How to turn Decimals into Fractions

Short answer: For a terminating decimal, write it as an integer over the appropriate power of 10 and simplify; for a repeating decimal use an algebraic trick (multiply and subtract) to get a rational fraction, then simplify.

Explanation & Steps

1) Terminating decimals (finite number of digits)

Steps:

1. Count how many digits are after the decimal point (call it $n$).
2. Multiply numerator and denominator so the decimal becomes an integer: put the digits as the numerator and $10^n$ as the denominator.
3. Simplify the fraction by dividing by the greatest common divisor (GCD).

Examples:

• $0.75$ → digits after decimal $n=2$, numerator $75$, denominator $10^2=100$:

$$\frac{75}{100}=\frac{3}{4}$$

• $0.2$ → $ \frac{2}{10}=\frac{1}{5}$
• $1.25$ → separate whole part or convert directly: $1.25=\frac{125}{100}=\frac{5}{4}=1\frac{1}{4}$

2) Repeating decimals

Let $x$ be the decimal. If it has a repeating block of length $r$ and possibly a non-repeating part of length $n$ after the decimal:

1. Multiply $x$ by $10^{n+r}$ and also by $10^n$.
2. Subtract the two equations to cancel the repeating part.
3. Solve for $x$ and simplify.

General formula: If $x=0.a_1a_2\ldots a_n\overline{b_1\ldots b_r}$ (non-repeating $a$ length $n$, repeating $b$ length $r$),
$$x=\frac{\text{integer formed by }a_1..a_n b_1..b_r – \text{integer formed by }a_1..a_n}{10^{n+r}-10^n}.$$

Examples:

• $0.\overline{3}$: let $x=0.\overline{3}$. Then $10x=3.\overline{3}$. Subtract:

$$10x-x=3\implies 9x=3\implies x=\frac{1}{3}.$$

• $0.16\overline{7}$: non-repeating $=16$ ($n=2$), repeating $=7$ ($r=1$). Let $x=0.167777\ldots$.

Multiply: $10^{3}x=1000x=167.777\ldots$ and $10^{2}x=100x=16.777\ldots$.
Subtract: $900x=151\Rightarrow x=\frac{151}{900}$ (simplify if possible).

3) Negative decimals and mixed numbers

• If the decimal is negative, put a minus sign: $-0.75=-\frac{3}{4}$.
• For numbers >1, either convert whole+fraction separately (e.g., $2.05=2+\frac{5}{100}=2+\frac{1}{20}=2\frac{1}{20}$) or convert full decimal then simplify.

Tips

• Always simplify the final fraction by dividing numerator and denominator by their GCD.
• For long repeating patterns, use the general formula above.
• If unsure how to simplify, use the Euclidean algorithm to find the GCD.

If you want, give me a specific decimal and I’ll convert it step-by-step.