¶Decimals and Percentages

Fractions, decimals, and percentages are three different mathematical languages used to describe the exact same concept: a part of a whole. Fluently translating between these three forms is a critical skill in elementary mathematics.

¶The Base-10 Decimal System

Our number system is based on powers of 10. Just as the places to the left of the decimal point represent ones, tens, hundreds, and so on, the places to the right of the decimal point represent fractions with denominators that are powers of 10:

• The first place to the right is the tenths place ($\frac{1}{10}$).
• The second place is the hundredths place ($\frac{1}{100}$).
• The third place is the thousandths place ($\frac{1}{1000}$).

Therefore, the decimal $0.45$ can be read as "forty-five hundredths," which immediately translates to the fraction $\frac{45}{100}$.

¶Understanding Percentages

The word percent literally translates from Latin per centum as "by the hundred" or "for every 100". Therefore, $X\%$ is exactly equal to the fraction $\frac{X}{100}$.

¶Translations

1. Fraction to Decimal: Perform long division, dividing the numerator by the denominator. Example: $\frac{5}{8} = 5 \div 8 = 0.625$

2. Decimal to Percentage: Multiply the decimal by 100 (which moves the decimal point two places to the right) and append the $$ symbol. Example: $0.625 \times 100 = 62.5\%$

3. Percentage to Decimal: Divide the percentage by 100 (which moves the decimal point two places to the left) and remove the $$ symbol. Example: $42\% = 42 \div 100 = 0.42$

¶Calculating with Percentages

When solving real-world problems involving percentages (like calculating tax, tips, or discounts), the most reliable method is to first convert the percentage into its decimal form, and then perform the arithmetic. When asked to find a percent of a number, the operation is multiplication.

*Example: Find a 15% tip on a $30 bill.*$15% = 0.15$$0.15 \times 30 = 4.5$The tip is$$4.50$.

¶References

• Khan Academy: Decimals, fractions, and percentages
• Common Core State Standards for Mathematics: CCSS.6.RP.A.3.c