Averages MCQs for UPSC Prelims

For equal distances at two speeds, the average speed is the harmonic mean: 2 x s1 x s2 / (s1 + s2) = 2 x 40 x 60 / (40 + 60) = 4800 / 100 = 48 km/hr. (Check: time A to B = 360/40 = 9 hr, time B to A = 360/60 = 6 hr; total distance 720 km in 15 hr = 48 km/hr.)

The reported total = 100 * 40 = 4000. One student's mark was read as 83 instead of the actual 53, so the total was overstated by 83 - 53 = 30. Corrected total = 4000 - 30 = 3970. Corrected mean = 3970 / 100 = 39.7. Answer (b) 39.7.

Value-I: the 11 numbers are consecutive integers, each at least -5. The average of 11 consecutive integers is the middle (6th) term, so to minimize it we start as low as allowed: -5, -4, ..., 5, whose average is 0. Any higher starting point raises the average, so the minimum average is 0. Value-II: the 11 numbers are consecutive non-negative integers. Starting at 0 gives 0, 1, ..., 10, whose product is 0 because of the factor 0. Starting at 1 or more gives a large positive product. So the minimum product is 0. Both values equal 0, so Value-I = Value-II. The official UPSC key confirms option (c).

Total runs after 50 innings = 50 x 46.4 = 2320. New average after 60 innings = 46.4 + 2.6 = 49, so total runs after 60 innings = 60 x 49 = 2940. Runs scored in the last 10 innings = 2940 - 2320 = 620. Average over those 10 innings = 620 / 10 = 62. Option 49 is the overall average not the last-ten average; 91 and 122 are arithmetic errors.

From the averages: A + B + C = 3 x 40 = 120 and B + D + E = 3 x 42 = 126. The required total is A + B + C + D + E + F = (A + B + C) + (B + D + E) - B + F. Since F = B, the -B and +F cancel, leaving 120 + 126 = 246. Average = 246/6 = 41 kg, option (c). The equality F = B is exactly what makes the problem solvable, so option (d), data inadequate, is wrong; options (a) and (b) result from averaging 40 and 42 incorrectly without handling the shared B.

Current total monthly income of the family = 5 x 10,000 = Rs. 50,000. One person's income rises by Rs. 1,20,000 per year, which is 1,20,000 / 12 = Rs. 10,000 per month. New total monthly income = 50,000 + 10,000 = Rs. 60,000. New average per person = 60,000 / 5 = Rs. 12,000. The trap answers 16,000 and 20,000 come from adding the yearly figure without converting to monthly. Hence Rs. 12,000.

For consecutive odd numbers the mean of any set equals its middle term. The first five numbers have mean 39, so the 3rd number is 39 and the five numbers are 35, 37, 39, 41, 43, making the first number 35. The thirteen consecutive odd numbers are 35, 37, ... up to 35 + 2 x 12 = 59. The mean of all thirteen is the 7th (middle) number = 35 + 2 x 6 = 47. Hence (a).

Every student in Class-A scores between 17 and 21, while every student in Class-B scores between 22 and 30. Any four students shifted from A to B therefore score below Class-B's current minimum of 22, so Class-B's average must fall - statement 1 is correct. For Class-A, the effect depends on which students leave: if the four shifted students scored near 17 (below A's average) the remaining average rises, but if they scored near 21 (above A's average) it falls. Since both are possible, statement 2 is not definite. Hence (a) 1 only. Options (b) and (c) wrongly treat statement 2 as certain, and (d) wrongly rejects statement 1.