5.4 Mathematical Operations

Why Mathematical Operations Matters

These questions combine arithmetic with symbolic reasoning. You will see an equation that is incorrect as written and a set of options for which sign substitution, interchange, or insertion makes it correct. The skill is mechanical evaluation under BODMAS — get the order right and the answer falls out.

BODMAS Refresher

The order of operations is Brackets → Of (powers / 'of') → Division → Multiplication → Addition → Subtraction. RRB distractors are built around violating this order — typically by tempting you to add before multiplying, or to subtract before adding.

Type 1 — Symbol Replacement

You are given an equation using stand-in symbols and told what each symbol actually means; you then evaluate.

Worked example. If A means +, B means −, C means ×, and D means ÷, evaluate 16 A 4 C 3 B 6 D 2.

• Substitute: 16 + 4 × 3 − 6 ÷ 2.
• BODMAS: division first → 6 ÷ 2 = 3, so the expression becomes 16 + 4 × 3 − 3.
• Multiplication next → 4 × 3 = 12, giving 16 + 12 − 3.
• Addition then subtraction → 28 − 3 = 25.
• Answer: 25.

Type 2 — Sign Interchange

You are given an incorrect equation and four options for which pair of signs to swap; choose the swap that makes the equation true. Test each option by substituting, evaluating under BODMAS, and comparing LHS to RHS.

Worked example. 5 + 3 × 4 − 2 ÷ 2 = 18 is false as written (LHS = 5 + 12 − 1 = 16). Which interchange makes it true?

• Interchange + and ×: 5 × 3 + 4 − 2 ÷ 2 → 15 + 4 − 1 = 18. Correct.
• Interchange + and −: 5 − 3 × 4 + 2 ÷ 2 → 5 − 12 + 1 = −6. Wrong.
• Interchange × and ÷: 5 + 3 ÷ 4 × 2 − 2 → non-integer. Wrong.
• Interchange − and ÷: 5 + 3 × 4 ÷ 2 − 2 → 5 + 6 − 2 = 9. Wrong.
• Answer: + and ×.

Type 3 — Balancing Equations

You are given ? + 3 × 4 − 2 = 14. What number replaces ??

• Evaluate the known part: 3 × 4 = 12, then 12 − 2 = 10.
• So ? + 10 = 14, giving ? = 4.
• Answer: 4.

Type 4 — Bracket and 'Of' Evaluation

When brackets or the word of appear, evaluate them first regardless of their position in the line.

Worked example. Evaluate 4 + 50% of 8 × 2.

• 'Of' binds before multiplication: 50% of 8 = 0.5 × 8 = 4.
• Now the expression is 4 + 4 × 2.
• Multiplication: 4 × 2 = 8.
• Addition: 4 + 8 = 12.

A common mistake is to compute 4 + 4 = 8 first, then 8 × 2 = 16. That violates BODMAS — 'of' must be resolved before any other operation.

Worked Example — Bracketed Expression

Worked example. Evaluate 2 × (5 + 3) − 4.

• Bracket first: 5 + 3 = 8.
• Multiplication: 2 × 8 = 16.
• Subtraction: 16 − 4 = 12.

Brackets override everything: even though multiplication normally precedes addition, the bracket forces the inner addition to happen first.

Worked Example — Two-Layer Balancing

Worked example. What number replaces ? in ? × 2 + 6 ÷ 3 = 12?

• Division: 6 ÷ 3 = 2. Now ? × 2 + 2 = 12.
• So ? × 2 = 10, giving ? = 5.
• Answer: 5.

Work from the operations you can evaluate (division here) inward to the unknown. Once the only unknown sits next to a single operation, solve directly.

Worked Example — Sign Interchange with No Brackets

Worked example. Which interchange makes 8 − 2 × 3 + 4 = 10 correct?

• As written: 2 × 3 = 6, so 8 − 6 + 4 = 6. Target is 10.
• Swap − and +: 8 + 2 × 3 − 4 → 8 + 6 − 4 = 10. Correct.
• Answer: swap − and +.

When the equation has no brackets, multiplication still precedes addition and subtraction; only the swapped signs change the order of the additions and subtractions.

Common Symbol Substitutions

Type 5 — Coded Numbers

Sometimes the digits themselves are coded as letters or symbols. Decode first, then evaluate.

Worked example. If A=1, B=2, C=3, D=4, evaluate ABC + DAB (read as two three-digit numbers).

• ABC = 123, DAB = 412.
• 123 + 412 = 535.

Exam Scenarios and Traps

Trap 1 — Left-to-right evaluation. 8 + 4 × 3 evaluated left-to-right gives (8+4)×3 = 36, but the correct BODMAS answer is 8 + 12 = 20. RRB will include 36 as a distractor.

Trap 2 — Forgetting that 'of' binds before multiplication. 50% of 200 × 3 is (0.5 × 200) × 3 = 300. The 'of' operation is treated as a bracket-level priority.

Trap 3 — Sign interchange with negative results. When swapping + and −, watch for intermediate negative values; they flip the final result. Write each step explicitly rather than doing it in your head.

Trap 4 — Multiple correct options. Some balancing questions have two options that both produce the RHS numerically, but only one preserves the integer or natural-number form expected. Re-read the question for that constraint.

A Five-Step Method

1. Decode any coded symbols to their real arithmetic meaning.
2. Substitute the proposed interchange or replacement into the equation.
3. Evaluate under BODMAS, one operation at a time, writing intermediate results.
4. Compare the LHS to the RHS; if equal, the option is correct.
5. If no option matches, recheck the order — most failures are BODMAS violations, not arithmetic errors.

Speed Tips

• For sign-interchange questions, start with the option that changes the largest term (usually a multiplication) — if that does not balance, the others rarely will.
• Keep the four BODMAS letters visible on your rough sheet; tick off each step as you complete it.
• Treat of as implicit multiplication that preceded explicit multiplication — it is a common RRB trap to evaluate it later.
• When numbers are coded as letters, decode all of them first into a clean arithmetic expression; never try to evaluate the letters directly.