Four disguises of one number

A single mathematical quantity can wear at least four costumes. You should learn to spot which costume you're looking at, and how to change one into another in seconds.

Concept

Consider the number $\tfrac{3}{8}$. The same value appears as:

• Fraction: $\tfrac{3}{8}$.
• Decimal: $0.375$ (from long division).
• Percent: $37.5\%$ (multiply decimal by $100$).
• Ratio (out of total): $3 : 8$ , though here we must remember $\tfrac{3}{8}$ is "$3$ parts out of $8$ total", not "$3 : 5$" (which would be $3$ vs $5$ remaining).

Why so many forms? Because each has its strengths.

• Fractions are exact and great for proportions ($\tfrac{1}{3}$ has a clean form, while $0.\overline{3}$ doesn't).
• Decimals are convenient on calculators and easy to compare ($0.32 < 0.37$ , just compare digits).
• Percents are easy to communicate ($87\%$ marks, $5\%$ discount).
• Ratios are perfect when no "whole" is fixed (recipes, mixtures).

Conversion drill.

Decimal $\to$ Fraction.

• Read the place value of the last digit. $0.625$ has $625$ in the thousandths place, so $= \dfrac{625}{1000}$. Reduce: $\dfrac{5}{8}$.

Percent $\to$ Fraction.

• Divide by $100$ and reduce. $45\% = \dfrac{45}{100} = \dfrac{9}{20}$.

Ratio $\to$ Percent.

• A ratio $a : b$ (out of total $a + b$) gives the fraction $\dfrac{a}{a+b}$ which becomes a percentage. Example: $3:5$ means $\dfrac{3}{8}$ of the total is the $a$-part, i.e., $37.5\%$.

Lowest-terms habit. Always reduce fractions to lowest terms; $\dfrac{50}{100}$ should be written $\dfrac{1}{2}$. Cleaner notation makes problems easier to spot.

Common conversions worth memorising.

Worked examples

Example 1. Express $\dfrac{7}{8}$ as a decimal and as a percent.

• $7 \div 8 = 0.875$.
• $\times 100$: $87.5\%$.

Example 2. Convert $62.5\%$ to a fraction in lowest terms.

• $\dfrac{62.5}{100} = \dfrac{625}{1000} = \dfrac{5}{8}$.

Example 3. A class has boys and girls in the ratio $7 : 5$. What percent are girls?

• Total $= 12$. Girls $= \dfrac{5}{12}$. As decimal $\approx 0.4167$.
• $\approx 41.67\%$.

Example 4. Order from smallest to largest: $0.4$, $\dfrac{2}{5}$, $35\%$, $\dfrac{3}{8}$.

• Convert to decimals: $0.4, 0.4, 0.35, 0.375$.
• Smallest $\to$ largest: $0.35, 0.375, 0.4, 0.4$.
• So $35\% < \dfrac{3}{8} < 0.4 = \dfrac{2}{5}$.

Try it yourself

1. Convert $\dfrac{9}{20}$ to a decimal and a percent.
2. Convert $0.84$ to a fraction in lowest terms.
3. Convert $80\%$ to a fraction.
4. Express a ratio of $4:5$ as a percent of the total.
5. Order from smallest: $\dfrac{2}{3}, 0.65, 70\%, \dfrac{17}{25}$.
6. A student scored $138$ out of $200$. Express as fraction, decimal, percent.
7. $0.6 \overline{6}$ as a fraction is?
8. A bag has $15$ red and $25$ blue marbles. What percent are red?

Activity

Four-column chart. On a sheet draw four columns: fraction, decimal, percent, ratio. Fill in $15$ rows by picking any quantity (say "marks out of $40$") and converting. After ten of these, the conversions become reflex.