Fractions, decimals and percentages are three ways of writing the same quantity, and you need to move between them easily. This note covers converting and ordering them, the four operations with fractions and mixed numbers, finding percentages of amounts, percentage change, and reverse percentages. These skills are used in shopping, finance, science and data.
Key ideas
Equivalence. A fraction, a decimal and a percentage can all represent the same value.
- Fraction to decimal: divide the numerator by the denominator. \(\tfrac58=5\div8=0.625\).
- Decimal to percentage: multiply by 100. \(0.625=62.5\%\).
- Percentage to decimal: divide by 100. \(35\%=0.35\).
- Decimal to fraction: use place value and simplify. \(0.35=\tfrac{35}{100}=\tfrac{7}{20}\).
- Some fractions give recurring decimals, such as \(\tfrac13=0.333\ldots\), written \(0.\dot3\).
To order a mixed set, convert all the values to decimals (or all to percentages) and compare them.
Fraction arithmetic.
- Adding and subtracting: write the fractions with a common denominator, then add or subtract the numerators and keep the denominator.
- Multiplying: multiply the numerators and multiply the denominators. Cancel common factors first to keep numbers small.
- Dividing: multiply by the reciprocal of the second fraction (turn it upside down).
- Mixed numbers: change each mixed number to an improper (top-heavy) fraction before multiplying or dividing. For example \(2\tfrac13=\tfrac73\). Change the answer back to a mixed number if it is large.
- Always simplify the final fraction by dividing the top and bottom by their HCF.
Percentages of amounts.
- To find a percentage of an amount, convert to a decimal and multiply: 15% of 240 is \(0.15\times240=36\).
- To write one quantity as a percentage of another, divide and multiply by 100: 18 out of 45 is \(\tfrac{18}{45}\times100=40\%\).
- To increase by \(p\%\), multiply by \(1+\tfrac{p}{100}\). To decrease by \(p\%\), multiply by \(1-\tfrac{p}{100}\). These are multipliers. An increase of 8% uses 1.08. A decrease of 20% uses 0.8.
Fraction of an amount. To find a fraction of a quantity, divide by the denominator and multiply by the numerator. \(\tfrac35\) of 45 kg is \(45\div5\times3=27\) kg. To write one quantity as a fraction of another, put the part over the whole and simplify, making sure both are in the same units first: 40 cm out of 2 m is \(\tfrac{40}{200}=\tfrac15\).
Why decimals help. Fractions with different denominators are hard to compare by eye, and percentages are the easiest form for comparing proportions of different totals. Converting everything to the same form first is the safest way to compare. Remember that a percentage can be more than 100%, for example a value that has more than doubled, and that a percentage change of \(-100\%\) means the value has fallen to zero.
Repeated changes. If a value is changed by a percentage twice, multiply by each multiplier in turn rather than adding the percentages. An increase of 10% followed by a decrease of 10% is \(\times1.1\times0.9=\times0.99\), which is a net decrease of 1%, not no change. This is because the second percentage is taken of a different amount.
Percentage change.
\[\text{percentage change}=\frac{\text{new value}-\text{original value}}{\text{original value}}\times100\%\]
A positive result is an increase and a negative result is a decrease. The change is always compared to the original value.
Reverse percentages. Sometimes you know the value after a change and need the original. The original is always 100% of the starting value, so divide by the multiplier: original \(=\dfrac{\text{new value}}{\text{multiplier}}\). The new value is not 100%, so do not take a percentage of it to go back.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Fraction \(\to\) decimal | Numerator \(\div\) denominator |
| Decimal \(\leftrightarrow\) percentage | \(\times100\) or \(\div100\) |
| \(\tfrac{a}{b}\div\tfrac{c}{d}=\tfrac{a}{b}\times\tfrac{d}{c}\) | Dividing by a fraction |
| Multiplier for increase of \(p\%\) | \(1+\tfrac{p}{100}\) |
| Multiplier for decrease of \(p\%\) | \(1-\tfrac{p}{100}\) |
| Percentage change \(=\dfrac{\text{change}}{\text{original}}\times100\%\) | Finding the percentage increase or decrease |
| Original \(=\dfrac{\text{new value}}{\text{multiplier}}\) | Reverse percentage |
Worked examples
Example 1 Put these in order, smallest first: \(\tfrac38\), 0.4, 35%, \(\tfrac{7}{20}\).
