Converting Between Fractions, Decimals, and Percentages

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

After reading this page you will be able to convert any fraction to a decimal or percentage, convert any decimal to a fraction or percentage, and convert any percentage to a fraction or decimal. You will understand why these three forms are interchangeable and when each form is most useful.

Theory

Three forms, one value

Fractions, decimals, and percentages are three different ways of writing the same number. Think of them as three languages for the same idea:

$\frac{3}{4} = 0.75 = 75\%$

Each form has its strengths. Fractions are exact and simplify well. Decimals work smoothly with calculators. Percentages make it easy to compare parts of a whole. Being able to switch between them fluently is one of the most practical math skills you can build.

Converting fractions to decimals

Divide the numerator by the denominator:

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

For example:

$\frac{5}{8} = 5 \div 8 = 0.625$

Some fractions produce terminating decimals (they end), while others produce repeating decimals (a pattern repeats forever):

• $\frac{1}{4} = 0.25$ (terminating)
• $\frac{1}{3} = 0.333\ldots = 0.\overline{3}$ (repeating)

A fraction in lowest terms produces a terminating decimal only when the denominator's prime factors are limited to 2 and 5.

Converting decimals to fractions

Write the decimal as a fraction over a power of 10, then simplify:

1. Count the decimal places.
2. Write the digits over the matching power of 10.
3. Simplify by dividing numerator and denominator by their GCD.

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

$0.45 = \frac{45}{100} = \frac{9}{20}$

$0.125 = \frac{125}{1000} = \frac{1}{8}$

For repeating decimals, use algebra. For example, to convert $0.\overline{3}$:

$$\begin{aligned} x &= 0.333\ldots \\ 10x &= 3.333\ldots \\ 10x - x &= 3 \\ 9x &= 3 \\ x &= \frac{3}{9} = \frac{1}{3} \end{aligned}$$

Converting fractions to percentages

Divide the numerator by the denominator, then multiply by 100:

$\frac{a}{b} \times 100\%$

For example:

$\frac{7}{20} = 7 \div 20 = 0.35 \quad \Rightarrow \quad 0.35 \times 100 = 35\%$

Shortcut for denominators that divide evenly into 100: If the denominator is a factor of 100, you can scale the fraction directly. Since $20 \times 5 = 100$:

$\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 35\%$

Converting percentages to fractions

Write the percentage over 100 and simplify:

$p\% = \frac{p}{100}$

For example:

$60\% = \frac{60}{100} = \frac{3}{5}$

$12.5\% = \frac{12.5}{100} = \frac{125}{1000} = \frac{1}{8}$

Converting decimals to percentages (and back)

Decimal to percentage: Multiply by 100 (shift the decimal point two places to the right):

$0.72 \rightarrow 72\%$

Percentage to decimal: Divide by 100 (shift the decimal point two places to the left):

$85\% \rightarrow 0.85$

Quick reference table

Fraction	Decimal	Percentage
$\frac{1}{2}$	0.5	50%
$\frac{1}{3}$	$0.\overline{3}$	$33.\overline{3}\%$
$\frac{1}{4}$	0.25	25%
$\frac{1}{5}$	0.2	20%
$\frac{2}{5}$	0.4	40%
$\frac{3}{4}$	0.75	75%
$\frac{1}{8}$	0.125	12.5%
$\frac{3}{8}$	0.375	37.5%

Memorising these common equivalences will save you time on tests and in everyday calculations.

Worked Examples

Example 1: Fraction to decimal and percentage (easy)

Problem: Convert $\frac{4}{5}$ to a decimal and a percentage.

Step 1: Divide to get the decimal. $4 \div 5 = 0.8$

Step 2: Multiply by 100 to get the percentage. $0.8 \times 100 = 80\%$

Answer: $\frac{4}{5} = 0.8 =$ 80%

Example 2: Decimal to fraction (medium)

Problem: Convert 0.36 to a fraction in lowest terms.

Step 1: Write as a fraction over a power of 10. There are 2 decimal places, so use 100. $0.36 = \frac{36}{100}$

Step 2: Find the GCD of 36 and 100. The GCD is 4. $\frac{36 \div 4}{100 \div 4} = \frac{9}{25}$

Step 3: Verify -- $9 \div 25 = 0.36$. Correct.

Answer: $0.36 = \dfrac{9}{25}$

Example 3: Percentage to fraction (medium)

Problem: Convert 62.5% to a fraction in lowest terms.

