BPSC 69th CCE Prelims General Studies · Question 143 of 144

Correct — B, Suresh’s land is more than Ramesh. Ramesh ends with 99 square yards and Suresh with 101. Step 1 — the transfer. Both men start with 100 square yards. Ramesh increases his holding by 10 per cent, and 10 per cent of 100 is 10 square yards, which comes out of Suresh's plot. Ramesh 110, Suresh 90. This step is unambiguous only because the two men start equal: 10 per cent of Ramesh's holding and 10 per cent of Suresh's holding are the same 10 square yards. Step 2 — the sale back. Two years later Ramesh sells back 10 per cent of his total plot. His total is now 110, not 100, so 10 per cent of it is 11 square yards, not 10. Ramesh 110 − 11 = 99, Suresh 90 + 11 = 101. Suresh is ahead by 2 square yards, which is option (b). The one-line version. Ramesh's land is multiplied by 1.10 and then by 0.90, and 1.10 × 0.90 = 0.99. A rise of ten per cent followed by a fall of ten per cent is not a wash — it is a net loss of exactly one per cent, because the fall is charged on the larger base the rise created. Ramesh keeps 99 per cent of his original 100, which is 99. And since no land leaves the pair, the two holdings must always add to 200, so Suresh is 200 − 99 = 101. You never have to track Suresh at all. Now the honest part, because both independent derivations marked this medium rather than high. The phrase 'the total plot' is loose: it could mean Ramesh's own total after the purchase, which is 110, or the two plots taken together, which is 200. Test the second reading. Ten per cent of 200 is 20; Ramesh 110 − 20 = 90 and Suresh 90 + 20 = 110. Suresh is still ahead, and the answer is still (b). The reading changes the margin — 2 square yards on one, 20 on the other — but it does not change which man ends up with more, and the question only asks which is more. That is what makes an item with a soft phrase still safely answerable. One last thing to notice: 'After 2 years' does no arithmetical work whatever. There is no rate of growth, no interest and no appreciation in the question; the two years are there to make the story sound like a transaction and to tempt you into looking for a compounding step that does not exist.

• (a)Ramesh’s land is more than Suresh — True after the first transaction and false after the second — the option for the candidate who computes Ramesh 110 against Suresh 90 and stops reading. It is also where you land if you take the sale back to be ten per cent of the strip Ramesh bought rather than of his plot: ten per cent of the 10 square yards acquired is 1, leaving Ramesh 109 and Suresh 91. That reading cannot stand, because the stem says ten per cent of the total plot, and a ten-square-yard strip is not anybody's total.
• (c)Both are equal — The designed trap on any question of this shape, and the one worth rehearsing until the reflex is gone. It assumes that adding ten per cent and then taking ten per cent away restores the starting position. It does not, because the two percentages are charged on different bases — 10 on a base of 100 going out to Ramesh, 11 on a base of 110 coming back. The holdings would only be equal again if he returned the same quantity he took, that is 10 square yards, which is ten per cent of his original plot and not of his total plot.
• (d)None of the above — Reserved for a candidate who reads 'the total plot' as ambiguous and concludes the question cannot be answered. The instinct is sound and the conclusion is wrong, because both defensible readings of that phrase — ten per cent of Ramesh's 110, or ten per cent of the combined 200 — leave Suresh with more land than Ramesh. When a stem is loose, test whether the looseness actually changes the answer before you reach for the catch-all; here it changes only the margin.

A percentage is never a quantity on its own — it is a quantity only once you name the base it is taken of. That is the whole content of this question, and of most of the percentage arithmetic BPSC and UPSC set. The cleanest way to handle successive percentage changes is to convert each into a multiplier: a rise of x per cent multiplies by (1 + x/100), a fall of x per cent multiplies by (1 − x/100), and successive changes multiply together in any order. A rise of x per cent followed by a fall of x per cent therefore gives (1 + x/100)(1 − x/100) = 1 − x²/10000, that is a net fall of x²/100 per cent, never a return to the start. Ten and ten gives a one per cent fall; twenty and twenty gives four per cent; twenty-five and twenty-five gives 6.25 per cent; fifty and fifty gives a full quarter. The same asymmetry read backwards answers the standard companion question: to undo a rise of x per cent you must cut by x/(100 + x), so undoing a 25 per cent rise takes a 20 per cent cut, and undoing a 10 per cent rise takes a cut of 1/11, about 9.09 per cent. A second idea does half the work here — conservation. Land is only moving between two owners, so the pair always totals 200 square yards; compute one man and the other follows by subtraction.

