Mathematical operations for SSC CHSL

Mathematical operations questions look like arithmetic, but they are really a test of discipline. The numbers are small and the sums are easy. What goes wrong is the order: a candidate starts calculating with the printed signs, or rewrites half the expression and computes the other half from memory.

The rule that fixes almost every error is rewrite first, calculate second. Write the whole new expression on your rough sheet, then apply BODMAS to it as if you had never seen the original.

These questions are a regular part of the reasoning section in both tiers. They are also among the most reliable marks there, because once the expression is rewritten there is nothing to interpret.

BODMAS, exactly

Brackets first, then orders (squares and roots), then division and multiplication, then addition and subtraction.

• Division and multiplication are worked left to right: 20 ÷ 4 × 3 = 5 × 3 = 15, not 20 ÷ 12.
• Addition and subtraction are also worked left to right.
• Examiners usually build the expression so that the answer is the same under any reasonable reading. If your answer is not among the options, suspect your rewriting before you suspect BODMAS.

The six question types

A seventh type, the missing number in a grid or figure, is covered at the end.

Worked questions

Q1. Substitution. If + means ×, × means −, − means ÷ and ÷ means +, find 24 − 6 + 5 ÷ 12 × 8. (a) 24 (b) 20 (c) 28 (d) 16

Rewrite sign by sign: − becomes ÷, + becomes ×, ÷ becomes +, × becomes −.
New expression: 24 ÷ 6 × 5 + 12 − 8.
Calculate: 24 ÷ 6 = 4; 4 × 5 = 20; 20 + 12 − 8 = 24. Answer: (a) 24.

Q2. Letter-coded signs. If P means +, Q means −, R means × and S means ÷, find 45 S 9 R 4 P 18 Q 7. (a) 29 (b) 31 (c) 33 (d) 18

Rewrite: 45 ÷ 9 × 4 + 18 − 7 = 5 × 4 + 18 − 7 = 20 + 11 = 31. Answer: (b) 31.

Q3. Sign combination. Select the set of signs that, replacing the # symbols in order, balances the equation: 18 # 6 # 3 # 5 # 4.
(a) ÷, ×, −, = (b) ×, ÷, +, = (c) +, −, ×, = (d) −, ÷, +, =

• (a) 18 ÷ 6 × 3 − 5 = 9 − 5 = 4. The right side is 4. ✓
• (b) 18 × 6 ÷ 3 + 5 = 36 + 5 = 41. ✗
• (c) 18 + 6 − 3 × 5 = 24 − 15 = 9. ✗
• (d) 18 − 6 ÷ 3 + 5 = 18 − 2 + 5 = 21. ✗

Answer: (a). Note where the = sign sits in each option; it tells you which number is the right-hand side.

Q4. Interchanging signs. Which two signs must be interchanged to make the equation correct? 15 × 3 + 4 ÷ 6 − 9 = 20
(a) × and ÷ (b) + and − (c) + and × (d) ÷ and −

• (a) 15 ÷ 3 + 4 × 6 − 9 = 5 + 24 − 9 = 20. ✓
• (b) 15 × 3 − 4 ÷ 6 + 9 gives about 53.3. ✗
• (c) 15 + 3 × 4 ÷ 6 − 9 = 15 + 2 − 9 = 8. ✗
• (d) 15 × 3 + 4 − 6 ÷ 9 gives about 48.3. ✗

Answer: (a). A result with a fraction is a quick sign that an option is wrong, since the target is a whole number.

Q5. Defined operation. If a ⊕ b = a × b + a − b, find (3 ⊕ 2) ⊕ 4. (a) 27 (b) 29 (c) 31 (d) 35

Inner bracket: 3 ⊕ 2 = 6 + 3 − 2 = 7.
Outer: 7 ⊕ 4 = 28 + 7 − 4 = 31. Answer: (c) 31.

