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UPSC CSE IAS · General Intelligence & Reasoning

Mathematical Operations

Solving equations after interchanging mathematical signs and numbers based on given rules.

This chapter is not arithmetic skill. Someone reprints an expression with the wrong symbols on purpose, hands you a key that says what each printed symbol really means, and asks you to evaluate. One running expression threads the first three cards: 6 + 2 − 1 × 3 ÷ 4, under the key + means ×, − means ÷, × means −, and ÷ means +. Rewrite first, then BODMAS — never left-to-right on the printed line, and never 'all divisions before multiplications' once × and ÷ share a level. The last two cards flip the same habit onto interchange questions: one frozen swap, tested on every option.

  • UPSC CSE IAS
  • Easy level
  • 5 concepts
  • 45 practice questions

1Rewrite the key before you compute

A substitution key tells you what each printed symbol really means. The running key is: + means ×, − means ÷, × means −, and ÷ means +. The printed expression is 6 + 2 − 1 × 3 ÷ 4. Those plus and minus signs are not plus and minus. The whole skill is to replace every symbol with its real operator first, then evaluate the rewritten expression. Computing while you still see the printed symbols mixes two alphabets and almost always flips the answer.

Rewrite the whole line in one pass, numbers staying put, operators swapping. 6 stays 6, the printed + becomes ×, 2 stays 2, the printed − becomes ÷, 1 stays 1, the printed × becomes −, 3 stays 3, the printed ÷ becomes +. The rewritten line is 6 × 2 ÷ 1 − 3 + 4. Stop. Do not resolve a multiplication in the middle of substituting, and do not leave one symbol untranslated because it 'looks like' the ordinary one. Only this rewritten line is evaluated — brackets, orders, ×÷ left to right, then +− left to right. The rewrite is the job here; BODMAS on 13 comes after the line is honest.

Figure. Two passes, not one: rewrite every printed symbol first, then evaluate only the rewritten line.

Order of attack

  1. Read the keyWrite each printed symbol beside its real operator so you cannot misread mid-expression. Running key: + → ×, − → ÷, × → −, ÷ → +.
  2. Rewrite onceCopy the numbers in place and swap every operator for the real one. Stop. Do not compute yet. 6 + 2 − 1 × 3 ÷ 4 becomes 6 × 2 ÷ 1 − 3 + 4.
  3. Then BODMASOnly the rewritten line is evaluated — brackets, orders, ×÷ left to right, then +− left to right. That walk lands on 13.

Symbol substitution with BODMAS

If + means ×, − means ÷, × means −, and ÷ means +, find the value of 6 + 2 − 1 × 3 ÷ 4.

  • Substitute: +→×, −→÷, ×→−, ÷→+6 × 2 ÷ 1 − 3 + 4
  • × and ÷ left to right: 6 × 212
  • 12 ÷ 112
  • + and − left to right: 12 − 3 + 413

Pro tip. Translate all symbols first, then strictly follow BODMAS — never mix substitution with calculation.

If + means ×, − means ÷, × means −, and ÷ means +, what is the value of 6 + 2 − 1 × 3 ÷ 4?
  1. 13
  2. 20
  3. 7

Rewrite to 6 × 2 ÷ 1 − 3 + 4. Then 12 ÷ 1 = 12, and 12 − 3 + 4 = 13. Answering 20 usually means someone multiplied or added left-to-right on the printed symbols without rewriting.

2BODMAS after the rewrite — not left to right

Once the real operators are on the page, evaluation order is ordinary BODMAS: Brackets, Orders (powers), Division and Multiplication left to right, then Addition and Subtraction left to right. Finish the running rewrite first: 6 × 2 ÷ 1 − 3 + 4. The × and ÷ stretch is 6 × 2 ÷ 1. Left to right that is 12, then 12 ÷ 1 = 12. Then the +− stretch: 12 − 3 + 4 = 13. That 13 is the value of the original printed line under the key.

Walk 6 + 4 × 2 slowly, because the options are built around it. There is no bracket in the line. Multiplication happens first: 4 × 2 = 8. Then addition: 6 + 8 = 14. The exam trap is applying operations left-to-right on a mixed line — for example treating 6 + 4 × 2 as (6 + 4) × 2 = 20 instead of 6 + 8 = 14. Underline the × and ÷ stretches before you touch + or −. 20 is almost always sitting in the options next to 14, waiting for anyone who invented a bracket the stem did not write.

Figure. Same digits, two evaluation orders: left-to-right invents a bracket and lands on 20; BODMAS resolves × first and lands on 14.

