33.3%
   - 2.5%

#### **Solutions to Practice Problems**

1. **Fractions to Percentages:**
   - $$ \frac{2}{5} = 0.4 \times 100 = 40\% $$
   - $$ \frac{7}{10} = 0.7 \times 100 = 70\% $$
   - $$ \frac{9}{16} = 0.5625 \times 100 = 56.25\% $$

2. **Percentages to Fractions:**
   - 80% = $$ \frac{80}{100} = \frac{4}{5} $$
   - 33.3% = $$ \frac{33.3}{100} = \frac{333}{1000} $$ (simplified as needed)
   - 2.5% = $$ \frac{2.5}{100} = \frac{25}{1000} = \frac{1}{40} $$

#### **Key Takeaways**

- **Percentage** is a way to express a number as a fraction of 100.
- **To convert a fraction to a percentage**, divide the numerator by the denominator, multiply by 100, and add the **\%** symbol.
- **To convert a percentage to a fraction**, remove the **\%** symbol, write the number over 100, and simplify the fraction.

Understanding how to convert between fractions and percentages is essential in business for interpreting data, calculating discounts, determining profit margins, and more.
To convert a fraction to a percentage, divide the numerator by the denominator, multiply by 100, and add the % symbol. For example, $$ \frac{3}{4} = 75\% $$.

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