6. Average and Ages Problem Framework

Why This Chapter Matters

Average and age questions often look verbal, but they simplify sharply once you convert them into totals and timelines. This chapter appears often in banking exams because the arithmetic is manageable but the interpretation can be tricky.

Core Ideas

• Average $=\frac{\text{sum of observations}}{\text{number of observations}}$.
• The average of consecutive natural numbers is simply the midpoint $\frac{\text{first}+\text{last}}{2}$.
• When one member enters or leaves a group, compare old total and new total instead of recalculating everything from scratch.
• If each value in a group rises or falls by the same amount, the average also rises or falls by the same amount.
• Age differences stay constant over time, but ratios change.
• Present-age equations are usually cleaner than future-age equations. Solve in the present first.
• Batsman-average questions are really total-score questions. One new innings changes both the sum and the count.

High-Value Formulas

How To Approach Questions

1. Convert the stated average into a total.
2. Track how many units are added, removed, or shifted.
3. For grouped data, find the group totals first and only then combine them.
4. For ages, set present ages first and move forward or backward in time together.
5. If a child is added to a family-age problem, increase the number of members from that point onward.

Worked Examples

Example 1

Prompt: The average of $12,18,24,30,36$ is what?

Approach: Total $=120$, count $=5$, so average $=24$.

Example 2

Prompt: A is $16$ years old and B is twice A's age. Find B's present age.

Approach: B's age $=2\times 16=32$ years.

Example 3

Prompt: The average of $10$ numbers is $15$ and that of $15$ numbers is $20$. Find the average of all $25$ numbers.

Approach: Use weighted totals: $\frac{10\times 15+15\times 20}{25}=18$.

Example 4

Prompt: Four years ago, A was $4$ years younger than B. Six years hence their ratio will be $16:17$. Find the sum of their ages four years ago.

Approach: Six years hence, let their ages be $16x$ and $17x$. The difference stays $4$, so $x=4$. Their total six years hence is $132$; four years ago means subtract $10$ years from each, so the sum was $112$.

Common Mistakes

• Averaging averages directly without weighting by the number of terms.
• Using changed ratios of ages as if they stay fixed forever.
• Forgetting that adding one person changes both the total and the count.
• Confusing the average increase with the total increase.
• Treating a future-age ratio as if the same ratio already holds today.

Quick Revision

Translate average questions into totals and age questions into present-age equations, then move through time only after the setup is clean.