BPSC 70th CCE Prelims General Studies · Question 83 of 143

Correct — A, 0. Write the substitution table out before touching the expression, because every error in this kind of question comes from doing the replacement in your head: ÷ becomes +, – becomes ÷, × becomes −, and + becomes ×. The long slash that separates the two brackets is the fraction bar of the expression itself, not one of the four redefined symbols, so it stays as an ordinary division. Now translate the numerator. (36 × 4) – 8 × 4 becomes (36 − 4) ÷ 8 − 4. Work the bracket first: 36 − 4 = 32. Then, by the order of operations, division before subtraction: 32 ÷ 8 = 4, and 4 − 4 = 0. The numerator is zero. Translate the denominator the same way. 4 + 8 × 2 + 16 ÷ 1 becomes 4 × 8 − 2 × 16 + 1. Multiplication before addition and subtraction gives 32 − 32 + 1 = 1. So the expression is 0 ÷ 1 = 0, which is option (a). Two features of this item are worth registering. The denominator coming out to exactly 1 is a strong signal that the substitution has been read the way the setter intended — a mis-substitution would almost certainly produce something messier. And the answer is unusually robust: since the numerator is exactly zero, the value is zero for any non-zero denominator, so even a candidate who mishandles the second bracket still arrives at 0, provided the first bracket is done correctly. The mark therefore rests entirely on the numerator.

• (b)10 — A plausible-looking whole number that no straightforward mis-substitution actually produces; it is filler. If your working returns 10, you have most likely applied the original symbols to part of the expression and the substituted ones to the rest, which is the commonest way this question goes wrong.
• (c)8 — The number 8 appears in the numerator, and an option that echoes a number visible in the question is always attractive to a candidate who has run out of time and is pattern-matching. Neither the substituted numerator nor the substituted denominator evaluates to 8.
• (d)12 — Also filler, and it catches the candidate who ignores the order of operations and works strictly left to right in places where the substituted expression demands division and multiplication first. A tidy answer is not evidence of a correct one.

Symbol-substitution questions test two separate abilities and reward the candidate who keeps them separate. The first is faithful translation: replacing each operator by the one the stem assigns to it, exactly once, without letting the visual habit attached to the printed sign leak back in. The second is the order of operations — BODMAS: brackets first, then orders (powers and roots), then division and multiplication, worked left to right, then addition and subtraction, also left to right. Division and multiplication rank equally, and so do addition and subtraction; the left-to-right rule breaks the tie. The trap in a substituted expression is that the symbol you can see and the operation you must perform are different, so the eye keeps applying the wrong precedence — a printed '+' that now means multiply must be done BEFORE a printed '×' that now means subtract, which is the reverse of what the page looks like it is saying. Writing the translated expression out in full on paper, before evaluating any part of it, removes that whole class of error at a stroke.

The method is mechanical, and the discipline is the entire difficulty. Step one: write the four substitutions in a column at the side of the rough sheet. Step two: rewrite the whole expression with the new operators in place, changing nothing else — not the numbers, not the brackets, not the fraction bar. Step three: only now evaluate, applying BODMAS to the rewritten expression. Candidates who skip step two and translate as they compute lose marks not because the arithmetic is hard but because they carry a stale operator across from one term to the next. There is also a strategic point here. A numerator that collapses to zero is a common design in these items, because it lets the setter build an elaborate-looking denominator that turns out not to matter; if you notice mid-way that the first bracket has come out to zero, you can mark the answer and move on without finishing the second. On a paper of 150 questions in 120 minutes, spotting that saves close to a minute, and minutes saved on mechanical items are what buy time for the ones that need thought.

• Substitutions given: ÷ means +, – means ÷, × means –, + means ×; the fraction bar separating the two brackets is not redefined and stays as ordinary division
• Numerator: (36 × 4) – 8 × 4 becomes (36 − 4) ÷ 8 − 4 = 32 ÷ 8 − 4 = 4 − 4 = 0
• Denominator: 4 + 8 × 2 + 16 ÷ 1 becomes 4 × 8 − 2 × 16 + 1 = 32 − 32 + 1 = 1
• Value = 0 ÷ 1 = 0; because the numerator is exactly zero, the result stays zero for any non-zero denominator
• BODMAS ranks brackets, then orders, then division and multiplication (left to right), then addition and subtraction (left to right) — the printed symbol's usual precedence must be ignored once it has been redefined
• Method that removes the whole class of error: write the four substitutions in a column, rewrite the entire expression with the new operators before computing anything, and only then apply BODMAS to the rewritten line

• Translating as you compute instead of rewriting the whole expression first, which lets a stale operator carry across terms
• Applying the precedence of the symbol you can see rather than of the operation it now stands for
• Redefining the fraction bar as well; only the four symbols named in the stem change meaning

BPSC embeds a short block of arithmetic reasoning inside the General Studies paper, and the symbol-substitution item is one of its regulars — always short, always mechanical, and always solvable in under a minute by a candidate with a method. UPSC set the same family of substitution puzzles in General Studies until 2011, in the form of coding-decoding and relabelled-vocabulary questions, and moved them to the CSAT paper thereafter; the underlying skill of answering strictly inside a redefined system is identical.

• practice — not a real PYQ

  If + means ÷, ÷ means ×, × means – and – means +, then the value of 18 + 6 × 4 – 3 ÷ 2 is

  • (a)3
  • (b)5
  • (c)9
  • (d)12

  Answer(b) 5 — substituting gives 18 ÷ 6 − 4 + 3 × 2; by BODMAS that is 3 − 4 + 6 = 5.

• practice — not a real PYQ

  In an expression, which of the following pairs of operations rank equally and are therefore worked from left to right ?

  • (a)Brackets and orders
  • (b)Division and multiplication
  • (c)Orders and addition
  • (d)Brackets and subtraction

  Answer(b) Division and multiplication — they share a rank, as do addition and subtraction, and each pair is resolved left to right.