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AFCAT Numerical Ability · Average

The Highest, the Lowest, and What Is Left

When the largest or smallest values are taken out, the old total minus the new total is their sum; given their difference as well, you can find each one.

All Average notes

Why this matters

Nearly a fifth of this page is HARD. Every question turns on one fact: the values taken out total the old total minus the new one. Marks go when the new total is built with the old count, and when a sum and a difference are added but not halved.

Concept 1 of 3: Removing the highest and the lowest

Taking out the highest and the lowest leaves n−2n - 2 values. The total before and the total after differ by exactly the two values removed.

Definition

  • Highest + lowest = old total − new total =nA−(n−2)A′= nA - (n - 2)A'.
  • If the removed values are given, the new average =nA−H−Ln−2= \dfrac{nA - H - L}{n - 2}.
  • Removing the top kk: subtract their total and divide by n−kn - k.
  • If the removed values average exactly AA, the rest still average AA.

The two extremes together

H+L=nA−(n−2)A′H + L = nA - (n - 2)A'

Worked example

The average of 20 scores is 55. Without the highest and the lowest, the other 18 average 53. Find the sum of the highest and the lowest.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Average · Highest and Lowest Values

The average marks of 3030 students is 4040. If the highest and lowest marks are removed, the average becomes 3939. What is the sum of the highest and lowest marks?

Count what is left

Removing the highest and the lowest of 20 leaves 18, so the new total is 18 times the new average, not 20 times.
Drill 19 more on removing the highest and the lowest

Concept 2 of 3: One extreme alone: the largest or the smallest

'The largest' is only a label. Removing it is removing one value, and its size is the difference between the two totals.

Definition

  • Removed value =nA−(n−1)A′= nA - (n - 1)A'.
  • In letters: without the largest LL, nn numbers averaging AA become nA−Ln−1\dfrac{nA - L}{n - 1}. If removing the smallest SS raises the average by dd, then S=A−(n−1)dS = A - (n - 1)d.
  • Check the label: the largest must be at least the average of the rest, and the smallest at most that average.

One value taken out

x=nA−(n−1)A′x = nA - (n - 1)A'

Worked example

The average of 10 numbers is 36. Without the largest, the other 9 average 32. Find the largest.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Average · Highest and Lowest Values

The average weight of 1515 boys is 6565 kg. When the heaviest boy leaves, the average becomes 6363 kg. What is the weight of the heaviest boy?

Check that it really is the extreme

If 6 numbers average 30 and leaving out 'the smallest' gives an average of 26, the number removed is 180−130=50180 - 130 = 50. That cannot be the smallest, so re-read the question.
Drill 10 more on one extreme alone: the largest or the smallest

Concept 3 of 3: Highest and lowest from their sum and difference

The two totals give H+LH + L; the question gives H−LH - L. Two numbers from their sum and difference: add and halve for the larger, subtract and halve for the smaller.

Definition

  • Step 1: H+L=nA−(n−2)A′H + L = nA - (n - 2)A'.
  • Step 2: H=sum+difference2H = \dfrac{\text{sum} + \text{difference}}{2} and L=sum−difference2L = \dfrac{\text{sum} - \text{difference}}{2}.
  • Sanity check: LL must not be above the average of the rest and HH not below it. Runs and goals must be whole numbers.

Two numbers from sum and difference

H=(H+L)+(H−L)2,L=(H+L)−(H−L)2H = \dfrac{(H + L) + (H - L)}{2}, \qquad L = \dfrac{(H + L) - (H - L)}{2}

Worked example

The average of 26 numbers is 48. The largest and the smallest differ by 70. Without them, the other 24 average 46. Find both.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Average · Highest and Lowest Values

A batsman has an average of 3232 runs in his 4545 innings. The difference between his maximum and minimum score is 120120. If these 22 innings are removed, his average for 4343 innings comes down to 3030. What was his minimum score?

Halve at the end

With H+L=144H + L = 144 and H−L=70H - L = 70, adding gives 2H=2142H = 214, so H=107H = 107. Stopping at 214 is the slip.
Drill 16 more on highest and lowest from their sum and difference

Summary — formulas & gotchas at a glance

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