SSC CGL Maths · Surds, Indices and Simplification

Missing Numbers and Letters

When a letter or a blank sits inside a BODMAS chain, simplify everything around it first, then undo what is left; when the whole expression is in letters, open the brackets and collect like terms, or substitute with care over signs.

All Surds, Indices and Simplification notes

Why this matters

Nine questions in the bank, eight EASY and one MODERATE, from the 2023 and 2024 shifts. They test the same BODMAS order as the chapter's first page, run backwards or done with letters, and the printed wrong options come from a sign slip with a negative value.

Concept 1 of 2: Finding a missing number in an expression

Most of the line has no unknown in it. Simplify all of that first, using BODMAS, and the unknown is left in a short equation. Then undo its operations in reverse order.

Definition

Work out every part that has no unknown.
Keep the unknown's own operation attached to it: in $4 \times A$ the 4 stays with A until the end.
Undo in reverse order: undo an addition by subtracting, a multiplication by dividing.
When the unknown is deep inside brackets, treat the whole bracket as one number first. In $30 - [\,\dots\,] = 28$ the bracket must be 2.
Fast route: test the options. Put each into the line and keep the one that balances. Start with a middle option; if it is too big or too small, that tells you which way to go.

Worked example

Find

x

56 \div 7 + 3 \times x - 45 \div 9 = 24

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 25 Sep 2024, Shift 3 · Q60Easy

Example 1 · Surds, Indices and Simplification · Missing Numbers and Letters

The value of N in the equation

9 \times \text{N of } 4 \div 4 - 15 \times 4 = 12

is:

Pulling the unknown away from its multiplier

20 - 3 \times y = 5

the 3 belongs to

y

3y = 15

, so

y = 5

. Working

20 - 3 = 17

first breaks BODMAS and gives

y = \frac{5}{17}

Skipping the check

An answer found by undoing steps is easily off by a sign. Put it back into the original line before marking it: the check takes a few seconds and catches the slip that the wrong options are built on.

Drill 4 more on finding a missing number in an expression

Concept 2 of 2: Simplifying and evaluating expressions in letters

Letters obey the same bracket and sign rules as numbers. Open each bracket from the inside and collect like terms as you go. To find a value, put each number in with its sign, inside brackets.

Definition

Like terms have the same letters to the same powers: $5p$ and $-2p$ make $3p$ ; $p$ and $q$ do not combine.
Open brackets from the inside out, flipping every sign after a minus, exactly as with numbers.
Substitute inside brackets: for $x = -3$ , write $(-3)^2 = 9$ and $(-3)^3 = -27$ .
An even power of a negative number is positive; an odd power stays negative.
Fast route: fix the sign of each term before working out its size; many wrong options differ from the key only in one sign.

Powers of a negative number

(-a)^{2} = a^{2} \qquad (-a)^{3} = -a^{3}

Worked example

Simplify

4a - [2b - \{5a - (3b - a)\}]

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2023 · Tier I, 14 Jul 2023, Shift 1 · Q69Easy

Example 2 · Surds, Indices and Simplification · Missing Numbers and Letters

What is the value of

(3x^3 + 5x^2y + 12xy^2 + 7y^3)

, when

x = -4

and

y = -1

Squaring a negative without brackets

-3^2

means

-(3^2) = -9

, but

(-3)^2 = 9

. When you put

x = -3

into

x^2

, write

(-3)^2

Adding unlike terms

3p + 2q

cannot be written as

5pq

5p

. Only terms with the same letters to the same powers combine.

Drill 3 more on simplifying and evaluating expressions in letters

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Simplifying and evaluating expressions in letters
Powers of a negative number
$(-a)^{2} = a^{2} \qquad (-a)^{3} = -a^{3}$

Watch out for (4)

Pulling the unknown away from its multiplier
→ Finding a missing number in an expression
Skipping the check
→ Finding a missing number in an expression
Squaring a negative without brackets
→ Simplifying and evaluating expressions in letters
Adding unlike terms
→ Simplifying and evaluating expressions in letters

Test yourself on Surds, Indices and Simplification

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Surds, Indices and Simplification chapter test Drill all 9 questions