🧮 Aptitude & Reasoning · Chapter 5

Averages and weighted averages

A batsman averaging 40 after 10 innings needs 95 in the next one to reach 45 — because an average is a total in disguise, and every innings has to be lifted by 5.

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The basics

An average (the mean) is the total shared out equally: five friends spend ₹1,500 at a dhaba, so the average is ₹300 each.

The problem it solves: describing a whole group with one number, and working backwards from that number to a missing value.

The mental model: an average is a total wearing a disguise. Average × count = total, and almost every question is solved by turning averages back into totals, adding or subtracting, and dividing again. Someone joins, leaves or is replaced? Compare the totals before and after.

When the groups are different sizes, plain averaging breaks. A weighted average gives each value a weight — the number of students, kilograms or credits behind it: (w₁x₁ + w₂x₂) ÷ (w₁ + w₂). It always lands nearer the heavier group.

For speed, use deviations: guess an assumed mean, note how far each value sits above or below it, and correct the guess by the average deviation.

Explained

A batsman has scored 400 runs in 10 innings: average 40. How many must he score in his 11th innings to lift the average to 45?

The instinct says 45. The answer is 95.

To average 45 over 11 innings he needs 11 × 45 = 495 runs. He has 400. 495 − 400 = 95. Every one of his 11 innings has to be lifted by 5, and the new innings pays for all of them: 45 + 10 × 5 = 95. An average is a total in disguise.

Settling the dhaba bill

Four friends stop at a dhaba on the Mumbai–Pune expressway. The bill is ₹1,200. Anil pays ₹250, Bina ₹300, Chetan ₹350, Divya ₹300.

Fair share is the average: 1,200 ÷ 4 = ₹300.

Nobody needs to redo the total to settle up. Look at how far each payment is from 300: Anil −50, Bina 0, Chetan +50, Divya 0. Anil pays Chetan ₹50, done.

Notice the deviations add to zero. They always do — that is what the average is: the balance point. That fact powers the fastest shortcut in this chapter.

The method, then the shortcuts

Standard method — totals. 30 students average 14 years. With the teacher, the average of 31 is 15. Teacher's age?

total before = 30 × 14 = 420
total after  = 31 × 15 = 465
teacher      = 465 − 420 = 45

Shortcut 1 — newcomer = new average + (old count × rise): 15 + 30 × 1 = 45. The teacher must be the new average and fund the 1-year rise for all 30 students.

Shortcut 2 — assumed mean. Average of 48, 52, 55, 45, 60? Assume 50. Deviations −2, +2, +5, −5, +10 sum to +10. 50 + 10/5 = 52.

When the shortcuts fail:

• Average of averages with unequal groups — 40 students at 60 and 60 at 70 average 66, not 65. Weight by the sizes
• The count changes twice — one leaves and one joins: back to totals
• Weights are hidden — speeds over equal distances are weighted by time (see ap-004), prices by quantity

Picture it: the average is the balance point

Deviations from the mean always cancel:

values:   250     300     300     350
           |       |       |       |
   ────────┼───────┼───────┼───────┼────
          −50      0       0      +50      → sum = 0
                       ▲
                 average = 300

Weighted average — two classes on a see-saw:

   40 students @ 60                  60 students @ 70
        ●━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━●
                         ▲
                        66
   distance 6                          distance 4

   40 × 6 = 240   =   60 × 4 = 240   ✓ balanced

The heavier side sits closer to the pivot. That is why the school average is pulled toward 70 — and why this is alligation (ap-003) run in reverse.

Solving any average question

If you never convert to totals, you will eventually average two averages and lose the mark.

The baby who kept the average

Three years ago, the average age of a family of five was 17 years. A baby has since been born, and today the average age of the six of them is still 17.

Bank PO prelims, 40 seconds

A batsman scores 87 runs in his 17th innings and thereby increases his average by 3. What is his average after the 17th innings?

Standard method: let the old average be x.

16x + 87 = 17(x + 3)
16x + 87 = 17x + 51
       x = 36   →  new average = 39

Topper's shortcut: the 87 had to cover the new average plus a 3-run lift for each of the 16 earlier innings.

new average = 87 − 16 × 3 = 87 − 48 = 39

Check: 16 × 36 = 576; 576 + 87 = 663; 663 ÷ 17 = 39 ✓

Ten seconds with the shortcut, and the check uses the totals method — so you trust the shortcut because you can see what it skips.

The traps examiners set

• Averaging averages — two sections of different sizes combine as a weighted average
• Forgetting the newcomer funds everyone's rise — to lift 10 innings' average by 5, the 11th must bring 45 + 10 × 5 = 95
• Ages move — 'three years ago' means every member has since aged; add 3 × count to the old total
• Dividing by the old count — after someone joins, divide by n + 1
• Consecutive odd/even numbers — they step by 2, so 7 consecutive odd numbers averaging 33 end at 33 + 3 × 2 = 39, not 36
• Mean vs median — one huge value (an owner's salary, a century) drags the mean far from the typical value

Words to know

In two minutes

• Average = total ÷ count, so total = average × count — convert to totals before doing anything else.
• When someone joins, leaves or is replaced, the change in total is all that matters.
• Group averages combine as a weighted average by group size, never as a plain average of averages.
• A weighted average sits between the values, nearer the heavier weight — it is alligation run in reverse.
• Raising an average of n items by d needs n × d extra in the total; a newcomer must bring old average + (n+1) × d.
• Adding a constant to every value adds it to the average; multiplying every value multiplies the average.
• Assumed-mean shortcut: pick a round guess, average the deviations, add the correction.

Test yourself

There are 10 questions, 3 puzzles and 5 recall cards for this lesson in the app — the recall cards come back on later days, just before you would forget.