How it works

  1. Finish the rewriteOnly evaluate after every printed symbol has been replaced by its real operator. Running line: 6 × 2 ÷ 1 − 3 + 4.
  2. Mark × and ÷Underline multiplication and division segments so they cannot be postponed behind addition. 6 × 2 ÷ 1 becomes 12, then 12.
  3. Then + and −Resolve addition and subtraction left to right on whatever remains. 12 − 3 + 4 = 13. On the trap line, 4 × 2 = 8 then 6 + 8 = 14, never 20.
BODMAS order after substitution
StageWhat you resolve
1. BracketsInnermost grouping first
2. OrdersPowers and roots
3. × and ÷Equal precedence, left to right
4. + and −Equal precedence, left to right

Left-to-right versus BODMAS

After substitution the expression is 6 + 4 × 2. What is its value under BODMAS?

  • Multiplication first: 4 × 28
  • Then addition: 6 + 814

Pro tip. If your option bank contains both 14 and 20, 20 is almost always the left-to-right trap on 6 + 4 × 2.

After substitution you hold 6 + 4 × 2. The correct value is
  1. 14
  2. 20
  3. 10

× before +: 4 × 2 = 8, then 6 + 8 = 14. 20 is (6 + 4) × 2 — left-to-right without BODMAS.

3Equal precedence runs left to right

Division and multiplication share one precedence level and are evaluated left to right. Addition and subtraction share the next level and are also left to right. After a rewrite such as 12 ÷ 6 × 3, you do not "do all divisions before multiplications" — you take 12 ÷ 6 first, then multiply by 3. 12 ÷ 6 = 2, then 2 × 3 = 6. Grouping as 12 ÷ (6 × 3) invents a bracket the line does not have and lands on 2/3; the options often plant 0.5 next to 6 as a cousin of that 'division first' reading.

The same rule applies when + and − sit together: 12 − 3 + 4 is (12 − 3) + 4 = 13, not 12 − (3 + 4) = 5. Equal precedence is not 'the more dramatic operator wins'; it is a left-to-right walk along one level. The running substitution already used this: 6 × 2 ÷ 1 was 12, then 12, not 'do the division first' as 6 × (2 ÷ 1) luckily giving the same 12 — a coincidence that 2 ÷ 1 = 2 hides. Change the 1 to a 4 and the two readings split.

Figure. × and ÷ share one level: walk left to right. 12 ÷ 6 × 3 becomes 2 × 3 = 6, not a division-first rewrite that invents 12 ÷ 18.

How it works

  1. Scan × and ÷While only × and ÷ remain in a stretch, resolve the leftmost operator, replace that pair, and continue. 12 ÷ 6 × 3 becomes 2 × 3, then 6.
  2. Then scan + and −Same left-to-right walk on the remaining addition and subtraction. 12 − 3 + 4 becomes 9 + 4, then 13 — never 12 − 7.

× and ÷ share a level

Evaluate 12 ÷ 6 × 3 under equal precedence (as it appears after a typical interchange rewrite).

  • Leftmost first: 12 ÷ 62
  • Then 2 × 36

Pro tip. Treating ÷ as always-before-× would give 12 ÷ (6 × 3) = 0.5 — a real wrong option when the stem mixes the two.

Under BODMAS, 12 ÷ 6 × 3 equals
  1. 6
  2. 0.5
  3. 2

× and ÷ are equal: 12 ÷ 6 = 2, then 2 × 3 = 6. 0.5 is the false "all divisions first / group the product" reading.

4Interchange, then test each option

Some stems propose a swap of two operators (and sometimes two numbers) and ask which equation becomes correct. The method is mechanical: apply that one swap uniformly to every option, evaluate each rewritten equation under BODMAS, and keep the option that balances. Do not invent a different swap per option. The stem's interchange is fixed; only the candidate equations change.

Work a tiny set under 'interchange + and ×'. Option A prints 6 + 2 × 3 = 12. After the swap it is 6 × 2 + 3. BODMAS: 12 + 3 = 15, which is not 12 — reject A. Option B prints 6 + 2 × 3 = 15. After the same swap it is again 6 × 2 + 3 = 15, and 15 = 15 — keep B. Notice that without the swap, option B is false (6 + 6 = 12, not 15); the interchange is what repairs it. A leftover printed sign means the rewrite is incomplete. Partial swaps ('only the first +') and post-hoc swaps ('evaluate first, then swap the answer') are how two options can look almost right.

Figure. Freeze + ↔ ×, then test every option. 6 + 2 × 3 becomes 6 × 2 + 3 = 15, which rejects the = 12 option and keeps the = 15 option.

Order of attack

  1. Freeze the swapWrite the stem's interchange once — e.g. + ↔ × — and reuse it on every option. Do not pick a fresh pair because an option 'looks closer'.
  2. Rewrite each optionReplace every occurrence of those signs (and numbers, if asked) in that option only. 6 + 2 × 3 becomes 6 × 2 + 3 under + ↔ ×.
  3. Evaluate under BODMASKeep a short result for each option; only one should make LHS equal RHS. 6 × 2 + 3 = 15 balances the = 15 option and not the = 12 option.
Interchange checklist
CheckFail mode if skipped
Same swap on every optionYou solve a different question per choice
Every matching sign swappedA leftover printed operator keeps the old meaning
BODMAS after each rewriteA true equation is rejected as "not 13"
A stem says "interchange ÷ and ×". On an option you must
  1. Swap ÷ with × everywhere in that option, then evaluate with BODMAS
  2. Swap only the first ÷ you see, and leave later × signs alone
  3. Evaluate the printed option first, then swap the answer

The interchange is uniform on the whole expression in each option. Partial swaps and post-hoc swaps are how two options can look "almost" right.