Q6. Coded inequality. A $ B means A is greater than B, A # B means A is less than B, and A @ B means A equals B. Given A $ B, B @ C and C # D, which conclusions follow? I: A $ C. II: D $ B.
(a) Only I (b) Only II (c) Both (d) Neither

Decode: A > B = C < D.
I: A > B and B = C, so A > C. Follows.
II: D > C and C = B, so D > B. Follows.
Answer: (c) Both.

Missing numbers in grids and figures

The rule usually runs along each row or down each column. Test it on two complete rows before trusting it.

Worked example (rows).

Row 1: 2² + 3² = 4 + 9 = 13. Row 2: 4² + 5² = 16 + 25 = 41. Row 3: 6² + 1² = 36 + 1 = 37.

Worked example (columns).

Column 1: 5² − 3² = 25 − 9 = 16. Column 2: 7² − 4² = 49 − 16 = 33. Column 3: 9² − 6² = 81 − 36 = 45.

For a triangle or circle with numbers at the corners and one in the centre, try in order: sum, product, sum of squares, then product minus or plus a constant.

Practice

1. If + means ×, − means +, × means ÷ and ÷ means −, find 40 × 8 + 3 − 6 ÷ 2. (a) 17 (b) 19 (c) 21 (d) 23
2. If P means ÷, Q means ×, R means + and S means −, find 36 P 6 Q 3 R 4 S 5. (a) 15 (b) 17 (c) 19 (d) 21
3. Which set of signs balances 8 # 4 # 2 # 6 # 22? (a) ×, ÷, +, = (b) +, ×, −, = (c) ÷, ×, +, = (d) +, −, ×, =
4. Which two signs must be interchanged? 24 ÷ 4 × 5 + 3 − 7 = 14 (a) ÷ and × (b) + and − (c) × and + (d) ÷ and −
5. If a $ b = a² − b, find 4 $ (3 $ 5). (a) 8 (b) 11 (c) 12 (d) 16
6. Grid rows: (3, 4, 25), (5, 12, 169), (8, 6, ?). Find ?. (a) 64 (b) 100 (c) 196 (d) 48
7. Triangles with corners (2, 3, 4) have 24 in the centre; (1, 5, 6) have 30; (3, 3, 3) have ?. (a) 9 (b) 18 (c) 27 (d) 30
8. A $ B means A is greater than B, A # B means A is less than B, A @ B means A equals B. Given P # Q, Q @ R and R # S, which is definitely true? (a) P $ S (b) P # S (c) S # Q (d) P @ R

Answers:

1. (b) 19. Rewrite: 40 ÷ 8 × 3 + 6 − 2 = 15 + 6 − 2 = 19.
2. (b) 17. 36 ÷ 6 × 3 + 4 − 5 = 18 + 4 − 5 = 17.
3. (a). 8 × 4 ÷ 2 + 6 = 16 + 6 = 22. Option (b) gives 10, (c) gives 10 and (d) gives 0.
4. (c). 24 ÷ 4 + 5 × 3 − 7 = 6 + 15 − 7 = 14. As printed, the left side is 26.
5. (c) 12. Inner: 3 $ 5 = 9 − 5 = 4. Outer: 4 $ 4 = 16 − 4 = 12.
6. (b) 100. Sum of squares: 9 + 16 = 25, 25 + 144 = 169, 64 + 36 = 100.
7. (c) 27. Product of the corners: 2 × 3 × 4 = 24, 1 × 5 × 6 = 30, 3 × 3 × 3 = 27.
8. (b) P # S. Decoded: P < Q = R < S, so P < S. S is greater than Q, not less, and P is less than R, not equal.

What to do next

• Solve 20 mixed questions a day for a week, always writing the rewritten expression in full.
• Time yourself: aim for under 40 seconds a question by the end of the week.
• Revise coding–decoding, which uses the same substitute-then-apply habit with letters.
• Then move to mirror and water images, the next quick-marks topic.