The four operations

The order of operations

A calculation such as $5+3\times 4$ could be read two ways: $8\times 4=32$ or $5+12=17$. Mathematicians agreed long ago on one order so that every written calculation has exactly one value. The agreed answer is 17.

The order is:

1. Brackets first. Anything inside brackets is worked out before it is used.
2. Indices next: powers and roots.
3. Division and multiplication, which have equal rank. Work them from left to right.
4. Addition and subtraction, which also have equal rank. Work them from left to right.

Many students remember this as BIDMAS. The danger in that word is that it seems to put D before M and A before S. It does not. $12÷3\times 2$ is $4\times 2=8$, working left to right, not $12÷6=2$. In the same way $10-4+3=9$, not $3$.

Why multiplication comes before addition. $3\times 4$ is shorthand for $4+4+4$, so $5+3\times 4$ means $5+4+4+4$. Doing the multiplication first simply respects what the shorthand stands for.

A fraction line acts as brackets. In $\frac{3+9}{8-2}$ the whole top and the whole bottom are worked out first: $\frac{12}{6}=2$. On a calculator you must type the brackets yourself, $(3+9)÷(8-2)$, or the calculator will do $3+9÷8-2$.

A question may ask you to insert brackets to make a statement true. Try each possible position and work out the value; a pair of brackets around something that would be done first anyway changes nothing.

Negative numbers

A number line makes adding and subtracting negative numbers concrete. Adding a positive number moves right; subtracting a positive number moves left.

• $-3+7=4$: start at $-3$ and move 7 to the right.
• $2-9=-7$: start at 2 and move 9 to the left.

Subtracting a negative number is the same as adding. $5-(-3)=8$. One way to see why: the difference between 5 and $-3$ on a number line is 8 steps, and subtraction measures that difference. Another: taking away a debt of 3 leaves you 3 better off. In the same way, adding a negative number is the same as subtracting: $6+(-10)=6-10=-4$.

Multiplying and dividing. When two numbers have the same sign the answer is positive; when the signs are different the answer is negative.

• $(-4)\times 3=-12$ and $4\times (-3)=-12$
• $(-4)\times (-3)=12$
• $(-20)÷5=-4$ and $(-20)÷(-5)=4$

Why is negative times negative positive? Look at the pattern $(-4)\times 2=-8$, $(-4)\times 1=-4$, $(-4)\times 0=0$. Each time the second number goes down by 1, the answer goes up by 4. Keep going: $(-4)\times (-1)=4$ and $(-4)\times (-2)=8$.

With several multiplications and divisions, count the negative signs: an even number of them gives a positive answer, an odd number gives a negative one.

Watch the difference between $(-3{)}^2=9$ and $-3^2=-9$. In the second, the square is done first (indices before the minus), and the minus is applied after.

Practical situations: temperature changes

Temperatures below freezing are the most common place negative numbers appear on the exam, along with bank balances, heights above and below sea level, and floors below ground.

The words tell you the operation:

• "rises by", "increases by" or "is warmer by": add;
• "falls by", "drops by" or "is colder by": subtract;
• "the difference between" or "how much warmer": subtract the lower value from the higher one, and the answer is a positive number of degrees.

The difference between $5^{°}\text{C}$ and $-8^{°}\text{C}$ is $5-(-8)=13$ degrees. A quick check: from $-8$ up to 0 is 8 degrees, and from 0 up to 5 is 5 more, so 13 in total.

Adding and subtracting fractions

You can only add or subtract fractions when they are the same kind of part, that is, when they have the same denominator. Three quarters and two fifths are different-sized pieces, so first rewrite both in pieces of a common size.

1. Find a common denominator. The lowest common multiple of the denominators keeps the numbers small.
2. Write each fraction as an equivalent fraction with that denominator, multiplying the top and bottom by the same number.
3. Add or subtract the numerators. The denominator stays the same.
4. Simplify, and change an improper fraction to a mixed number if the question asks for one.

$$\frac{3}{4}+\frac{2}{5}=\frac{15}{20}+\frac{8}{20}=\frac{23}{20}=1\frac{3}{20}$$

Never add the tops and add the bottoms. $\frac{1}{2}+\frac{1}{2}$ is clearly 1, but adding tops and bottoms would give $\frac{2}{4}$, which is a half.

