Math · Aptitude · Chapter 09

👥 Ages & Averages

Age problems and average value calculations.

💡 Two Topics, One Idea: Algebra of Time

Averages — sum of numbers ÷ count of numbers.

Average = Sum / Count

If average changes when one item is replaced/added, the change tells you the missing value.

Age problems are usually solved by setting up an equation. Let present age = x. Then x years ago → (x − n), x years later → (x + n).

Replacement of one item

New − Old = total change in sum

Add/remove n items

Sum changes by (avg × n)

Ages — "n years ago"

Use (x − n) for past age

Ages — "n years hence"

Use (x + n) for future age

⚡ Average — replacement trick

The average weight of 10 students increases by 2 kg when a new student replaces one weighing 50 kg. What's the new student's weight?

Total weight increased by 2 × 10 = 20 kg. So new = 50 + 20 = 70 kg. Simple.

⚡ Ages — ratio change

"Father's age is 4 times son's. After 10 years, it'll be 2 times. Find present ages."

Let son = x, father = 4x. After 10 yrs: father = 4x + 10, son = x + 10.
4x + 10 = 2(x + 10) → 4x + 10 = 2x + 20 → 2x = 10 → x = 5. Son is 5, father is 20.

⚡ Bowler average trick

"Cricketer's average in 64 innings is 22 runs. To increase average to 24 in next innings, how many runs needed?"

Current total = 22 × 64 = 1408. New target average × new count = 24 × 65 = 1560. Required = 1560 − 1408 = 152 runs.

🎬

Average Visualized

Animation

Bars show each value. The red dashed line is the average — it's the "fair share" if you redistribute equally.

🧮

Average & Age Calculators

Calculator

📊 Average of up to 5 numbers

Enter values (use 0 to skip):

Average = 30

⏳ Father-son age

If father is × son's age now, and after years father will be × son:

Son = 5 · Father = 20

Practice (SSC): The average age of 11 players is 25 years. If the age of the coach is included, the average rises by 1. Find coach's age.

Sum of 11 players = 25 × 11 = 275. After including coach, count = 12, avg = 26. New sum = 26 × 12 = 312. Coach's age = 312 − 275 = 37 years.
Shortcut: Coach = new_avg + (new_avg − old_avg) × (n) = 26 + 1×11 = 37.

Practice (Banking): 5 years ago, the average age of A, B, C was 25 years. Today the average of A, B, C, D, E is 30 years. Find the average age of D and E today.

5 years ago, sum of A+B+C = 25 × 3 = 75. Today, sum of A+B+C = 75 + 5×3 = 90. Today A+B+C+D+E = 30 × 5 = 150. So D+E = 150 − 90 = 60. Average of D, E = 60/2 = 30 years.