Quick Revision

BODMAS Order and Traps

Order: the bar (vinculum), brackets ( ) { } [ ], then ‘of’, then ÷ and × left to right, then + and − left to right.

[SSC Tip] Most wrong options are built by doing × before ÷.

Multiplying: the decimal places add up. 0.3 × 0.02 = 0.006.
Dividing: shift both points until the divisor is whole. 1.2 ÷ 0.03 = 120 ÷ 3 = 40.
Percentages as fractions: 12.5% = $\frac{1}{8}$ , 37.5% = $\frac{3}{8}$ , 62.5% = $\frac{5}{8}$ , $16\frac{2}{3}$ % = $\frac{1}{6}$ , 6.25% = $\frac{1}{16}$ .
x% of y = y% of x: 18% of 50 = 50% of 18 = 9.
× 25 is × 100 ÷ 4, and × 125 is × 1000 ÷ 8.
Patterns: 0.1 + 0.11 + 0.111 + 0.1111 = 0.4321.

Same top: the smaller bottom wins. Same bottom: the bigger top wins.
Two fractions: cross-multiply. $\frac{a}{b} > \frac{c}{d}$ when ad > bc.
Four options: turn each into a decimal to two or three places.
Fractions close to 1: compare the gap below 1. Fractions close to $\frac{1}{2}$ : compare the extra above $\frac{1}{2}$ .
For a proper fraction, adding the same number to top and bottom makes it bigger: $\frac{3}{5} < \frac{4}{6} < \frac{5}{7}$ .
Mixed numbers: compare the whole parts first.

[SSC Tip] Check whether the question wants ascending or descending order.

Evaluate a continued fraction from the bottom: add, flip, add, flip. $1 + \cfrac{1}{2 + \cfrac{1}{3}} = 1 + \frac{3}{7} = \frac{10}{7}$ .
To solve for x, peel layers off from the top.
$\frac{1}{n(n + k)} = \frac{1}{k}\left(\frac{1}{n} - \frac{1}{n + k}\right)$ , so long sums cancel to first minus last.
Spot hidden products: 20 = 4 × 5, 30 = 5 × 6, then 42, 56, 72, 90.
$\left(1 - \frac{1}{2}\right) \cdots \left(1 - \frac{1}{n}\right) = \frac{1}{n}$ and $\left(1 + \frac{1}{2}\right) \cdots \left(1 + \frac{1}{n}\right) = \frac{n + 1}{2}$ .

Know squares to 30 and cubes to 15 by heart.

Prime factors: halve every power for a square root, divide by 3 for a cube root. 5184 = $2^6 \times 3^4$ , so its root is $2^3 \times 3^2$ = 72.
Cube root of a perfect cube: the last digit gives the last digit (2 and 8 swap, 3 and 7 swap). Drop the last three digits, and what is left gives the first digit: $\sqrt[3]{148877} = 53$ .
Decimals: $\sqrt{0.0729} = 0.27$ and $\sqrt[3]{0.000343} = 0.07$ .
Trap: $\sqrt{0.4}$ is about 0.632, not 0.2.
Mixed numbers: make them improper first.

Finite nests: start with the innermost root and work outwards.
$\sqrt{x + \sqrt{x + \cdots}}$ : write x as n(n + 1); the answer is the bigger number. $\sqrt{42 + \sqrt{42 + \cdots}} = 7$ , as 42 = 6 × 7.
$\sqrt{x - \sqrt{x - \cdots}}$ : the answer is the smaller number. $\sqrt{42 - \sqrt{42 - \cdots}} = 6$ .
Endless $\sqrt{x\sqrt{x\sqrt{x \cdots}}} = x$ .
With n square roots, $\sqrt{x\sqrt{x \cdots \sqrt{x}}} = x^{\frac{2^n - 1}{2^n}}$ , so three roots give $x^{\frac{7}{8}}$ .
Endless cube roots: $\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x \cdots}}} = \sqrt{x}$ .

