Average and problems on ages are two of the easiest scoring topics in the Lok Sewa Maths and IQ (aptitude) section. Most questions can be solved in under a minute if you know the basic formula, a few shortcuts and the trick of working with totals. This note gives the formulas, the standard question types and fully worked examples for each.
1. What is an average?
The average (arithmetic mean) of a group of numbers is their sum divided by how many numbers there are.
- Average = Sum of observations ÷ Number of observations
- Sum = Average × Number of observations
- Number = Sum ÷ Average
Example: Find the average of 12, 15, 18, 21 and 24. Sum = 12 + 15 + 18 + 21 + 24 = 90. Number = 5. Average = 90 ÷ 5 = 18.
The key idea in almost every average problem is to convert averages into totals (Sum = Average × Number), work with the totals, and convert back at the end.
2. Useful shortcut formulas
| Series | Average |
|---|---|
| First n natural numbers (1, 2, 3, … n) | (n + 1) ÷ 2 |
| First n even numbers (2, 4, 6, … 2n) | n + 1 |
| First n odd numbers (1, 3, 5, …) | n |
| Any numbers in equal steps (arithmetic series) | (first term + last term) ÷ 2 |
| Odd number of consecutive numbers | the middle number |
| Squares of first n natural numbers | (n + 1)(2n + 1) ÷ 6 |
Example: Average of the first 50 natural numbers = (50 + 1) ÷ 2 = 25.5. Average of the first 10 even numbers = 10 + 1 = 11.
Example: The average of 7 consecutive numbers is 20. Since 20 is the middle (4th) number, the numbers are 17 to 23, so the largest is 23.
3. Effect of changing every number
| If every number is … | New average |
|---|---|
| increased by k | old average + k |
| decreased by k | old average − k |
| multiplied by k | old average × k |
| divided by k | old average ÷ k |
Example: The average of 10 numbers is 25. If 4 is added to each number, the new average is 25 + 4 = 29. If each number is multiplied by 3, the new average is 25 × 3 = 75.
4. Adding, removing and replacing a member
Worked example 1 (adding): The average age of 30 students is 14 years. When the teacher’s age is included, the average becomes 15. Find the teacher’s age.
- Total age of 30 students = 30 × 14 = 420
- Total age of 31 people = 31 × 15 = 465
- Teacher’s age = 465 − 420 = 45 years
Worked example 2 (removing): The average of 5 numbers is 40. If one number is removed, the average of the rest is 38. Find the removed number.
- Sum of 5 numbers = 5 × 40 = 200
- Sum of 4 numbers = 4 × 38 = 152
- Removed number = 200 − 152 = 48
Worked example 3 (replacing): The average weight of 8 people increases by 2.5 kg when a new person comes in place of one who weighs 65 kg. Find the weight of the new person.
- Total increase = 8 × 2.5 = 20 kg
- This extra weight is brought by the new person.
- New person’s weight = 65 + 20 = 85 kg
- Shortcut: New = Old + (Number × Change in average). If the average falls, subtract instead.
5. Combined (weighted) average and the batsman problem
When two groups are joined, do not simply average the two averages. Use the totals: Combined average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).
Worked example 4: Class A has 40 students with an average of 60 marks and class B has 60 students with an average of 70. Combined average = (40 × 60 + 60 × 70) ÷ 100 = (2,400 + 4,200) ÷ 100 = 66.
Worked example 5 (batsman): A batsman scores 87 runs in his 17th innings and so increases his average by 3. Find his average after the 17th innings.
- Let the average after 16 innings be x. Total after 16 innings = 16x.
- After the 17th innings, total = 16x + 87, and the new average is x + 3, so total = 17(x + 3).
- 16x + 87 = 17x + 51, so x = 36.
- New average = 36 + 3 = 39.
- Shortcut: New average = Score − (old number of innings × increase) = 87 − 16 × 3 = 39.
6. Problems on ages: basic ideas
- If a person’s present age is x, then n years ago the age was x − n and n years hence (after n years) it will be x + n.
- The difference between the ages of two people never changes.
- If ages are in the ratio a : b, take them as ax and bx.
- If A is k times as old as B, take B = x and A = kx.
- The sum of the ages of a group increases by (number of people × number of years) after that many years.
7. Worked examples on ages
Worked example 6 (ratio): The present ages of A and B are in the ratio 4 : 5. After 5 years the ratio will be 5 : 6. Find their present ages.
- Let the ages be 4x and 5x.
- (4x + 5) ÷ (5x + 5) = 5 ÷ 6
- Cross-multiply: 6(4x + 5) = 5(5x + 5), so 24x + 30 = 25x + 25
- x = 5. Present ages: A = 20 years, B = 25 years. (Check: 25 : 30 = 5 : 6)
Worked example 7 (times as old): A father is 3 times as old as his son. After 10 years he will be twice as old as the son. Find their present ages.
- Let the son be x, so the father is 3x.
- After 10 years: 3x + 10 = 2(x + 10) = 2x + 20
- x = 10. Son = 10 years, father = 30 years.
Worked example 8 (sum and past ratio): The sum of the ages of A and B is 50 years. Five years ago their ages were in the ratio 1 : 3. Find their present ages.
- Five years ago the sum was 50 − 5 − 5 = 40.
- Split 40 in the ratio 1 : 3: 10 and 30.
- Present ages: A = 10 + 5 = 15, B = 30 + 5 = 35.
Worked example 9 (hence and ago): A man’s age after 15 years will be 4 times his age 15 years ago. Find his present age. x + 15 = 4(x − 15), so x + 15 = 4x − 60, 3x = 75 and x = 25 years.
Worked example 10 (average and ages): The average age of a family of 5 is 24 years. The youngest member is 8 years old. Find the average age of the family at the birth of the youngest member.
- Total age now = 5 × 24 = 120. The other 4 members together = 120 − 8 = 112.
- 8 years ago each of them was 8 years younger: 112 − 4 × 8 = 80.
- At that time the family had 4 members, so the average = 80 ÷ 4 = 20 years.
8. Quick exam tips
- Always change averages into totals first.
- In replacement questions use New = Old ± (Number × Change).
- For age questions, form one equation using the information about the future or the past.
- Check your answer by putting it back into the question; this takes seconds and saves marks.
- When options are given, test them in the ratio condition; this is often faster than solving.
📌 Key points to remember
- Average = Sum ÷ Number, and Sum = Average × Number.
- Average of first n natural numbers = (n + 1) ÷ 2; first n even numbers = n + 1; first n odd numbers = n.
- Adding k to every number adds k to the average; multiplying by k multiplies the average by k.
- Replacement: new value = old value + (number × increase in average).
- Combined average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), not the simple mean of the two averages.
- n years ago age = x − n; n years hence age = x + n.
- The difference between two people’s ages always stays the same.
- For ratio a : b, take ages as ax and bx and form an equation.
Test yourself: Average and Problems on Ages
50 questions check what you read about What is an average?, Adding, removing and replacing a member and Worked examples on ages. Each attempt picks 25 at random.