The Four Operations and Ordering

Order of operations: the BIDMAS rules

The four operations are addition, subtraction, multiplication and division. When more than one of them appears in the same calculation everyone has to agree on the order, or the same sum gives different answers. That agreed order is BIDMAS: brackets, indices, division and multiplication, then addition and subtraction.

Division and multiplication have equal priority, so when both appear you work from left to right. Addition and subtraction are also equal to each other, and are done last. So $6÷2\times 3$ is $3\times 3=9$, not $6÷6=1$.

Brackets override everything. $(4+6)÷2$ is $10÷2=5$, while $4+6÷2$ is $4+3=7$. Show the value at each stage on its own line: Cambridge gives a method mark for the substituted line, so the working scores even when the final number slips.

Negative numbers in calculations

A negative number is written with a minus sign in front of it, and the sign is part of the number: $-7$ is seven below zero. On the number line, adding moves you right and subtracting moves you left. Adding a negative therefore moves left, and subtracting a negative moves right: $7-(-3)=7+3=10$.

Multiplying or dividing two numbers with the same sign gives a positive answer, and two different signs give a negative answer. So $(-3)\times (-4)=12$ and $(-8)÷(-2)=4$, while $(-3)\times 5=-15$.

A minus sign in front of a bracket multiplies everything inside it: $-(4-9)=-4+9=5$. Watch the difference between $-3^2$ and $(-3{)}^2$. Without brackets the index applies to the 3 first, so $-3^2=-9$, while $(-3{)}^2=9$.

Fractions: adding, subtracting, multiplying and dividing

To add or subtract fractions you first write them with a common denominator, then add or subtract the numerators only. $\frac{2}{3}+\frac{1}{6}=\frac{4}{6}+\frac{1}{6}=\frac{5}{6}$. The denominator is the size of the piece, not a quantity, which is why it stays unchanged.

Multiplying is the easy one: multiply the numerators and multiply the denominators. $\frac{2}{3}\times \frac{3}{5}=\frac{6}{15}=\frac{2}{5}$. To divide, turn the second fraction upside down and multiply instead: $\frac{3}{4}÷\frac{2}{5}=\frac{3}{4}\times \frac{5}{2}=\frac{15}{8}=1\frac{7}{8}$.

Mixed numbers must be turned into improper fractions before you multiply or divide. $2\frac{1}{2}=\frac{2\times 2+1}{2}=\frac{5}{2}$. Going the other way, $\frac{17}{5}=3\frac{2}{5}$ because 17 divided by 5 is 3 remainder 2. Leave the answer as a fraction in its lowest terms unless the question asks for a decimal.

• $\frac{3}{4}-\frac{5}{6}=\frac{9}{12}-\frac{10}{12}=-\frac{1}{12}$
• $\frac{1}{2}\times \frac{3}{7}=\frac{3}{14}$
• $4÷\frac{1}{2}=4\times 2=8$
• $\frac{11}{4}=2\frac{3}{4}$

Decimals and the four operations

Decimals behave like integers as long as you keep the digits lined up. For addition and subtraction, write the numbers with the decimal points underneath each other and fill any gaps with zeros: $3-0.75=3.00-0.75=2.25$.

For multiplication, ignore the decimal points, multiply the whole numbers, then put the points back. The answer has as many decimal places as the two numbers had between them: $0.4\times 0.3$, so $4\times 3=12$ with two decimal places, gives $0.12$.

To divide by a decimal, make the divisor a whole number by multiplying both numbers by the same power of ten: $1.2÷0.4=12÷4=3$. The value is unchanged because both numbers were scaled in the same way.

Ordering quantities and the comparison symbols

To order quantities by magnitude means putting them in size order, and the symbol you choose says how two quantities compare. Cambridge expects all six: $=$ for is equal to, $\ne$ for is not equal to, $>$ for is greater than, $<$ for is less than, $\ge$ for is greater than or equal to, and $\le$ for is less than or equal to.

Decimals are compared place by place from the left, so a number with more digits after the point is not automatically bigger. 0.58 and 0.583... differ at the third decimal place, so 0.583... is larger. To compare a fraction with a decimal, convert one of them: $\frac{5}{8}=0.625$, so $0.63>\frac{5}{8}$.

Negative numbers reverse the usual feeling. $-0.75$ is further from zero than $-0.5$, but further from zero on the negative side means smaller, so $-0.75<-0.5$. In the list $-2.5,-\frac{1}{3},0,0.4,\frac{7}{10}$ the negative with the largest magnitude comes first, because $-2.5$ is the smallest value in the list.

Putting it together: multi-step problems

Most exam questions hide two or three operations inside a sentence, and the marks are for the order you do them in as much as for the answer. Write the calculation as a single line of arithmetic, with brackets where the question implies them, before you reach for the calculator. That fixes the order of operations on the page where the examiner can see it.

A shop example. Pens cost $1.25 each. Jamil buys 3 pens and pays with a $5 note. The change is $5-3\times 1.25=5-3.75=1.25$, so $1.25. The multiplication comes first because it works out the total cost, and the subtraction then answers the question that was asked.

A temperature that changes in equal steps becomes a multiplication. Start at $-6$ °C, rise $4$ °C an hour for 3 hours, then fall $5$ °C: $-6+3\times 4-5=-6+12-5=1$, so the final temperature is $1$ °C.

An average is a bracket question in disguise, because the total has to be found before it is divided: $\frac{4+7+9+12}{4}=\frac{32}{4}=8$.

• $18÷(2+4)\times 3=18÷6\times 3=9$
• $\frac{1}{2}+\frac{2}{3}\times \frac{3}{4}$: multiply first, so $\frac{1}{2}+\frac{1}{2}=1$
• $(-3{)}^2-4\times 2=9-8=1$
• A rise of 3 °C followed by a fall of 8 °C is $3-8=-5$ °C