This note covers the four operations with integers, fractions and decimals, the order in which to carry out a calculation, and percentages, including increase, decrease, reverse percentages and compound change. These skills are used in almost every other topic, from algebra to statistics, so accuracy here pays off everywhere. Content for the Higher tier only is marked.
Key ideas
Integers and order of operations. Integers are whole numbers, including negatives and zero. Carry out a calculation in this order: brackets, then powers and roots, then multiplication and division (left to right), then addition and subtraction (left to right). Use the word BIDMAS to remember it. For negatives: adding a negative is the same as subtracting, and subtracting a negative is the same as adding. A negative times a negative gives a positive, and a negative times a positive gives a negative. The same sign rules apply to division.
Fractions.
- Equivalent fractions: multiply or divide the top and bottom by the same number. Always give a final answer in its simplest form.
- To add or subtract, first write both fractions with a common denominator. Then add or subtract the numerators only.
- To multiply, multiply the numerators and multiply the denominators. Cancel common factors first to keep numbers small.
- To divide, turn the second fraction upside down and multiply.
- For mixed numbers, change to improper fractions first, calculate, then change back. For example, \(2\tfrac35=\tfrac{13}{5}\).
Decimals. Line up place values when adding or subtracting. To multiply, ignore the decimal points, multiply as whole numbers, then give the answer the total number of decimal places. To divide by a decimal, multiply both numbers by a power of 10 to make the divisor a whole number. A fraction becomes a decimal by dividing the top by the bottom, and a terminating decimal becomes a fraction using place value, such as \(0.35=\tfrac{35}{100}=\tfrac{7}{20}\).
Recurring decimals (Higher). A dot above a digit marks a repeating digit, and a dot over the first and last digits marks a repeating block, so \(0.\dot4\dot5=0.454545\ldots\) To convert, let \(x\) be the decimal, multiply by 10, 100 and so on to line up the repeating parts, then subtract so that they cancel. For \(x=0.\dot4\dot5\): \(100x=45.4545\ldots\), so \(99x=45\) and \(x=\tfrac{45}{99}=\tfrac{5}{11}\).
Percentages. A percentage means "out of 100". To convert a percentage to a decimal, divide by 100. To find a percentage of an amount, multiply by the decimal. To write one quantity as a percentage of another, divide and multiply by 100. For percentage change, use \[\text{percentage change}=\frac{\text{change}}{\text{original}}\times100\%.\] The best method for repeated or reverse problems is the multiplier. An increase of \(r\%\) has multiplier \(1+\tfrac{r}{100}\), and a decrease of \(r\%\) has multiplier \(1-\tfrac{r}{100}\). For example, a 15% decrease has multiplier 0.85.
Reverse percentages. If you are given the amount after a change, divide by the multiplier to get back to the original. Do not take the percentage off the new value, because the percentage applies to the original.
Compound change. When the same percentage change applies each year, the amount after \(n\) years is original \(\times\ (\text{multiplier})^n\). Simple interest, by contrast, adds the same amount every year, calculated on the original sum only.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| BIDMAS | Order of operations: brackets, indices, division and multiplication, addition and subtraction. |
| \(\dfrac ab+\dfrac cd=\dfrac{ad+bc}{bd}\) | Adding fractions with different denominators. |
| \(\dfrac ab\div\dfrac cd=\dfrac ab\times\dfrac dc\) | Dividing by a fraction. |
| \(\dfrac{\text{change}}{\text{original}}\times100\%\) | Percentage increase or decrease. |
| Multiplier \(=1\pm\dfrac{r}{100}\) | Increase (+) or decrease (\(-\)) by \(r\%\). |
| \(\text{original}=\dfrac{\text{final}}{\text{multiplier}}\) | Reverse percentage. |
| \(A=P\left(1+\dfrac{r}{100}\right)^n\) | Compound growth of an amount \(P\) over \(n\) years. |
| \(I=\dfrac{PRT}{100}\) | Simple interest: rate \(R\%\) a year for \(T\) years on principal \(P\). |
Worked examples
Example 1 Work out \(18-3\times(5-9)^2\div4\).
