Converting Between Fractions and Decimals

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

After this guide, you will be able to convert any fraction to a decimal using division, and convert any terminating or repeating decimal back to a fraction. You will also learn to recognize which fractions produce terminating decimals and which produce repeating decimals, giving you flexibility to work with numbers in whichever form is most convenient.

Theory

Fractions and decimals are two forms of the same number

Every fraction is a division problem waiting to happen. The fraction bar means "divided by":

$\frac{a}{b} = a \div b$

So $\frac{3}{4} = 3 \div 4 = 0.75$. The fraction and the decimal are two different ways to write the same value. Being able to switch between them is essential for comparing numbers, solving equations, and working with real-world measurements.

Converting fractions to decimals

The division method:

Divide the numerator by the denominator using long division (or a calculator). This always works.

$\frac{7}{8} = 7 \div 8 = 0.875$

The equivalent-fraction method (when the denominator is a factor of 10, 100, or 1000):

If you can rewrite the fraction with a denominator of 10, 100, or 1000, the decimal is immediate:

$\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10} = 0.6$

$\frac{7}{25} = \frac{7 \times 4}{25 \times 4} = \frac{28}{100} = 0.28$

This shortcut is fast but only works when the denominator has no prime factors other than 2 and 5.

Terminating vs. repeating decimals

Terminating decimals end after a finite number of digits:

$\frac{1}{4} = 0.25, \quad \frac{3}{8} = 0.375, \quad \frac{9}{20} = 0.45$

A fraction (in lowest terms) produces a terminating decimal when its denominator has only the prime factors 2 and/or 5.

Repeating decimals have one or more digits that cycle infinitely:

$\frac{1}{3} = 0.\overline{3} = 0.333\ldots$

$\frac{5}{6} = 0.8\overline{3} = 0.8333\ldots$

$\frac{2}{7} = 0.\overline{285714} = 0.285714285714\ldots$

A fraction produces a repeating decimal when its denominator (in lowest terms) has a prime factor other than 2 or 5.

The overline notation $\overline{3}$ means the digit 3 repeats forever. When multiple digits repeat, the bar covers all of them: $\overline{285714}$.

Converting terminating decimals to fractions

Step 1: Write the decimal as a fraction over the appropriate power of 10.

• 1 decimal place: denominator is 10
• 2 decimal places: denominator is 100
• 3 decimal places: denominator is 1000

$0.35 = \frac{35}{100}$

Step 2: Simplify by dividing numerator and denominator by their GCD.

$\frac{35}{100} = \frac{35 \div 5}{100 \div 5} = \frac{7}{20}$

Converting repeating decimals to fractions

For a single repeating digit like $0.\overline{3}$:

Let $x = 0.\overline{3}$

$10x = 3.\overline{3}$

Subtract: $10x - x = 3.\overline{3} - 0.\overline{3}$

$9x = 3$

$x = \frac{3}{9} = \frac{1}{3}$

For a repeating block like $0.\overline{12}$:

Let $x = 0.\overline{12}$

$100x = 12.\overline{12}$

$100x - x = 12$

$99x = 12$

$x = \frac{12}{99} = \frac{4}{33}$

The general pattern: if $n$ digits repeat, multiply by $10^n$, subtract the original, and solve for $x$.

Common fraction-decimal equivalents worth memorizing

These appear so frequently that knowing them by heart will save you time:

Fraction	Decimal	Fraction	Decimal
$\frac{1}{2}$	0.5	$\frac{1}{3}$	$0.\overline{3}$
$\frac{1}{4}$	0.25	$\frac{2}{3}$	$0.\overline{6}$
$\frac{3}{4}$	0.75	$\frac{1}{5}$	0.2
$\frac{1}{8}$	0.125	$\frac{1}{6}$	$0.1\overline{6}$
$\frac{3}{8}$	0.375	$\frac{5}{6}$	$0.8\overline{3}$

Worked Examples

Example 1: Fraction to decimal using division (easy)

Problem: Convert $\dfrac{3}{8}$ to a decimal.

Step 1: Divide 3 by 8 using long division. $3 \div 8 = 0.375$

(8 goes into 30 three times with remainder 6; 8 goes into 60 seven times with remainder 4; 8 goes into 40 five times with no remainder.)

Answer: $0.375$

Example 2: Fraction to decimal using equivalent fractions (easy)

Problem: Convert $\dfrac{9}{25}$ to a decimal.

Step 1: Rewrite with a denominator of 100 (since $25 \times 4 = 100$). $\frac{9}{25} = \frac{9 \times 4}{25 \times 4} = \frac{36}{100}$

Step 2: Write as a decimal. $\frac{36}{100} = 0.36$

Answer: $0.36$

Example 3: Terminating decimal to fraction (medium)

Problem: Convert $0.625$ to a fraction in lowest terms.

