Ages Problems; Averages

Ages Problems

4.1 General Approach

Let the current age of a person = x years

Age n years ago = x - n
Age n years hence = x + n

4.2 Common Problem Types

Type 1: Ratio of Current Ages

If ratio of ages of A and B is a:b and sum = S:

A's age = S × a/(a+b)
B's age = S × b/(a+b)

Type 2: Ages n years ago/hence

Read the problem carefully. Set up equations based on what ratio/difference is given.

Type 3: Age when another was born

If A is currently 'a' years old and B is 'b' years old (b < a):

Age of A when B was born = a - b

4.3 Sample Formulas

Problem Pattern	Setup
Present ages in ratio a:b	Ages = ak, bk (where k is common multiple)
Sum of present ages = S	ak + bk = S → k = S/(a+b)
n years later, ratio changes	Set new ratio equation with (ak+n)/(bk+n)
n years ago, given ratio	(ak-n)/(bk-n) = given ratio

4.4 Key Tips

Always denote unknowns as variables (x, y or ak, bk)
Form TWO equations for TWO unknowns
Check if the answer makes sense (ages can't be negative)
If ratio is given for a future/past time, express using a single variable

Example: The ratio of A's age to B's age is 4:5. 5 years ago the ratio was 3:4. Find their current ages.

Let current ages = 4k and 5k
5 years ago: (4k-5)/(5k-5) = 3/4

4(4k-5) = 3(5k-5)

16k - 20 = 15k - 15
k = 5
A = 20, B = 25

Averages

5.1 Basic Formula

Average = Sum of all observations / Total number of observations

Or equivalently:

Sum = Average × Total number of observations

5.2 Extended Concepts

Simple Average

Average of n numbers = (a₁ + a₂ + ... + aₙ) / n

Weighted Average

When different groups have different averages:
Weighted Average = (n₁ × A₁ + n₂ × A₂ + ...) / (n₁ + n₂ + ...)
Where n₁, n₂ = number of items and A₁, A₂ = respective averages

5.3 Key Results — Averages

Result	Formula
Average of first n Natural Numbers	(n+1)/2
Average of first n Even Numbers	n+1
Average of first n Odd Numbers	n
Average of consecutive numbers	(first + last) / 2
Average of n consecutive even/odd	middle number

5.4 Effect of Replacing a Number

If a number 'x' in a group is replaced by 'y':
- New sum = Old sum - x + y
- Change in average = (y - x) / n

Example: Average of 5 numbers is 20. If one number (25) is replaced by 35:

New Average = 20 + (35 - 25)/5 = 20 + 2 = 22

5.5 Increase/Decrease in Total When Average Changes

If average of group drops by 'd' when 'k' elements are removed:
- Sum removed = k × (old average) + n × (-d)
More precisely: New total = Old total - removed sum; New count = n - k New average = New total / New count

5.6 Important Tricks

Trick 1: If one observation is changed and average changes:

(New value - Old value) = Change in average × Total count

Trick 2: Average speed when distances are equal:

Average Speed = 2S₁S₂ / (S₁ + S₂) (Harmonic Mean)
NOT (S₁ + S₂)/2

Trick 3: Average of consecutive numbers in AP with common difference d:

Average = (First term + Last term) / 2