Step 1: Write over 100. $62.5\% = \frac{62.5}{100}$

Step 2: Eliminate the decimal by multiplying numerator and denominator by 10. $\frac{62.5 \times 10}{100 \times 10} = \frac{625}{1000}$

Step 3: Simplify. The GCD of 625 and 1000 is 125. $\frac{625 \div 125}{1000 \div 125} = \frac{5}{8}$

Answer: $62.5\% = \dfrac{5}{8}$

Example 4: Repeating decimal to fraction (challenging)

Problem: Convert $0.\overline{27}$ (0.272727...) to a fraction.

Step 1: Let $x = 0.272727\ldots$

Step 2: Multiply by 100 (because the repeating block has 2 digits). $100x = 27.2727\ldots$

Step 3: Subtract the original equation. $100x - x = 27.2727\ldots - 0.2727\ldots$ $99x = 27$

Step 4: Solve for $x$. $x = \frac{27}{99} = \frac{3}{11}$

Step 5: Verify -- $3 \div 11 = 0.2727\ldots$ Correct.

Answer: $0.\overline{27} = \dfrac{3}{11}$

Example 5: Comparing values in different forms (challenging)

Problem: Which is largest: $\frac{7}{12}$, $0.58$, or $57.5\%$?

Step 1: Convert all values to decimals for easy comparison.

• $\frac{7}{12} = 7 \div 12 = 0.58\overline{3}$
• $0.58 = 0.58$
• $57.5\% = 0.575$

Step 2: Compare. $0.575 < 0.58 < 0.58\overline{3}$

Answer: $\dfrac{7}{12}$ is the largest.

Common Mistakes

Mistake 1: Moving the decimal point the wrong direction

❌ Convert 0.6 to a percentage: $0.6 \div 100 = 0.006 = 0.006\%$.

✅ $0.6 \times 100 = 60\%$

Why this matters: To go from a decimal to a percentage, you multiply by 100 (move the point right). To go from a percentage to a decimal, you divide by 100 (move the point left). Mixing up the direction is one of the most common conversion errors.

Mistake 2: Not simplifying the fraction

❌ $0.75 = \frac{75}{100}$ (left unsimplified).

✅ $\frac{75}{100} = \frac{3}{4}$

Why this matters: While $\frac{75}{100}$ is technically correct, most teachers expect fractions in lowest terms. Always check whether the numerator and denominator share a common factor and divide both by the GCD.

Mistake 3: Incorrect power of 10 for the decimal places

❌ $0.125 = \frac{125}{100}$

✅ $0.125 = \frac{125}{1000}$ (three decimal places = denominator of 1000)

Why this matters: The number of decimal places determines which power of 10 to use. One decimal place uses 10, two uses 100, three uses 1000, and so on. Using the wrong denominator changes the value entirely.

Practice Problems

Try these on your own before checking the answers:

1. Convert $\frac{5}{16}$ to a decimal and a percentage.
2. Convert 0.84 to a fraction in lowest terms.
3. Convert 37.5% to a fraction in lowest terms.
4. Convert $0.\overline{6}$ to a fraction.
5. Put these in order from smallest to largest: $\frac{2}{3}$, $0.65$, $67\%$.

Click to see answers

1. $5 \div 16 = 0.3125$. $0.3125 \times 100 = 31.25\%$
2. $0.84 = \frac{84}{100} = \frac{21}{25}$
3. $37.5\% = \frac{375}{1000} = \frac{3}{8}$
4. Let $x = 0.\overline{6}$. $10x = 6.\overline{6}$. $9x = 6$. $x = \frac{6}{9} = \frac{2}{3}$
5. $\frac{2}{3} \approx 0.667$, $0.65$, $67\% = 0.67$. Order: $0.65 < \frac{2}{3} < 67\%$

Summary

• Fraction to decimal: Divide numerator by denominator.
• Decimal to fraction: Write over the correct power of 10, then simplify.
• Fraction to percentage: Divide numerator by denominator, then multiply by 100.
• Percentage to fraction: Write over 100, then simplify.
• Decimal to percentage: Multiply by 100. Percentage to decimal: Divide by 100.
• Memorise common equivalences ($\frac{1}{4} = 0.25 = 25\%$, etc.) to save time.

Related Topics

• Percentages — Complete Guide
• Fractions — Complete Guide
• How to Calculate Percentage of a Number

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