Step 1: Write over 1000 (three decimal places). $0.625 = \frac{625}{1000}$

Step 2: Find the GCD of 625 and 1000. $625 = 5^4, \quad 1000 = 2^3 \times 5^3$ $\gcd = 5^3 = 125$

Step 3: Simplify. $\frac{625 \div 125}{1000 \div 125} = \frac{5}{8}$

Answer: $\dfrac{5}{8}$

Example 4: Repeating decimal to fraction (medium)

Problem: Convert $0.\overline{45}$ to a fraction.

Step 1: Let $x = 0.\overline{45} = 0.454545\ldots$

Step 2: Since two digits repeat, multiply by 100. $100x = 45.454545\ldots$

Step 3: Subtract the original equation. $100x - x = 45.454545\ldots - 0.454545\ldots$ $99x = 45$

Step 4: Solve and simplify. $x = \frac{45}{99} = \frac{45 \div 9}{99 \div 9} = \frac{5}{11}$

Answer: $\dfrac{5}{11}$

Example 5: Mixed number to decimal and back (challenging)

Problem: Convert $2\dfrac{5}{12}$ to a decimal, then confirm by converting back.

Step 1: Convert the fraction part to a decimal. $5 \div 12 = 0.41\overline{6}$

Step 2: Add the whole number. $2\frac{5}{12} = 2.41\overline{6}$

Step 3: Convert back -- separate the whole number and decimal. The decimal part is $0.41\overline{6}$. Let $x = 0.41\overline{6}$.

$10x = 4.1\overline{6}$ $100x = 41.\overline{6}$ $1000x = 416.\overline{6}$ $1000x - 100x = 416.\overline{6} - 41.\overline{6} = 375$ $900x = 375$ $x = \frac{375}{900} = \frac{5}{12}$

Adding back the whole number: $2\frac{5}{12}$. Confirmed!

Answer: $2.41\overline{6}$

Common Mistakes

Mistake 1: Dividing the denominator by the numerator instead of the other way

❌ $\frac{3}{4} = 4 \div 3 = 1.333\ldots$

✅ $\frac{3}{4} = 3 \div 4 = 0.75$

Why this matters: The fraction bar means numerator divided by denominator, not the reverse. Always divide the top number by the bottom number.

Mistake 2: Forgetting to simplify after converting a decimal to a fraction

❌ $0.4 = \frac{4}{10}$ (left as final answer)

✅ $0.4 = \frac{4}{10} = \frac{2}{5}$

Why this matters: Fractions should always be expressed in lowest terms. After writing the decimal over a power of 10, check for common factors.

Mistake 3: Using the wrong power of 10 for the denominator

❌ $0.125 = \frac{125}{100}$ (only two zeros instead of three)

✅ $0.125 = \frac{125}{1000}$

Why this matters: The number of decimal places tells you which power of 10 to use. Three decimal places means 1000, not 100. Using the wrong denominator changes the value entirely.

Practice Problems

Try these on your own before checking the answers:

1. Convert $\dfrac{5}{16}$ to a decimal.
2. Convert $0.875$ to a fraction in lowest terms.
3. Convert $\dfrac{4}{11}$ to a decimal.
4. Convert $0.\overline{72}$ to a fraction in lowest terms.
5. Which fractions produce terminating decimals: $\dfrac{3}{15}$, $\dfrac{7}{12}$, $\dfrac{9}{40}$?

Click to see answers

1. $5 \div 16 = 0.3125$
2. $\frac{875}{1000}$. $\gcd(875, 1000) = 125$. $\frac{875 \div 125}{1000 \div 125} = \frac{7}{8}$
3. $4 \div 11 = 0.\overline{36} = 0.363636\ldots$
4. Let $x = 0.\overline{72}$. $99x = 72$. $x = \frac{72}{99} = \frac{8}{11}$
5. Simplify first: $\frac{3}{15} = \frac{1}{5}$ (denominator $5$ -- terminates). $\frac{7}{12}$ (denominator $12 = 2^2 \times 3$ -- repeats because of the factor 3). $\frac{9}{40}$ (denominator $40 = 2^3 \times 5$ -- terminates). So $\frac{3}{15}$ and $\frac{9}{40}$ produce terminating decimals.

Summary

• To convert a fraction to a decimal, divide the numerator by the denominator.
• A fraction in lowest terms produces a terminating decimal when the denominator has only the prime factors 2 and 5, and a repeating decimal otherwise.
• To convert a terminating decimal to a fraction, write it over the appropriate power of 10 and simplify.
• To convert a repeating decimal to a fraction, use the algebraic method: multiply by $10^n$ (where $n$ is the number of repeating digits), subtract, and solve.
• Memorize common fraction-decimal equivalents to speed up your work.

Related Topics

• Fractions -- Complete Guide
• How to Simplify Fractions to Lowest Terms
• How to Multiply and Divide Fractions Step by Step

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