In an examination paper where maximum marks are 500, A got 10% marks less than B, B got 25% marks more than C, and C got 20% marks less than D. If A got 360 marks, what marks did D get ?
- (a)65%
- (b)70%
- (c)75%
- (d)80%
Answer
Why
Correct — D, (d) 80%.
BEFORE THE ARITHMETIC, A NOTE ON WHAT THE PAPER ASKS. The stem says 'what marks did D get ?' — which invites a count out of the stated maximum of 500 — but all four options are PERCENTAGES. That mismatch between the question sentence and the option set is the paper's own, and it is printed the same way in both language columns. The card reproduces it as set rather than repairing it. The reconciliation is straightforward: D scored 400 marks, which out of a maximum of 500 is 80 per cent, and 80% is the keyed option. A candidate meeting this in the hall should compute the mark, convert it against the maximum the stem supplies, and select the matching percentage.
THE CHAIN, TAKEN ONE LINK AT A TIME. Every relation in the stem is 'X compared with Y', so each one is read as a multiplier on Y.
'A got 10% marks less than B' means A = 0.90 x B. 'B got 25% marks more than C' means B = 1.25 x C. 'C got 20% marks less than D' means C = 0.80 x D.
Now work backwards from the one number given.
A = 360, and A = 0.90 B, so B = 360 / 0.90 = 400. B = 400, and B = 1.25 C, so C = 400 / 1.25 = 320. C = 320, and C = 0.80 D, so D = 320 / 0.80 = 400.
D scored 400 marks. Out of 500 that is 400/500 = 0.80, or 80 PER CENT. Option (d).
THE SHORTCUT WORTH LEARNING, because it removes two of the three steps. Compose the multipliers instead of applying them one at a time:
A = 0.90 x B = 0.90 x (1.25 x C) = 0.90 x 1.25 x (0.80 x D)
and 1.25 x 0.80 = 1 exactly, so
A = 0.90 x D
A is simply ninety per cent of D. Hence D = 360 / 0.90 = 400, in one step. The reason the middle two factors cancel is worth stating on its own: an increase of 25 per cent multiplies by 5/4 and a decrease of 20 per cent multiplies by 4/5, and those are reciprocals. So B and D scored exactly the same, 400 each, and a candidate who spots that has finished the problem before starting it. The general rule behind it is that an increase of x per cent is exactly reversed by a decrease of 100x/(100+x) per cent — 25 is undone by 20, 100 by 50, 50 by 33⅓.
WORKING ENTIRELY IN PERCENTAGES, which is quicker still and stays in the units the options use. A's 360 marks out of 500 is 72 per cent. Since A is ninety per cent of D, D's percentage is 72 / 0.90 = 80 per cent. No mark count is needed at any stage.
THE FULL PICTURE, as a check that every condition of the stem holds:
A = 360 = 72% of 500 B = 400 = 80% of 500 — and 360 is indeed 10% less than 400 C = 320 = 64% of 500 — and 400 is indeed 25% more than 320 D = 400 = 80% of 500 — and 320 is indeed 20% less than 400
Every relation checks out, and 400 lies within the maximum of 500, so the answer is consistent.
BACK-SUBSTITUTION SETTLES THE OPTIONS DIRECTLY, using A = 0.90 x D:
D = 65% of 500 = 325 gives A = 292.5, not 360 D = 70% of 500 = 350 gives A = 315, not 360 D = 75% of 500 = 375 gives A = 337.5, not 360 D = 80% of 500 = 400 gives A = 360, which matches
The percentages are printed with no space before the sign, and the stem carries a space before its question mark, as this booklet sets them.
Why the others are wrong
- (a)65% — Sixty-five per cent of 500 is 325 marks. Testing it against the chain settles the matter at once: because the twenty-five per cent rise and the twenty per cent fall cancel exactly, A must be ninety per cent of D, so a D of 325 would give A = 0.90 x 325 = 292.5 marks, not the 360 the stem states. The option is also the lowest on the list, and a moment's structural reasoning shows the answer cannot be low. A scored 360 out of 500, which is already 72 per cent, and A is the SMALLEST of the four scores in the chain as it works out — so D, standing at the top of the chain and equal to B, must exceed A's percentage. That single observation removes the two options at or near 72 per cent from below. Testing a candidate answer by substituting it back into the stem's own relations is the most reliable check available on a chained-percentage item, because it uses the conditions directly rather than repeating the calculation that might have gone wrong.
- (b)70% — Seventy per cent of 500 is 350 marks, which would make A equal to 0.90 x 350 = 315, again short of the stated 360. It is also arithmetically impossible in a more direct way: 350 is LESS than A's own 360, and the chain requires D to be larger than A, since A is ninety per cent of D. Any option corresponding to fewer than 360 marks can therefore be discarded on sight, and both 65% and 70% fall below that line — 325 and 350 marks respectively. The option is on the list because it is a round number in the right general region, and because a candidate who applies one of the three percentage changes in the wrong direction can easily land in the low seventies. The defence is to fix the direction of each relation before any arithmetic: 'A got less than B' means B is the larger, so B is found by DIVIDING A by 0.90, never by multiplying.
