There are two circles of radii r1 and r2 (r1 < r2). The area of the bigger circle is 693/2 cm2. The difference of their circumferences is 22 cm. What is the sum of the diameters of the two circles ?
- (a)17·5 cm
- (b)22 cm
- (c)28·5 cm
- (d)35 cm
Answer
Why
Correct — D, (d) 35 cm.
Two facts are given about two circles, and each yields one equation. Solve them in the order the paper supplies them and the answer needs no algebra beyond a subtraction.
STEP 1 — the bigger radius, from the area. The stem prints the area of the bigger circle as a stacked fraction, 693 over 2, followed by cm with a raised 2; written inline that is 693/2 cm2, or 346·5 square centimetres. Using pi = 22/7, which is what the numbers in this item are chosen for:
(22/7) x r2^2 = 693/2 r2^2 = (693 x 7) / (2 x 22) = 4851/44 = 110·25 r2 = 10·5 cm
STEP 2 — the difference of the radii, from the difference of the circumferences. This is the step that saves the work, because the difference of two circumferences depends only on the difference of the radii:
2 x pi x r2 - 2 x pi x r1 = 2 x pi x (r2 - r1) = 22 (44/7) x (r2 - r1) = 22 r2 - r1 = 22 x 7 / 44 = 3·5 cm
There is no need to find each circumference separately, and no need for the area of the smaller circle at all.
STEP 3 — the smaller radius, and the answer.
r1 = 10·5 - 3·5 = 7 cm sum of the diameters = 2 x r1 + 2 x r2 = 2 x (7 + 10·5) = 2 x 17·5 = 35 cm
CHECK. A circle of radius 7 has circumference 44 and a circle of radius 10·5 has circumference 66; the difference is 22, as given. The area of the larger is (22/7) x 110·25 = 346·5, which is 693/2, also as given. Both conditions hold, so 35 cm is right, and it is option (d).
The one insight in the item is the linearity in step 2: circumference is proportional to radius, so differences of circumferences translate straight into differences of radii, while AREA is proportional to the square of the radius and does not behave that way. Mixing the two is what the wrong options are built to catch.
Why the others are wrong
- (a)17·5 cm — This is the sum of the RADII, 7 + 10·5, and it is the answer to a question one step short of the one asked. The stem asks for the sum of the DIAMETERS, which is twice as large. It is the strongest trap in the set because every piece of reasoning behind it is correct: the candidate has found both radii, added them properly, and then failed to read the last five words of the stem. The defence is mechanical — after solving, go back and reread what quantity was asked for. The raised middle dot in '17·5' is the booklet's own decimal point, printed that way throughout this item.
- (b)22 cm — This is simply the difference of the circumferences, given in the stem, offered back as though it were the answer. It is the option for a candidate who has run out of time and picks a number that is definitely in the question. It also catches a subtler error: reading 'the difference of their circumferences is 22 cm' as though 22 were a difference of diameters or of radii. It is neither. Dividing 22 by 2 x pi gives the difference of the radii, 3·5 cm, and that is the only thing this datum yields directly.
- (c)28·5 cm — This value does not arise from any step of the correct chain. The quantities the data actually produce are 10·5, 7, 3·5, 17·5 and 35, and 28·5 is none of them and no simple combination of the given numbers. Its function in the option set is positional: it is the only option other than (a) that carries a raised middle-dot decimal, so a candidate who has arrived at a value ending in ·5 and is unsure whether to double it finds a plausible-looking decimal both below and just short of the true answer. Treat an option that cannot be reached by any route as a sign that the chain has been reconstructed wrongly rather than as a near miss to be rounded to.
Concept
The item rests on how the two measures of a circle scale with its radius, which is the single most productive fact in circle problems.
Circumference C = 2 x pi x r — LINEAR in r. Area A = pi x r^2 — QUADRATIC in r.
Because C is linear, differences and sums of circumferences pass straight through to differences and sums of radii: C2 - C1 = 2 x pi x (r2 - r1). Nothing analogous holds for areas. A2 - A1 = pi x (r2 - r1) x (r2 + r1), so a difference of areas gives you a PRODUCT of a difference and a sum, which is why area data are almost always given for a single circle, as here, and circumference data for a pair.
That asymmetry is also why percentage questions about circles behave counter-intuitively: halving the radius halves the circumference but quarters the area, so the area falls by 75%, not by 50%.
A second habit this item rewards is choosing the value of pi from the numbers on the page. When a circle problem prints 693, 22, or any multiple of 7, it has been constructed for pi = 22/7 and the arithmetic will come out exactly. When it prints 3·14 or leaves pi symbolic, it has not. Here 693 = 7 x 99 and the circumference difference is 22 itself, so 22/7 is signalled twice.
