Fractions, Decimals, and Percents

Three ways to write the same number

The number "half" can be written as:

• Fraction: $\dfrac{1}{2}$
• Decimal: $0.5$
• Percent: $50\%$

Same value, different notation. Fluently switching between them saves a lot of grief later.

Converting between forms

Fraction → decimal. Divide top by bottom. $\dfrac{3}{4} = 3 \div 4 = 0.75$.

Decimal → fraction. Read the decimal, write the fraction, simplify. $0.6 = \dfrac{6}{10} = \dfrac{3}{5}$.

Decimal → percent. Multiply by 100 (i.e. move the decimal two places right). $0.42 = 42\%$.

Percent → decimal. Divide by 100 (move the decimal two places left). $35\% = 0.35$.

Operations with fractions

Adding/subtracting requires a common denominator.

$\dfrac{1}{3} + \dfrac{1}{4} = \dfrac{4}{12} + \dfrac{3}{12} = \dfrac{7}{12}$

Multiplying is straight across — top times top, bottom times bottom.

$\dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{8}{15}$

Dividing is multiplying by the reciprocal — flip the second fraction.

$\dfrac{2}{3} \div \dfrac{4}{5} = \dfrac{2}{3} \times \dfrac{5}{4} = \dfrac{10}{12} = \dfrac{5}{6}$

The percent of a number

To find $p\%$ of $n$, multiply: $\dfrac{p}{100} \times n$.

Example. $20\%$ of $80$ is $0.20 \times 80 = 16$.

Example — percent change. A $50 item is on sale for $40. The discount is $10. As a percent of the original price: $\dfrac{10}{50} = 0.20 = 20\%$ off.

Key takeaways

• Fractions, decimals, and percents are three ways to write the same value.
• Adding or subtracting fractions needs a common denominator.
• Divide a fraction by multiplying by its reciprocal.
• $p\%$ of $n$ is $\dfrac{p}{100} \cdot n$.