Number

Fractions, indices, standard form, surds, rounding and estimation: the fluency the whole test is built on.

9 min read

Number underpins every module. Because the ESAT is non-calculator, the marks here come from being fast and exact with fractions, powers and surds rather than from anything conceptually hard.

Make sure you can move freely between fractions, decimals and percentages, apply BIDMAS, and handle negative numbers without slips, as these appear inside questions on every topic.

Order of operations (BIDMAS): Brackets, Indices, Division/Multiplication (left→right), Addition/Subtraction (left→right). A power applies before a sign, so $- 3^{2} = - 9$ but $(- 3)^{2} = 9$ .

Fractions. To add or subtract, use a common denominator; to multiply, multiply tops and bottoms; to divide, multiply by the reciprocal ("flip the second fraction"). Always cancel before multiplying to keep numbers small.

Worked example

Evaluate $\frac{3}{4} \div \frac{9}{10}$ .

$\frac{3}{4} \times \frac{10}{9} = \frac{3 \times 10}{4 \times 9} = \frac{30}{36} = \frac{5}{6} .$

Cancelling first ( $\frac{3}{9} = \frac{1}{3}$ , $\frac{10}{4} = \frac{5}{2}$ ) gives $\frac{1}{3} \times \frac{5}{2} = \frac{5}{6}$ with no large numbers.

Recurring decimals convert to fractions: if $x = 0. \overline{36}$ then $100 x = 36. \overline{36}$ , so $99 x = 36$ and $x = \frac{36}{99} = \frac{4}{11}$ .

Exam tip

Keep answers as exact fractions or surds for as long as possible; rounding early is the most common way to lose a non-calculator question.

Every integer above 1 has a unique prime factorisation (the fundamental theorem of arithmetic), e.g. $360 = 2^{3} \times 3^{2} \times 5$ .

From prime factorisations: the HCF takes the *lowest* power of each shared prime; the LCM takes the *highest* power of every prime that appears.

Worked example

Find the HCF and LCM of $24$ and $36$ .

$24 = 2^{3} \times 3, 36 = 2^{2} \times 3^{2} .$

HCF $= 2^{2} \times 3 = 12$ . LCM $= 2^{3} \times 3^{2} = 72$ .

Check: $HCF \times LCM = 12 \times 72 = 864 = 24 \times 36.$

Index laws: $a^{m} a^{n} = a^{m + n}$ , $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ , $(a^{m})^{n} = a^{mn}$ , $a^{0} = 1$ , $a^{- n} = \frac{1}{a ^{n}}$ , and $a^{1/ n} = n a$ so $a^{m / n} = (n a)^{m}$ .

Worked example

Simplify $\frac{( 2 x ^{2} ) ^{3}}{4 x}$ .

$\frac{8 x ^{6}}{4 x} = 2 x^{6 - 1} = 2 x^{5} .$

Standard form writes a number as $a \times 1 0^{n}$ with $1 \leq a < 10$ . Multiply/divide by handling the $a$ parts and the powers of 10 separately.

Worked example

$(3 \times 1 0^{5}) \times (4 \times 1 0^{- 2}) = (3 \times 4) \times 1 0^{5 - 2} = 12 \times 1 0^{3} = 1.2 \times 1 0^{4} .$

Note the final fix-up so that $1 \leq a < 10$ .

A surd is an irrational root left in exact form. Simplify using $ab = a b$ : pull out the largest square factor, e.g. $12 = 4 \times 3 = 23$ .

Rationalising the denominator removes the surd from the bottom. For a single surd, multiply top and bottom by it; for $a + b$ , multiply by the conjugate $a - b$ .

Worked example

Rationalise $\frac{3}{2 - 5}$ .

Multiply by $\frac{2 + 5}{2 + 5}$ :

$\frac{3 ( 2 + 5 )}{( 2 - 5 ) ( 2 + 5 )} = \frac{3 ( 2 + 5 )}{4 - 5} = \frac{3 ( 2 + 5 )}{- 1} = - 6 - 35 .$

Exam tip

Learn the squares up to $1 5^{2} = 225$ and the common surd simplifications. Spotting $50 = 52$ instantly saves real time when you can't reach for a calculator.

Round to a number of decimal places (d.p.) or significant figures (s.f.). The first significant figure is the first non-zero digit, so $0.004072$ to 2 s.f. is $0.0041$ .

Error bounds: a length given as $7 cm$ to the nearest cm lies in $6.5 \leq x < 7.5$ . The upper bound is "halfway to the next value", not $7.4$ .

Worked example

Estimate $\frac{498 \times 0.21}{0.0048}$ by rounding to 1 s.f.

$\frac{500 \times 0.2}{0.005} = \frac{100}{0.005} = 20000.$

Exam tip

In a multiple-choice test, a quick estimate often eliminates two or three options immediately. Round hard, get the order of magnitude, then choose.

Writing $a^{- n}$ as a negative number instead of a reciprocal: $2^{- 3} = \frac{1}{8}$ , not $- 8$ .
Forgetting to restore standard form so that $1 \leq a < 10$ .
Taking the upper bound as $7.4$ instead of $7.5$ .
Adding fractions by adding numerators and denominators separately.

Number

Overview

Fractions, decimals & order of operations

Primes, factors, HCF & LCM

Index laws & standard form

Surds

Rounding, bounds & estimation

Common mistakes