Decimals as fractions (and back)

Decimals are simply fractions whose denominators are powers of ten. Once you see this, switching between forms takes seconds.

Idea

Decimal → fraction. Read the decimal aloud and write it as a fraction with the matching denominator. For example, $0.7$ reads "seven tenths" $=\tfrac{7}{10}$. And $0.45$ reads "forty-five hundredths" $=\tfrac{45}{100} = \tfrac{9}{20}$ (after simplification).

A trick: count the digits after the point. That tells you the power of ten in the denominator. $0.123$ has $3$ digits after the point, so it equals $\tfrac{123}{1000}$.

Fraction → decimal. If the denominator is already $10, 100, 1000, \dots$, write the numerator with appropriate decimal places. So $\tfrac{47}{100} = 0.47$.

If the denominator is not a power of ten, try to make it one by multiplying numerator and denominator by the same number. E.g., $\tfrac{3}{5} = \tfrac{6}{10} = 0.6$. Or $\tfrac{7}{25} = \tfrac{28}{100} = 0.28$. (Some fractions like $\tfrac{1}{3} = 0.333\dots$ are non-terminating , beyond this chapter.)

A handful of equivalents are so common they should be memorised:

• $\tfrac{1}{2} = 0.5$
• $\tfrac{1}{4} = 0.25$, $\tfrac{3}{4} = 0.75$
• $\tfrac{1}{5} = 0.2$, $\tfrac{2}{5} = 0.4$, $\tfrac{3}{5} = 0.6$, $\tfrac{4}{5} = 0.8$
• $\tfrac{1}{10} = 0.1$

These pop up in shopping, recipes, and measurements all the time.

Worked examples

Example 1. Convert $0.36$ to a fraction in simplest form.

$0.36 = \tfrac{36}{100}$. Divide top and bottom by $4$: $\tfrac{9}{25}$.

Example 2. Convert $\tfrac{7}{20}$ to a decimal.

Multiply top and bottom by $5$: $\tfrac{35}{100} = 0.35$.

Example 3. Convert $2.4$ to a mixed fraction in simplest form.

$2.4 = 2 + 0.4 = 2 + \tfrac{4}{10} = 2 + \tfrac{2}{5} = 2\tfrac{2}{5}$.

Example 4. Order from smallest to largest: $\tfrac{3}{4}$, $0.7$, $\tfrac{1}{2}$.

Convert all to decimals: $\tfrac{3}{4} = 0.75$, $0.7$, $\tfrac{1}{2} = 0.5$. Order: $0.5 < 0.7 < 0.75$, i.e., $\tfrac{1}{2} < 0.7 < \tfrac{3}{4}$.

Try it yourself

1. Convert $0.9$ to a fraction.
2. Convert $\tfrac{1}{5}$ to a decimal.
3. Convert $0.125$ to a fraction in simplest form.
4. Convert $\tfrac{3}{8}$ to a decimal (hint: multiply by $\tfrac{125}{125}$).
5. Convert $3.06$ to a mixed fraction.
6. Convert $\tfrac{9}{4}$ to a decimal.
7. Which is bigger, $0.3$ or $\tfrac{1}{4}$?
8. Write $0.05$ as a fraction and simplify.

Activity

Make a "decimal–fraction" matching game. Write decimals on $10$ cards and their equivalent fractions on another $10$. Shuffle and match. Time yourself , how quickly can you pair them?