Decimal Representation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Decimal Representation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Writing fractions as digits to the right of a decimal point, using place values of tenths, hundredths, thousandths, etc.

Just like 234 = 200 + 30 + 4, we have 2.34 = 2 + 0.3 + 0.04.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Decimals extend place value using powers of \frac{1}{10}: tenths, hundredths, thousandths...

Common stuck point: More digits after decimal doesn't mean larger (0.5 > 0.125).

Sense of Study hint: Compare decimals by lining up the decimal points vertically and adding trailing zeros so both have the same number of digits.

Worked Examples

Example 1

easy
Convert \dfrac{5}{8} to a decimal by long division, and determine whether it terminates or repeats.

Solution

  1. 1
    Divide 5 \div 8: 8 goes into 50 six times (48), remainder 2. So far: 0.6.
  2. 2
    8 goes into 20 twice (16), remainder 4. Decimal: 0.62.
  3. 3
    8 goes into 40 five times (40), remainder 0. Decimal: 0.625.
  4. 4
    Remainder is 0, so the decimal terminates: \dfrac{5}{8} = 0.625.

Answer

\dfrac{5}{8} = 0.625 (terminating decimal)
A fraction in lowest terms terminates as a decimal if and only if its denominator has no prime factors other than 2 and 5. Here 8 = 2^3, so the decimal terminates.

Example 2

medium
Convert 0.\overline{142857} to a fraction in simplest form.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Write each fraction as a decimal and classify as terminating or repeating: (a) \dfrac{3}{4}, (b) \dfrac{2}{9}.

Example 2

medium
Convert 0.3\overline{6} to a fraction in simplest form. (Note: only the 6 repeats.)
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Background Knowledge

These ideas may be useful before you work through the harder examples.

place valuefractions