- (c)75% — Seventy-five per cent of 500 is 375 marks, which does clear the floor set by A's 360 — so this is the one wrong option that survives the quick structural check and it is correspondingly the most dangerous. It fails on the arithmetic: A must be ninety per cent of D, and 0.90 x 375 = 337.5, not 360. Where does the number come from? Most often from applying a percentage to the wrong base. 'B got 25% more than C' means the 25 per cent is measured on C, so B = 1.25C and C = B/1.25 = 0.8B; a candidate who instead takes 25 per cent of B and subtracts it gets C = 0.75B, which brings 75 into the working and can carry it through to the answer. That confusion of base is the single commonest error in percentage problems: the phrase 'more than' or 'less than' always names the base, and the base is the quantity that follows it. Read the sentence, mark the base, then write the equation.
Concept
CHAINED PERCENTAGE COMPARISONS are the subject here, and the technique that handles them all is to convert every verbal comparison into a MULTIPLIER and then compose the multipliers.
TRANSLATING THE THREE STANDARD PHRASES. 'X is p% more than Y' becomes X = Y x (1 + p/100). 'X is p% less than Y' becomes X = Y x (1 - p/100). 'X is p% of Y' becomes X = Y x (p/100). In every case the quantity named AFTER 'than' or 'of' is the BASE, and the base is what the percentage is measured on. Identifying the base correctly is most of the skill.
WHY DIRECTION MATTERS. If A is ten per cent less than B, then A = 0.9B, and to recover B from A one DIVIDES: B = A/0.9, which is about 1.111 x A. It is not true that B is ten per cent more than A — it is 11.1 per cent more. The asymmetry catches candidates constantly, and it is worth stating as a rule: percentage relations are not symmetric, because the two directions use different bases.
THE REVERSAL RULE that follows from this. An increase of x per cent is exactly undone by a decrease of 100x/(100+x) per cent, and a decrease of x per cent is exactly undone by an increase of 100x/(100-x) per cent. The pairs worth memorising are +25% is undone by -20% (multiply by 5/4, then by 4/5) +50% is undone by -33⅓% +100% is undone by -50% +10% is undone by -9 1/11% The first of these appears in this very problem and makes two of its three steps cancel.
COMPOSING MULTIPLIERS. Successive percentage changes multiply; they never add. Two successive changes of a% and b% give a net multiplier of (1 + a/100)(1 + b/100), which expands to a net percentage change of a + b + ab/100. That is why a twenty per cent rise followed by a twenty per cent fall does NOT return to the start but leaves 96 per cent of the original: 20 - 20 - 400/100 = -4 per cent.
WORKING FORWARDS AND BACKWARDS. When the known quantity sits at one end of a chain and the unknown at the other, decide first which way the chain runs. Here the chain is defined from A downwards to D, but the value given is A, so every step must be inverted — each multiplier becomes a division. Writing the three relations out as equations before touching the arithmetic prevents the commonest failure, which is inverting some steps and not others.
THE PERCENTAGE-ONLY METHOD. In a problem where a maximum is given, it is often quicker to convert the one known value into a percentage at the start and to work throughout in percentages, since the multipliers act identically on marks and on percentages of the same maximum. Here A = 360/500 = 72 per cent, and D = 72/0.9 = 80 per cent, with no mark counts computed at all.
CHECKING AN ANSWER ON A CHAINED PROBLEM. Two checks are always available and both are quick. Substitute the chosen answer back into the original relations and verify each one in turn. And apply a bounding argument before calculating — asking which of the four quantities must be largest and which smallest — which frequently eliminates half the options without any arithmetic.
Quantitative aptitude is the largest strand on this APFC paper, about a third of Part B, and percentages are among the most frequently tested topics within it. This item is a chained comparison — four quantities linked by three relations, one value supplied, one asked for — which is the standard way the topic is set.
The item also carries an inconsistency of its own, and it is worth naming plainly because a student will notice it. The question sentence asks for 'marks' out of a maximum of 500, while every option is expressed as a percentage. The same mismatch appears in the Hindi column, so it is not an artefact of one column's typesetting. The practical effect on a candidate in the hall is small — the answer is computed as 400 marks and converted against the stated maximum to 80 per cent, and only one option matches — but the effect on a candidate's confidence can be larger, and the right response is to convert rather than to hunt for a misreading of the stem.
The mathematics rewards one specific habit: composing the multipliers before doing any arithmetic. Written out, the chain gives A = 0.90 x 1.25 x 0.80 x D, and the middle two factors are reciprocals, so A = 0.90 x D and the problem collapses into a single division. A candidate who works link by link will still arrive at 400, but through three divisions instead of one, with three chances to invert a step wrongly. Seeing that a 25 per cent rise and a 20 per cent fall cancel is the kind of small numerical fact that repays being learned once.