$a^m \times a^n = a^{m + n}$ , $a^m \div a^n = a^{m - n}$ , $(a^m)^n = a^{mn}$ , $(ab)^n = a^n b^n$ .
$a^0 = 1$ ( $a \neq 0$ ), $a^{-n} = \frac{1}{a^n}$ and $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ .
$a^{\frac{m}{n}} = (\sqrt[n]{a})^m$ , so $27^{\frac{2}{3}} = 9$ .
Equal bases: if $a^x = a^y$ (a not 0, 1 or −1), then x = y.
Write numbers as prime powers first: 8 = $2^3$ , 27 = $3^3$ , 0.001 = $10^{-3}$ .
Trap: $2^{3^2} = 2^9$ , not $(2^3)^2 = 2^6$ .
A negative power never makes the answer negative.

Make the powers equal using their HCF: $2^{300} = 8^{100}$ and $3^{200} = 9^{100}$ , so $3^{200}$ is larger.
Or make the bases equal: $4^{15} = 2^{30}$ is smaller than $8^{11} = 2^{33}$ .
Roots of different orders: raise all of them to the LCM of the orders. For $\sqrt{2}$ , $\sqrt[3]{3}$ and $\sqrt[4]{4}$ , the 12th powers are 64, 81 and 64, so $\sqrt[3]{3}$ is the largest.
Among $\sqrt[n]{n}$ for whole numbers, $\sqrt[3]{3}$ is the largest.
Same power: the bigger base wins.

$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ and $\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}$ .
$\sqrt{a} + \sqrt{b}$ is not $\sqrt{a + b}$ : $\sqrt{9} + \sqrt{16} = 7$ , but $\sqrt{25} = 5$ .
Simplify by taking out square factors: $\sqrt{72} = 6\sqrt{2}$ and $\sqrt{108} = 6\sqrt{3}$ .
Only like surds add: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$ , but $\sqrt{2} + \sqrt{3}$ stays as it is.
Move a number inside the root to compare: $3\sqrt{2} = \sqrt{18}$ and $2\sqrt{5} = \sqrt{20}$ .
Bring surds of different orders to one order using indices: $\sqrt[3]{2} = \sqrt[6]{4}$ .

One surd at the bottom: multiply by it. $\frac{6}{\sqrt{3}} = 2\sqrt{3}$ .
Two terms: multiply by the conjugate. $\frac{1}{\sqrt{a} + \sqrt{b}} = \frac{\sqrt{a} - \sqrt{b}}{a - b}$ .
$(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$ , always rational.
$\frac{1}{\sqrt{n} + \sqrt{n + 1}} = \sqrt{n + 1} - \sqrt{n}$ , so long sums cancel: up to $\frac{1}{\sqrt{99} + \sqrt{100}}$ the sum is 10 − 1 = 9.
If $x = 2 + \sqrt{3}$ , then $\frac{1}{x} = 2 - \sqrt{3}$ .
[Tier 2] Three terms: group two of them and use a conjugate twice.

Same total under the roots: $\sqrt{a} + \sqrt{b}$ is biggest when a and b are closest, so $\sqrt{4} + \sqrt{5} > \sqrt{2} + \sqrt{7}$ .
Differences shrink as numbers grow, since $\sqrt{n + 1} - \sqrt{n} = \frac{1}{\sqrt{n + 1} + \sqrt{n}}$ .
$\sqrt{a + 2\sqrt{b}} = \sqrt{x} + \sqrt{y}$ where x + y = a and xy = b: $\sqrt{8 + 2\sqrt{15}} = \sqrt{5} + \sqrt{3}$ .
Minus form: make the inner part $2\sqrt{b}$ and put the larger root first: $\sqrt{8 - \sqrt{60}} = \sqrt{8 - 2\sqrt{15}} = \sqrt{5} - \sqrt{3}$ .

$a^2 - b^2 = (a + b)(a - b)$ : $63^2 - 37^2 = 100 \times 26$ .
$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$ ; in a fraction the long bracket cancels.
$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$ .
If a + b + c = 0: $a^3 + b^3 + c^3 = 3abc$ .
Near a base: 98 × 102 = $100^2 - 2^2$ .
With decimals, name a, b and c first.

Round to easy values: 49.98 → 50, 19.97% → 20%, $\sqrt{899}$ → 30.
Round one number up and one down, so errors cancel.
Options are usually far apart, so rough is enough.
Digit-sum check: keep adding the digits until one digit is left. The answer's digit sum must match the product of the digit sums: for 238 × 47, 4 × 2 = 8, and 11,186 → 17 → 8.
Try the last digit first. The check rules options out but cannot prove one right.