Averages questions, with answers

The questions, with answers

1. 1.What is the one habit that solves most averages questions?

   Total = average × count, so translate averages into totals and work with the totals. If the average of 8 numbers is 25, their sum is 200; if one of them is removed and the average of the rest is 24, the remaining seven sum to 168, so the removed number is 200 - 168 = 32. Every step is an addition or subtraction of totals; the division back to an average happens once, at the end. Candidates who try to reason about averages directly ("removing a number below the average raises it, so...") get the direction right and the number wrong. Write the totals.

2. 2.How do you find the value of a new member who changes the average?

   Compute the old total, the new total, and subtract. The average weight of 12 students is 50 kg; when one more joins, the average rises by 0.5 kg. Old total 12 × 50 = 600; new total 13 × 50.5 = 656.5; the newcomer weighs 56.5 kg. There is a shortcut worth knowing: the newcomer equals the new average plus the rise multiplied by the old count — 50.5 + 0.5 × 12 = 56.5 — because they must supply their own share of the new average plus the extra half-kilogram for each of the twelve others. The same logic in reverse handles a member leaving: if the average rises by d when someone leaves, the leaver was the old average minus d times the new count; if it falls by d, the old average plus d times the new count.

3. 3.How do you correct an average after a wrongly entered value?

   Adjust the total by the difference between the correct and the wrong value, then divide again. The average marks of 40 students were calculated as 68, but one score was entered as 27 instead of 72. The recorded total was 40 × 68 = 2,720; the correct total is 2720 + (72 - 27) = 2,765; the correct average is 2765 / 40 = 69.125. The shortcut: the average moves by the error divided by the count, 45 / 40 = 1.125, in the direction of the correction. Two errors are handled the same way — add the net correction to the total. Don't recompute from scratch; the question gives you the wrong average precisely so that you can work from it.

4. 4.What is the average of consecutive numbers, and how do you use it?

   The average of any evenly spaced set is the middle term (odd count) or the mean of the two middle terms (even count), because the values are symmetric around the centre. So the average of 9 consecutive integers is the fifth one: if the average is 30, the numbers are 26 to 34 and the largest is 34. For 8 consecutive integers with average 20.5, the middle pair is 20 and 21, and the set runs from 17 to 24. The same fact gives the averages of the first n natural numbers, (n + 1) / 2, and of consecutive even or odd numbers, without summing anything. It also flags an impossible question: an even count of consecutive integers cannot have an integer average.

5. 5.What is a weighted average, and when is a plain average wrong?

   A weighted average multiplies each value by its weight — a count, a quantity, a time — before averaging: sum of (value × weight) divided by sum of weights. A plain average is wrong whenever the groups are unequal. If a class of 30 averages 60 marks and a class of 20 averages 75, the combined average is (30 × 60 + 20 × 75) / 50 = (1800 + 1500) / 50 = 66, not 67.5. Average speed over equal distances is the same trap in disguise, and so is the price per kilogram of two mixed lots. The check: if the question mentions two different sizes, weight by them.

6. 6.How do you combine the averages of two groups, or find one group's average from the whole?

   Through totals again. A company's 60 employees have an average salary of Rs 30,000; the 20 managers among them average Rs 45,000. Total pay is 60 × 30,000 = 18,00,000; the managers account for 20 × 45,000 = 9,00,000; so the other 40 employees total 9,00,000 and average Rs 22,500. The combined average always lies between the two group averages, closer to the larger group, which is a quick sanity check — and if it does not, you have miscounted a group. Alligation gives the group sizes when the three averages are known: distances from the combined average to each group's average are in inverse ratio to the group sizes.

7. 7.How do the cricket-average questions work?

   They are new-member questions with the average as the unknown. A batsman has an average of 40 after 10 innings; in the 11th he scores 84 and his average rises by 4. Check: old total 400, new total 484, new average 484 / 11 = 44 — a rise of 4, consistent. The question shapes are: given the new score and the rise, find the old average (let it be x: 10x + 84 = 11(x + 4), so x = 40); or given the old average and the rise, find the score (score = new average + rise × old innings count = 44 + 4 × 10 = 84). Watch for "not out" innings, which the question may tell you to exclude from the count.

8. 8.What is the average of ages after some years, or of a group with a member replaced?

   Ages: every member ages the same amount, so the average rises by exactly that many years — a family's average age of 26 today is 31 in five years, whatever the ages are, provided the family is unchanged. Replacement: when a member is swapped, the total changes by (new value - old value), and the average changes by that difference divided by the count. If replacing a 60 kg student with a new one raises a 20-student average by 1.5 kg, the total rose by 30, so the newcomer weighs 90 kg. Both shapes are the total habit again: a uniform change moves the average by the same amount; a single change moves it by the change divided by the count.

How the diagnostic asks it

One question from the Aptitude bank, exactly as a sitting would show it. The bank has 4 on averages and 60 across Aptitude.