The option set is built so that structure alone does much of the work. Two of the four percentages correspond to fewer than 360 marks, and since A is ninety per cent of D, D must exceed A. Reading the options as mark counts before comparing them therefore removes half the list.
Key facts
- 'X is p% less than Y' means X = Y x (1 - p/100), and Y is recovered from X by dividing, not by adding p% back.
- 'X is p% more than Y' means X = Y x (1 + p/100), with Y as the base.
- Successive percentage changes multiply rather than add; two changes of a% and b% give a net change of a + b + ab/100.
- An increase of 25% multiplies by 5/4 and a decrease of 20% multiplies by 4/5, so the two cancel exactly.
- An increase of x% is exactly reversed by a decrease of 100x/(100+x)%.
- In this question the composition gives A = 0.90 x 1.25 x 0.80 x D = 0.90 x D, so B and D scored the same.
- A scored 360, so B = 400, C = 320 and D = 400.
- Out of a maximum of 500, D's 400 marks are 80 per cent.
- Where a maximum is given, the chain of multipliers can be applied to percentages directly: A is 72% of 500, and 72 / 0.90 = 80.
- Back-substitution is the reliable check on a chained problem: each candidate answer is put into the stem's own relations and tested.
Study next
Common traps
- Applying a percentage to the wrong base. 'A is 10% less than B' measures the ten per cent on B, not on A.
- Assuming the relation is symmetric. If A is 10% less than B, B is 11.1% more than A, not 10% more.
- Adding successive percentage changes instead of multiplying them.
- Inverting some links of a chain and not others when working backwards from a known value.
- Missing the mismatch between a stem asking for marks and options expressed as percentages, and answering in the wrong unit.
- Overlooking the cancellation of a 25 per cent rise against a 20 per cent fall, which turns a three-step problem into a one-step one.
Percentage items on EPFO papers arrive most often as a chain of two or three comparisons among named quantities, with one value supplied and another asked for, dressed as marks, salaries, populations, prices or production figures. The dressing changes; the method does not. Translate each sentence into a multiplier, noting carefully which quantity is the base, compose the multipliers before computing anything, and look for reciprocal pairs that cancel. Where a maximum or a total is stated, decide at the outset whether to work in absolute values or in percentages and stay in that unit until the end — and then check what unit the OPTIONS are in, because the two need not match, as this item shows. Finish with a bounding check: work out which quantity in the chain must be largest, and discard every option that contradicts it.
Related PYQs
EPFO_APFC_2016_Q105In an office, 40% of the employees are men and the rest women. Half of the employees are tall and half short. If 10% of the employees are men and short, and 40 employees are women and tall, the number of tall men employees is
- (a) 60
- (b) 50
- (c) 40
- (d) 30
Answer(a) 60
The office in which 40 per cent of employees are men and half are tall — the paper's other multi-condition percentage item, where the discipline is again to translate each sentence before computing anything.
EPFO_APFC_2016_Q113If the radius of a circle is reduced by 50%, its area will be reduced by
- (a) 30%
- (b) 50%
- (c) 60%
- (d) 75%
Answer(d) 75%
By what percentage the area of a circle falls when its radius is cut by half — percentage change applied to a two-dimensional quantity, and the same lesson that a percentage must be attached to the right base.
EPFO_EOAO_2020_Q11840% of the students in a class are from India and 50% are girls. If 25% of the Indian students are girls, what percentage of non-Indian students are boys ?
- (a) 33·33%
- (b) 40%
- (c) 25%
- (d) 20%
Answer(a) 33·33%
What percentage of non-Indian students are boys, given percentages of Indians and of girls in a class — a later EPFO percentage item built as a chain of conditions on overlapping groups.
Practice
- practice — not a real PYQ
P's salary is 20% less than Q's, and Q's salary is 25% more than R's. If R earns ₹ 40,000, what is P's salary ?
- (a)₹ 36,000
- (b)₹ 40,000
- (c)₹ 44,000
- (d)₹ 50,000
Answer(b) ₹ 40,000 — Q = 1.25 x 40,000 = ₹ 50,000, and P = 0.80 x 50,000 = ₹ 40,000. The multipliers 1.25 and 0.80 are reciprocals, so P and R earn exactly the same; ₹ 50,000 is Q's salary and is the trap for a candidate who stops one step early.
- practice — not a real PYQ
The price of a commodity is raised by 20% and the new price is later reduced by 20%. Compared with the original price, the final price is
- (a)the same
- (b)4% lower
- (c)4% higher
- (d)40% lower
Answer(b) 4% lower — successive percentage changes multiply, so the net multiplier is 1.20 x 0.80 = 0.96, leaving 96 per cent of the original price. Equal percentage rises and falls do not cancel, because the fall is measured on the larger, already increased base.