5Find the one swap that repairs the equation

The twin of option-testing is the repair question: one equation is given, and you must find which single interchange of signs makes it true. Guessing from the look of the numbers fails; each candidate swap is applied to that one equation and checked under BODMAS. Take the printed equation 6 + 2 × 3 = 15. As printed it is false: 6 + 6 = 12, not 15. The options name pairs. Try + with ×: the equation becomes 6 × 2 + 3 = 15, and 12 + 3 = 15 — it balances, so that pair is the repair. Try + with − only if you still need a second trial; here you can stop.

Organise the trials — rejected swaps with their values — so you do not re-test the same interchange. Double-check the symbol key or the named pair before you commit; a single misread sign flips every trial. 'Close' is not a pass: 14 against a right-hand side of 15 is a rejected swap, not a rounding story. The same BODMAS that made 6 + 4 × 2 equal 14, not 20, is the judge here.

Figure. The printed line 6 + 2 × 3 = 15 is false (12 ≠ 15). The candidate pair + ↔ × rewrites it to 6 × 2 + 3 = 15, which balances.

Order of attack

  1. List the candidate swapsTake the options as named pairs (÷ with ×, + with −, …) rather than inventing pairs. The given equation stays fixed; only the pair changes.
  2. Apply one pair at a timeRewrite the given equation under that pair alone; do not mix two trials. + ↔ × on 6 + 2 × 3 = 15 gives 6 × 2 + 3 = 15.
  3. Accept only a balanceLHS must equal RHS after BODMAS. 12 + 3 = 15 works; do not stop on 'close'. Keep going until one pair works.
To find which sign interchange makes a given equation correct, you should
  1. Apply each candidate swap to the equation and evaluate under BODMAS until one balances
  2. Change signs until the left side looks larger than the right
  3. Evaluate left-to-right ignoring × and ÷ precedence, then pick any swap

Repair questions are disciplined trials under BODMAS. Cosmetic left-to-right checks invent false "solutions" that the options were written to catch.

Notes

  • Sign Substitution: A key tells you what each symbol means (e.g. @ means +, # means -). Replace every symbol with its real operator, then evaluate using standard BODMAS order.
  • Sign Interchange: You are asked to swap two operators (and sometimes two numbers) and then find which equation becomes correct. Apply the swap to each option and test validity.
  • BODMAS Discipline: After substitution, always resolve Brackets, then Orders (powers), then Division/Multiplication left to right, then Addition/Subtraction; skipping order causes wrong answers.
  • Correct-the-Equation Type: Find the single interchange of signs that makes a given equation true; test each candidate swap rather than guessing.
  • Common trap: Applying operations left-to-right without BODMAS - e.g. after substitution 6 + 4 × 2 must be 6 + 8 = 14, not 20.

Formulas

  • Order of evaluation: BODMAS = Brackets, Orders, Division/Multiplication, Addition/Subtraction.
  • Division and multiplication share equal precedence, evaluated left to right.
  • Addition and subtraction share equal precedence, evaluated left to right.
  • After sign substitution, re-parenthesise mentally to respect precedence before computing.
  • For interchange questions, apply the swap uniformly to all symbols/numbers in the expression.

Exam traps & shortcuts

  • Rewrite the whole expression with real operators first, then apply BODMAS - do not compute during substitution.
  • For 'which equation is correct after swapping', plug the swap into each option and evaluate; only one balances.
  • Underline multiplication/division parts to remind yourself to compute them before addition/subtraction.
  • Double-check by re-reading the symbol key; a single misread symbol flips the answer.

Reference tables

Two question shapes, one evaluation rule.

What this chapter asks
ShapeWhat you do
Symbol substitutionRewrite every symbol from the key, then BODMAS
Sign / number interchangeApply the named swap, then BODMAS; keep the balance
Either shapeNever evaluate left-to-right across × or ÷

Recap

Read only this the night before.

Rewrite first
Substitute every symbol from the key before any arithmetic. Mixing the two alphabets is the silent error.
BODMAS
After the rewrite: brackets, orders, ×÷ left to right, then +− left to right. 6 + 4 × 2 is 14, not 20.
Equal precedence
× and ÷ share a level; + and − share the next. 12 ÷ 6 × 3 is 6, not 0.5.
Interchange trials
One named swap, applied uniformly, then BODMAS on each candidate. Organise rejects so you do not re-test.
Re-read the key
A single misread symbol flips the whole answer. Check the key once more before you mark.

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