Fractions, Decimals and Percentages

The language of fractions

A fraction is a division written down: $\frac{3}{4}$ means 3 divided by 4. The numerator is the number on top and the denominator is the number underneath. A proper fraction has a numerator smaller than its denominator, so its value is less than 1: $\frac{3}{4}$, $\frac{5}{8}$ and $\frac{9}{10}$ are all proper.

An improper fraction has a numerator greater than or equal to the denominator, so its value is 1 or more: $\frac{7}{4}$, $\frac{9}{8}$ and $\frac{5}{5}$. A mixed number writes the same value as a whole number with a proper fraction beside it, so $\frac{7}{4}$ becomes $1\frac{3}{4}$. The two forms are interchangeable, and a question can ask for either one.

To change a mixed number into an improper fraction, multiply the whole number by the denominator and add the numerator: $2\frac{3}{5}=\frac{2\times 5+3}{5}=\frac{13}{5}$. To go the other way, divide the numerator by the denominator and write the remainder as a fraction: $\frac{17}{5}=3\frac{2}{5}$, because $17=3\times 5+2$.

Fractions that look different can be worth exactly the same. $\frac{1}{2}$, $\frac{2}{4}$, $\frac{3}{6}$ and $\frac{50}{100}$ are all one half. Equivalent fractions are made by multiplying or dividing the top and the bottom by the same number. A fraction is in its simplest form when no number other than 1 divides both parts, so divide both by their highest common factor: $\frac{18}{24}=\frac{3}{4}$ after dividing by 6.

Converting between fractions, decimals and percentages

Every fraction becomes a decimal when you do the division it stands for: $\frac{3}{8}=3÷8=0.375$. Every decimal becomes a percentage when you multiply by 100, so 0.375 is 37.5%. That gives the two-step route from any fraction to a percentage: divide, then multiply by 100.

Going the other way, a percentage is a fraction with 100 as its denominator. 45% means $\frac{45}{100}$, which cancels to $\frac{9}{20}$, and as a decimal it is 0.45. Dividing by 100 moves the digits two places to the right of the decimal point: 8% = 0.08, 60% = 0.6 and 125% = 1.25.

Some conversions come up so often that you should not have to work them out. The table below is worth learning by heart, because it saves time in a paper where every minute counts.

The awkward ones are thirds and sixths, which never terminate. $\frac{1}{3}=0.333\dots$ and $\frac{1}{6}=0.1666\dots$, so as percentages they are $33\frac{1}{3}\%$ and $16\frac{2}{3}\%$. Core candidates may leave them as fractions, while Extended candidates write them with recurring decimal notation.

Comparing fractions, decimals and percentages

To compare two fractions, give them the same denominator. For $\frac{5}{7}$ and $\frac{7}{10}$ the lowest common multiple of 7 and 10 is 70, so $\frac{5}{7}=\frac{50}{70}$ and $\frac{7}{10}=\frac{49}{70}$. Since 50 is bigger than 49, $\frac{5}{7}$ is the larger fraction.

A second method is to turn both into decimals: $5÷7=0.714285\dots$ and $7÷10=0.7$, which reaches the same conclusion. Either method scores; the common denominator keeps you exact, and the decimal method is quicker when the numbers are awkward.

When a question mixes the three forms, convert everything to one form first. To put $\frac{3}{4}$, 0.7 and 72% in order, write all three as decimals: 0.75, 0.7 and 0.72. The order is now clear: 0.7, then 72%, then $\frac{3}{4}$.

Watch out for comparing digits instead of values. 0.7 is not smaller than 0.65 because 7 is bigger than 6; line the decimal points up and compare place by place. The same trap catches students who see 0.5 and 0.48 and assume the longer decimal must be larger.

• $\frac{3}{4}=0.75=75\%$
• $0.7=70\%$, so 0.7 is smaller than $\frac{3}{4}$
• $72\%=0.72$, which sits between the other two
• Smallest first: 0.7, 72%, $\frac{3}{4}$

Finding a percentage of a quantity

The reliable method is the multiplier. A percentage is a fraction out of 100, so 15% of 240 is $240\times \frac{15}{100}=240\times 0.15=36$. Write the percentage as a decimal and multiply, and that one step covers every case, including percentages above 100.

Without a calculator, build the percentage out of pieces you can do in your head. 10% is the number divided by 10, 5% is half of that, and 1% is the number divided by 100. For 15% of 240: 10% is 24 and 5% is 12, so 15% is $24+12=36$.

Percentages over 100% behave in exactly the same way. 120% of 45 is $45\times 1.2=54$, which is bigger than 45 because you are taking more than the whole amount. Cambridge lists percentages over 100% among the percentage calculations you may be asked for.

In a word problem, read what the percentage is being taken of. '8% of the students walk to school' means 8% of the whole school roll, not 8% of the students who cycle. Underline the quantity the percentage applies to before you start calculating.

• 15% of 240: $240\times 0.15=36$
• 8% of 350: $350\times 0.08=28$
• 37.5% of 96: $96\times 0.375=36$
• 120% of 45: $45\times 1.2=54$

Writing one quantity as a percentage of another

To express one quantity as a percentage of another, divide the first by the second and multiply by 100: $\frac{A}{B}\times 100$. A student who scores 27 out of 45 has $\frac{27}{45}\times 100=60\%$.