Brackets first: \(5-9=-4\).
Powers next: \((-4)^2=16\), because a negative squared is positive.
Now \(18-3\times16\div4\). Multiply and divide from left to right: \(3\times16=48\), then \(48\div4=12\).
Finally, \(18-12=6\).
Answer: 6
Example 2 (a) Work out \(2\tfrac35-1\tfrac34\). (b) Work out \(\tfrac34\div\tfrac56\).
(a) Change to improper fractions: \(\tfrac{13}{5}-\tfrac74\).
The common denominator is 20: \(\tfrac{52}{20}-\tfrac{35}{20}=\tfrac{17}{20}\). This is already in its simplest form.
(b) Turn the second fraction upside down and multiply: \(\tfrac34\times\tfrac65=\tfrac{18}{20}\).
Divide the top and bottom by 2: \(\tfrac{9}{10}\).
You can check both on your calculator using its fraction key.
Answer: (a) \(\tfrac{17}{20}\); (b) \(\tfrac9{10}\)
Example 3 (a) A jacket costing £240 is reduced by 15%. Find the sale price. (b) After a 20% increase, a train fare is £54. Find the original fare. (c) (Higher) £2500 is invested at 3.5% compound interest per year. Find its value after 4 years.
(a) The multiplier for a 15% decrease is \(1-0.15=0.85\). Then \(240\times0.85=204\).
(b) The multiplier for a 20% increase is 1.2. The original is \(54\div1.2=45\).
(c) The multiplier is \(1+0.035=1.035\). The value after 4 years is \(2500\times1.035^4\).
\(1.035^4\approx1.14752\), so \(2500\times1.14752\approx2868.8\). Using the unrounded multiplier gives £2868.81 to the nearest penny.
Answer: (a) £204; (b) £45; (c) £2868.81 to the nearest penny
Example 4 (Higher) Write \(0.3\dot2\) as a fraction in its simplest form.
Let \(x=0.3222\ldots\) Then \(10x=3.222\ldots\) and \(100x=32.222\ldots\)
Subtract: \(100x-10x=32.222\ldots-3.222\ldots\), so \(90x=29\).
Therefore \(x=\tfrac{29}{90}\). The numbers 29 and 90 share no common factor, so this is in simplest form.
Answer: \(\dfrac{29}{90}\)
Common errors
- Adding fractions by adding the tops and the bottoms, such as \(\tfrac12+\tfrac13=\tfrac25\). You need a common denominator, giving \(\tfrac56\).
- Working out \((-4)^2\) as \(-16\). The brackets mean the negative is squared too. Without brackets, \(-4^2\) in a calculator needs care.
- Doing a reverse percentage by taking 20% off the final price. Divide by the multiplier instead.
- Calculating addition before multiplication, for example reading \(2+3\times4\) as 20 rather than 14.
- Treating compound interest as simple interest, or rounding the multiplier too early. Keep the full value until the last step.
Exam tip: Show each stage of working, such as the multiplier or the common denominator, even when your calculator can do the whole step. It also lets you spot a slip in your own method.
Check yourself
- Work out \(-7-(-4)+2\times(-3)\).
Answer
Multiplication first: \(2\times(-3)=-6\). Then \(-7+4-6=-9\). - Work out \(\tfrac23+\tfrac38\), giving your answer as a mixed number.
Answer
Common denominator 24: \(\tfrac{16}{24}+\tfrac{9}{24}=\tfrac{25}{24}=1\tfrac1{24}\). - (Higher) Write \(0.\dot7\dot2\) as a fraction in its simplest form.
Answer
\(100x=72.7272\ldots\), so \(99x=72\) and \(x=\tfrac{72}{99}=\tfrac{8}{11}\). - A phone costs £320 and its price rises to £368. Find the percentage increase.
Answer
Change \(=368-320=48\). Percentage increase \(=\tfrac{48}{320}\times100=15\%\).