Improper fractions and mixed numbers. $\frac{23}{20}$ is an improper fraction (the top is bigger than the bottom). As a mixed number it is $1\frac{3}{20}$, because 20 goes into 23 once with 3 left over. To go the other way, $2\frac{3}{5}=\frac{2\times 5+3}{5}=\frac{13}{5}$.

Mixed numbers. The safest method on the exam is to change every mixed number to an improper fraction first, then work as above. You can add whole numbers and fractions separately, but in a subtraction the fraction part can come out negative, which is where marks are usually lost.

Multiplying and dividing fractions

Multiplying. Multiply the numerators together and the denominators together. $\frac{2}{3}\times \frac{4}{5}=\frac{8}{15}$. This works because $\frac{2}{3}$ of $\frac{4}{5}$ means split $\frac{4}{5}$ into 3 equal parts and take 2 of them.

Cancel before you multiply to keep the numbers small: in $\frac{4}{15}\times \frac{5}{8}$, the 4 and the 8 share a factor of 4, and the 5 and the 15 share a factor of 5, so it becomes $\frac{1}{3}\times \frac{1}{2}=\frac{1}{6}$.

Dividing. To divide by a fraction, multiply by its reciprocal (turn the second fraction upside down). Why? Dividing asks "how many of these fit in?" How many quarters fit into 3? Each whole holds 4 quarters, so 12. That is $3÷\frac{1}{4}=3\times 4=12$.

$$\frac{5}{6}÷\frac{2}{9}=\frac{5}{6}\times \frac{9}{2}=\frac{45}{12}=\frac{15}{4}=3\frac{3}{4}$$

With mixed numbers, convert first. $1\frac{1}{2}\times 2\frac{1}{3}$ is not $2\frac{1}{6}$. Multiplying the whole numbers and the fractions separately misses two of the four pieces of the product. Change to improper fractions: $\frac{3}{2}\times \frac{7}{3}=\frac{21}{6}=\frac{7}{2}=3\frac{1}{2}$.

On a non-calculator paper, write every line: the improper fractions, the common denominator or the reciprocal, the unsimplified answer and the final answer. The method marks are for those lines.

Decimals without a calculator

Adding and subtracting. Line up the decimal points, filling gaps with zeros: $6.4-2.75$ is $6.40-2.75=3.65$.

Multiplying. Ignore the decimal points, multiply the whole numbers, then put back as many decimal places as there were in the question altogether.
$$0.3\times 0.04:3\times 4=12,\text{and there are 3 decimal places, so}0.012$$
This works because $0.3\times 0.04=\frac{3}{10}\times \frac{4}{100}=\frac{12}{1000}$.

Dividing by a decimal. Multiply both numbers by the same power of 10 until you are dividing by a whole number. The answer does not change, because $\frac{a}{b}$ and $\frac{10a}{10b}$ are equivalent fractions.
$$2.8÷0.07=280÷7=40$$

Sense check. Multiplying by a number less than 1 makes a positive number smaller; dividing by a number less than 1 makes it bigger. So $5.6÷0.07$ must be much more than 5.6. If your answer is smaller, the decimal point is in the wrong place.

Using a calculator with brackets

On Papers 3 and 4 you can use a calculator, but it only does what you type. It follows the same order of operations, so a fraction line or a hidden bracket must be typed as brackets.

For $\frac{12.4+{3.75}^2}{6.1-2.38}$ type $(12.4+{3.75}^2)÷(6.1-2.38)$. Without the brackets the calculator works out $12.4+\frac{{3.75}^2}{6.1}-2.38$, a different number.

A good habit is to write down the value of the top and the bottom separately (here $26.4625$ and $3.72$), then divide. If your final answer is wrong, the examiner can still award a method mark for those values.

Negative numbers need the negative key, and $(-2.5{)}^2$ needs brackets around $-2.5$: without them most calculators square 2.5 and then make it negative.