Keep the order straight, because swapping the two numbers gives a completely different answer. The quantity that follows the word 'of' goes on the bottom. 'What percentage of the class are girls?' puts the number of girls on top; 'what percentage of the girls play hockey?' puts the hockey players on top.

The answer can be more than 100%. A shop that takes 480 in a week against a target of 400 has $\frac{480}{400}\times 100=120\%$ of target. An answer above 100% is not a mistake; it means the quantity was larger than the one it was compared with.

Always check that the answer is sensible. If the first quantity is smaller than the second, the percentage must be less than 100, and if it is much smaller the percentage should be small. 18 out of 240 is 7.5%, which looks right because 18 is a small part of 240.

• 27 out of 45: $\frac{27}{45}\times 100=60\%$
• 18 out of 240: $\frac{18}{240}\times 100=7.5\%$
• 480 against a target of 400: $\frac{480}{400}\times 100=120\%$

Percentage increase and decrease

An increase of 15% multiplies by $1+0.15=1.15$. A decrease of 15% multiplies by $1-0.15=0.85$. Increasing 240 by 15% gives $240\times 1.15=276$, and decreasing 240 by 15% gives $240\times 0.85=204$. One multiplication replaces the two-step 'find the change, then add it or subtract it' method.

The multiplier for an increase is always greater than 1 and the multiplier for a decrease is always less than 1, so you can see at a glance whether your answer should be bigger or smaller than the original. A 30% discount on an item priced 80 gives $80\times 0.7=56$.

The syllabus names the contexts Cambridge uses: deposit, discount, profit and loss, and earnings. A deposit is a percentage of the price paid up front, a discount is a percentage taken off, and profit is the difference between what an item costs and what it sells for.

Profit or loss as a percentage is always worked out on the cost price. Buy at 45 and sell at 54: the profit is 9, so as a percentage of the cost price it is $\frac{9}{45}\times 100=20\%$. Earnings work the same way: a rate of pay rising from 12.50 an hour to 13.75 an hour has risen by 1.25, which is $\frac{1.25}{12.50}\times 100=10\%$.

Simple interest, compound interest and repeated change

Simple interest is paid on the original amount only, so it is the same every year. For a principal P at a rate of R% per year for T years, the interest is $P\times \frac{R}{100}\times T$. 600 invested at 4% simple interest earns $600\times 0.04\times 5=120$ over 5 years, so the investment is worth 720. Cambridge states that formulas are not given, so this one has to be learned.

Compound interest is paid on the original amount plus the interest already added, so each year's interest is larger than the year before. The multiplier method does the whole calculation in one line: $600\times {1.04}^5$. That comes to 729.99, compared with 720 for simple interest, and the extra 9.99 is the interest earned on the interest.

For 3 years the multiplier is ${1.04}^3$, not $1.04\times 3$. That single error is the most expensive one in this topic: it turns 5624.32 into 5600. The index counts how many times the interest is added, so it always equals the number of years.

Compound interest is also how depreciation is worked out, except that the multiplier is below 1. A machine worth 12000 that loses 15% of its value each year is worth $12000\times {0.85}^2=8670$ after two years.

Extended candidates also meet repeated percentage change, where the rate is different each time. A price of 250 that rises 20% and then falls 10% becomes $250\times 1.2\times 0.9=270$. Multiply the multipliers together in the order the changes happen, and never add the percentages.

Extended only: recurring decimals and reverse percentages

A terminating decimal stops after a fixed number of digits, such as 0.375. A recurring decimal never stops but repeats a block of digits for ever, such as 0.454545... Every fraction produces one or the other, because the remainders when you divide must eventually repeat. Core candidates are not expected to use recurring decimal notation or to convert a recurring decimal to a fraction; both are Extended.

Extended candidates write a recurring decimal with a dot over the repeating digit, or over the first and last digits of a repeating block: $0.\dot{3}=0.333\dots$, $0.\dot{4}\dot{5}=0.454545\dots$ and $0.1\dot{6}=0.1666\dots$.

To turn a recurring decimal into a fraction, call it $x$, multiply by a power of 10 until the repeating parts line up, then subtract. For $x=0.\dot{4}\dot{5}$: $100x=45.454545\dots$, so $100x-x=45$, giving $99x=45$ and $x=\frac{45}{99}=\frac{5}{11}$.

A one-digit repeating block needs a multiplier of 10, and a two-digit block needs 100. For $x=0.\dot{3}$, $10x=3.333\dots$, so $9x=3$ and $x=\frac{1}{3}$. When some digits come before the repeating block, such as $0.1\dot{6}$, multiply by 10 first: $10x=1.666\dots$, so $10x-x=1.5$, $9x=1.5$ and $x=\frac{1}{6}$.

The other Extended addition is the reverse percentage, where you are given the amount after a change and must find the original. If 96 is the price after a 20% increase, then 96 is 120% of the original, so the original is $96÷1.2=80$. Divide by the multiplier, and never take the percentage off the new figure: a 25% discount that leaves a price of 63 means the original price was $63÷0.75=84$.