Finally, the checking discipline. With two circles there are two given conditions, and a candidate who has found r1 = 7 and r2 = 10·5 can verify both in a few seconds: circumferences 44 and 66, difference 22; larger area (22/7) x 110·25 = 346·5 = 693/2. A solution that satisfies both givens is right, whatever the option list looks like.
Mensuration sits inside the large quantitative strand of this paper, and circle problems are its most common single form. The construction here is typical: two circles, one datum about each of the two measures, and a question about a third quantity that is neither of them. The design guarantees that a candidate must convert between measures rather than read an answer off.
Three pieces of typography in this item do not survive into plain text and are worth naming, because a card that silently tidied them would be describing a different question. The radii are printed as r with subscript 1 and subscript 2, written inline here as r1 and r2, and the inequality is printed as (r1 < r2). The area is a genuine two-level stacked fraction — 693 above a rule, 2 below it — followed by cm with a superscript 2, which is written inline with a solidus as 693/2 cm2. And the decimal points in options (a) and (c) are RAISED MIDDLE DOTS, not full stops: the booklet prints 17·5 and 28·5, and those characters are reproduced exactly. The raised middle dot is an older British typographic convention for the decimal point and appears at several places in this booklet; it means exactly what a full stop would mean.
The stem also states r1 < r2 explicitly rather than leaving it to be inferred, which matters: without it, 'the difference of their circumferences' would be ambiguous in sign and the smaller radius could be taken as 14 rather than 7.
Key facts
- Circumference is linear in the radius, so a difference of circumferences converts directly: C2 - C1 = 2 x pi x (r2 - r1).
- Area is quadratic in the radius, so a difference of areas gives pi x (r2 - r1) x (r2 + r1) — a product, not a plain difference.
- From the area 693/2 = 346·5 square centimetres with pi = 22/7, the larger radius is 10·5 cm.
- From a circumference difference of 22 cm, the radii differ by 22 x 7 / 44 = 3·5 cm, so the smaller radius is 7 cm.
- Sum of the diameters = 2 x (7 + 10·5) = 35 cm; the sum of the RADII, 17·5 cm, is the printed near-answer.
- A circle problem printing 693, 22 or any multiple of 7 is built for pi = 22/7 and will come out exactly.
- Verification for two circles: radii 7 and 10·5 give circumferences 44 and 66, difference 22, and larger area 346·5 — both givens satisfied.
Study next
Common traps
- Answering with the sum of the radii, 17·5 cm, when the stem asks for the sum of the diameters. Reread the last clause of the stem after solving.
- Treating the given 22 cm as a difference of radii or of diameters. It is a difference of circumferences, and dividing by 2 x pi gives 3·5 cm.
- Trying to use the area of the smaller circle. It is never needed; the circumference difference alone links the two radii.
- Applying difference reasoning to areas as though area were linear in the radius. It is not, which is why only one area is given.
Mensuration items on EPFO papers stay with the standard figures — circle, rectangle, triangle, cylinder — and construct difficulty by asking for a quantity one conversion away from the data. Expect a datum about area paired with a datum about perimeter, a percentage change in one dimension with a question about a derived measure, or a quantity given for one figure and asked for another. The numbers are chosen so that pi = 22/7 divides cleanly, which is a reliable signal. The two habits that pay are converting every given into a radius or a side before doing anything else, and rereading the final clause of the stem before choosing, because the commonest wrong option in this family is the correct answer to a question one step earlier.
Related PYQs
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%
The other circle item on this paper — the percentage reduction in area when the radius is halved — which turns on exactly the linear-versus-quadratic distinction used here.
EPFO_APFC_2016_Q19What is the perimeter of the figure shown below ? AJ = 10 cm, JI = 12 cm, AB = x, CD = x + 1, EF = x + 2, GH = x + 3, BC = DE = FG = HI = y
- (a) 44 cm
- (b) 48 cm
- (c) 54 cm
- (d) 58 cm
Answer(a) 44 cm
The paper's other perimeter question, a staircase-shaped figure whose horizontal runs and vertical rises each sum to a single given length; both items reward converting the givens before computing.
Practice
- practice — not a real PYQ
The circumferences of two circles differ by 44 cm. By how much do their radii differ ? (Take pi = 22/7)
- (a)3·5 cm
- (b)7 cm
- (c)14 cm
- (d)22 cm
Answer(b) 7 cm — the difference of circumferences is 2 x pi x (r2 - r1), so r2 - r1 = 44 x 7 / (2 x 22) = 7 cm. No information about either circle individually is needed, because circumference is linear in the radius.
- practice — not a real PYQ
If the radius of a circle is increased by 20%, its area increases by
- (a)20%
- (b)40%
- (c)44%
- (d)120%
Answer(c) 44% — area is proportional to the square of the radius, so the new area is (1·2)^2 = 1·44 times the old, an increase of 44%. Answering 20% treats area as linear in the radius, which is the same error that makes circle percentage questions counter-intuitive.