AQA GCSE Mathematics

Place value and ordering

Each digit in a number is worth ten times the digit to its right, so in $4\,706.35$ the 7 stands for 700 and the 5 for five hundredths. Place value is what lets you compare decimals and multiply or divide by powers of ten: multiplying by 10 moves every digit one place to the left, and dividing by 100 moves it two places to the right.

To order decimals, write them with the same number of decimal places and compare digit by digit from the left. To order fractions, either convert each to a decimal or rewrite them over a common denominator. Negative numbers order from the number line: the further left, the smaller, so $-8 < -3 < 0$.

Worked example

Put $0.45,\ 0.405,\ \tfrac{4}{9},\ 0.5$ in ascending order.

So the order is $0.405,\ \tfrac{4}{9},\ 0.45,\ 0.5$.

Inequality symbols

The open end of the symbol faces the larger number. The statement $-2 \le x < 5$ means $x$ can equal $-2$ but cannot equal 5.

The four operations with integers and decimals

Adding and subtracting negatives

Subtracting a negative number is the same as adding the positive: $-7 - (-3) = -7 + 3 = -4$. Two like signs in a product or quotient give a positive result and two unlike signs give a negative one, so $(-6) \times (-4) = 24$ and $18 \div (-3) = -6$.

Long multiplication

Multiply by each digit of the second number in turn, keeping track of place value, then add.

Multiplying decimals

Ignore the decimal points, multiply as whole numbers, then put back as many decimal places as the two original numbers had between them. For $3.4 \times 0.25$, the whole-number product is $34 \times 25 = 850$; the factors have one and two decimal places, so the answer has three, which gives $0.850 = 0.85$.

Dividing by a decimal

Multiply both numbers by the same power of ten until the divisor is a whole number, because the quotient does not change when both parts are scaled equally. For $7.56 \div 0.12$ multiply both by 100 to get $756 \div 12 = 63$.

Long division works by finding how many times the divisor goes into each successive part of the dividend. When there is a remainder, either write it as a remainder, as a fraction of the divisor, or carry on into decimal places by adding zeros after the decimal point.

Fractions

Any fraction can be written as an equivalent fraction by multiplying or dividing the numerator and denominator by the same number. A fraction is in its simplest form when the numerator and denominator have no common factor except 1. An improper fraction has a numerator larger than its denominator, and a mixed number combines a whole number with a proper fraction, so $2\tfrac{1}{3} = \tfrac{7}{3}$.

Adding and subtracting needs a common denominator, because you can only combine parts of the same size.

With mixed numbers, either add the whole numbers and fractions separately or convert to improper fractions first. Converting is safer when the subtraction would otherwise need borrowing.

Multiplying fractions multiplies the numerators and the denominators. Cancelling a common factor before multiplying keeps the numbers small: $\tfrac{4}{9} \times \tfrac{3}{10} = \tfrac{2}{3} \times \tfrac{1}{5} = \tfrac{2}{15}$ after cancelling 3 and then 2.

Dividing by a fraction means multiplying by its reciprocal, which is the fraction turned upside down. This works because $\tfrac{2}{5}$ means "divide by 5, multiply by 2", so undoing it means multiplying by 5 and dividing by 2.

All of these work the same way for negative fractions, using the sign rules above.

Priority of operations

A calculation is evaluated in a fixed order so that everyone gets the same answer: brackets first, then powers and roots, then multiplication and division from left to right, then addition and subtraction from left to right. A fraction bar acts as a bracket around its numerator and around its denominator, and a root sign covers everything under it.

A reciprocal is $1$ divided by the number, so the reciprocal of 8 is $\tfrac{1}{8}$ and the reciprocal of $\tfrac{2}{5}$ is $\tfrac{5}{2}$. Zero has no reciprocal because you cannot divide by zero.

Inverse operations

Addition and subtraction undo each other, as do multiplication and division, squaring and square rooting, and cubing and cube rooting. Using an inverse lets you check an answer (if $17.5 \times 4 = 70$ then $70 \div 4$ should be $17.5$) and solve for an unknown, and it is the idea behind cancelling in an expression: $\dfrac{7 \times 12}{12} = 7$ because dividing by 12 undoes multiplying by 12.