Understanding Fractions, Decimals and Percentages

Before you begin

You should be comfortable with times tables and simplifying fractions by dividing the top and bottom by the same number.

The intuition

A fraction like $\frac{3}{4}$ means "3 out of 4 equal parts". The decimal $0.75$ and the percentage $75\%$ describe exactly the same amount — they are just different notations.

Definitions

• Fraction: $\frac{a}{b}$, where $a$ is the numerator and $b$ is the denominator.
• Decimal: a number written in base 10 using a decimal point.
• Percentage: a number expressed as a part of $100$.
• Equivalent: two forms that represent the same value.

Conversion method

• Fraction to decimal: divide the numerator by the denominator. $\frac{3}{4} = 0.75$.
• Decimal to percentage: multiply by $100$. $0.75 \times 100 = 75\%$.
• Percentage to fraction: write over $100$ and simplify. $75\% = \frac{75}{100} = \frac{3}{4}$.
• Fraction to percentage: multiply by $100$. $\frac{3}{4} = 75\%$.

Step-by-step method

1. Identify the form you have and the form you need.
2. Apply the relevant conversion.
3. Simplify fractions by dividing the numerator and denominator by their highest common factor.

Worked examples

Foundation example

Convert $\frac{2}{5}$ to a decimal and a percentage. $\frac{2}{5} = 0.4$ $0.4 \times 100 = 40\%$

Developing example

Convert $0.625$ to a fraction in its simplest form. $0.625 = \frac{625}{1000}$ Divide top and bottom by $125$: $\frac{625}{1000} = \frac{5}{8}$

Advanced example

A student scores $18$ out of $24$ on a problem set. Express this as a percentage to 1 decimal place. $\frac{18}{24} = 0.75$ $0.75 \times 100 = 75\%$ Answer: $75.0\%$.

Common mistakes

• Forgetting to simplify: $\frac{625}{1000}$ should be reduced to $\frac{5}{8}$.
• Wrong direction: to go from a decimal to a percentage, multiply by $100$, not divide.
• Misreading the whole: $18$ out of $24$ means $\frac{18}{24}$, not $\frac{24}{18}$.

Concept checklist

• I can convert between fractions, decimals and percentages.
• I can simplify fractions fully.
• I can express one quantity as a percentage of another.

Practice questions

1. Convert $\frac{7}{20}$ to a decimal.
2. Convert $0.35$ to a percentage.
3. Convert $45\%$ to a fraction in simplest form.
4. Convert $0.875$ to a fraction in simplest form.
5. Express $15$ out of $40$ as a percentage.
6. Convert $\frac{5}{8}$ to a percentage.

Challenge questions

1. Write $0.\overline{3}$ (a recurring decimal) as a fraction.
2. A price increases from $80$ to $92$. Express the increase as a percentage of the original price.

Frequently asked questions

How do I convert a recurring decimal to a fraction? Let $x$ equal the recurring decimal, multiply to shift the repeating block, then subtract to remove the repeating part and solve for $x$.

Is a percentage always out of $100$? Yes — "per cent" means "per $100$".

Related concepts

• Percentage increase, decrease and reverse percentages
• Ratio and direct proportion explained
• Rounding numbers to decimal places and significant figures