Under time pressure, resist the urge to build a table. Note that the answer only asks who has more, not by how much, so you need one number, not four. Multiply Ramesh out — 100 × 1.1 × 0.9 = 99 — see that it is below 100, and you are done: whatever Ramesh has lost, Suresh has gained, so Suresh is ahead. The single discriminating fact is that 1.1 × 0.9 = 0.99 and not 1. If you can hold that one product in mind, you will never mark 'both are equal' on any version of this question, and you will also spot at once that the order of the two operations does not matter: 0.9 × 1.1 is the same 0.99, so a man who sold first and bought back later would end at 99 too. Two features of the stem are there to slow you down rather than to be used. 'After 2 years' introduces no rate and no compounding. And the two men starting on identical 100-square-yard plots removes what would otherwise be a real ambiguity in the first sentence, since ten per cent more of Ramesh's land and ten per cent of Suresh's land happen to be the same ten square yards. The looseness that survives is in the second sentence, and as shown above it does not reach the answer.

• Stage by stage: both start at 100; Ramesh takes 10 per cent of 100 = 10, giving 110 and 90; he returns 10 per cent of 110 = 11, giving 99 and 101
• Multiplier form: 1.10 × 0.90 = 0.99, so Ramesh keeps 99 per cent of his original holding whatever that holding was, and the order of the two operations is irrelevant
• General rule: a rise of x per cent followed by a fall of x per cent gives a net fall of x²/100 per cent — 1 per cent for x = 10, 4 per cent for x = 20, 6.25 per cent for x = 25, 25 per cent for x = 50
• To reverse a rise of x per cent you must cut by x/(100 + x): a 25 per cent rise needs a 20 per cent cut, a 10 per cent rise needs a cut of about 9.09 per cent
• The two holdings always sum to 200 square yards because the land only changes hands, so computing one man determines the other
• The answer is robust to the ambiguity in 'the total plot': reading it as the combined 200 square yards gives Ramesh 90 and Suresh 110, and Suresh is still the larger holder
• 'After 2 years' carries no arithmetic — the question states no growth rate, interest or appreciation of any kind

• Assuming a rise of x per cent and a fall of x per cent cancel — they always leave a net fall of x²/100 per cent
• Charging the second percentage on the original figure instead of on the new one
• Reading 'After 2 years' as a signal to compound, when the question supplies no rate at all

BPSC dresses percentage arithmetic as a short story about two named people and then asks only for a comparison — who has more, who gains, which is correct — so the numbers can usually be finished in one line and the marks turn on noticing which base each percentage is charged on. UPSC set the same material in General Studies Paper I until 2010, but preferred to ask for the number itself: by what percentage should a price be cut to restore it, or by what percentage is one discounted figure below another. Since the CSAT arrived in 2011 this content sits in the qualifying Paper II, which is why the BPSC General Studies paper is now the place a UPSC aspirant meets it under marking pressure.

• practice — not a real PYQ

  The price of a commodity is first increased by 20% and then decreased by 20%. The net change in the price is

  • (a)No change
  • (b)4% increase
  • (c)4% decrease
  • (d)20% decrease

  Answer(c) 4% decrease — the multipliers give 1.20 × 0.80 = 0.96, and the general rule x²/100 gives 400/100 = 4 per cent down.

• practice — not a real PYQ

  A's salary is 25% more than B's salary. By what per cent is B's salary less than A's salary?

  • (a)20%
  • (b)25%
  • (c)16.67%
  • (d)33.33%

  Answer(a) 20% — take B = 100, so A = 125; the gap of 25 is measured against A's 125, giving 25/125 = 20 per cent.