Convert each to a decimal. \(\tfrac38=3\div8=0.375\). \(0.4\) is already a decimal. \(35\%=0.35\). \(\tfrac{7}{20}=7\div20=0.35\).
Order: 0.35, 0.35, 0.375, 0.4.
Answer: 35% and \(\tfrac{7}{20}\) (equal), then \(\tfrac38\), then 0.4.
Example 2 Work out (a) \(2\tfrac13+1\tfrac34\) and (b) \(4\tfrac12\div1\tfrac15\). Give each answer as a mixed number in its simplest form.
(a) Change to improper fractions: \(2\tfrac13=\tfrac73\) and \(1\tfrac34=\tfrac74\). The common denominator is 12: \(\tfrac73=\tfrac{28}{12}\) and \(\tfrac74=\tfrac{21}{12}\).
Add: \(\tfrac{28}{12}+\tfrac{21}{12}=\tfrac{49}{12}=4\tfrac{1}{12}\).
(b) \(4\tfrac12=\tfrac92\) and \(1\tfrac15=\tfrac65\). Multiply by the reciprocal: \(\tfrac92\times\tfrac56=\tfrac{45}{12}\).
Simplify by dividing top and bottom by 3: \(\tfrac{45}{12}=\tfrac{15}{4}=3\tfrac34\).
Answer: (a) \(4\tfrac1{12}\), (b) \(3\tfrac34\).
Example 3 (a) A laptop costs €640. After a price rise it costs €736. Find the percentage increase. (b) In a sale, all prices are reduced by 20%. A jacket now costs €52. What was its original price?
(a) Change \(=736-640=96\). Percentage increase \(=\tfrac{96}{640}\times100=15\%\).
(b) A decrease of 20% means the sale price is 80% of the original, so the multiplier is 0.8.
Original \(=52\div0.8=65\).
Check: \(65\times0.8=52\).
Answer: (a) 15% increase, (b) €65.
Common errors
- Adding fractions by adding the numerators and the denominators, such as writing \(\tfrac12+\tfrac13=\tfrac25\). Use a common denominator: the correct answer is \(\tfrac56\).
- Multiplying mixed numbers by treating the whole numbers and fractions separately. Convert to improper fractions first.
- Dividing by a fraction without flipping the second fraction, or flipping the first one.
- Working out percentage change as a fraction of the new value instead of the original value.
- For a reverse percentage, taking 20% of the sale price and adding it on. The original is found by dividing by the multiplier (0.8), not by adding 20% of the new price.
Exam tip: Use the fraction key on your calculator to check a fraction calculation, but still write down the steps. For percentages, write the multiplier first, for example "\(\times1.08\)", so the method is clear.
Check yourself
- Write 0.35 as a fraction in its simplest form, and \(\tfrac58\) as a percentage.
Answer
\(0.35=\tfrac{35}{100}=\tfrac{7}{20}\). \(\tfrac58=0.625=62.5\%\). - Work out \(\tfrac56-\tfrac38\).
Answer
Common denominator 24: \(\tfrac{20}{24}-\tfrac{9}{24}=\tfrac{11}{24}\). - Find 15% of 240, and write 18 out of 45 as a percentage.
Answer
\(0.15\times240=36\). \(\tfrac{18}{45}\times100=40\%\). - After an 8% increase, a price is €135. What was the original price?
Answer
The multiplier is 1.08. Original \(=135\div1.08=125\), so the original price was €125.