Average, Mixture & Alligation — IBPS PO Quant

Average and mixture are two views of the same idea: balancing values around a mean. An average is the balance point of a set; alligation is the shortcut for finding the ratio in which two things must be mixed to hit a target average. IBPS asks 1–3 of these directly and folds averages into nearly every DI set. The tools are small — think in total sums for averages, use the alligation cross for any two-group blend — and once they click, these are one-line answers.

1. What IBPS actually asks

• Average: find/adjust a mean; the effect of adding, removing or replacing a member.
• Weighted average: combine two groups of different sizes.
• Mixture/alligation: the ratio to mix two ingredients (milk/water, two prices, two concentrations) for a target; replacement problems ("x litres removed and replaced…").

2. Averages — think in total sums

$\text{Average}=\frac{\text{sum}}{\text{count}}\Rightarrow \text{sum}=\text{average}\times \text{count}$

The fastest average problems are solved by tracking the total sum, not the mean:

• Adding a member: new sum = old sum + new value; recompute.
• Change trick: if the average of n numbers rises by d when one number is replaced, the new number is higher by n·d than the old one. ("Average of 8 rises by 2 ⇒ the replacement is 16 more.")
• Age/weight-of-group problems are pure sum bookkeeping.

3. Weighted average

Two groups of sizes n₁, n₂ with averages a₁, a₂ combine to:

$\text{Weighted avg}=\frac{n_1{a}_1+n_2{a}_2}{n_1+n_2}$

This is the same math as alligation — which just rearranges it to find a missing ratio or value.

4. Alligation — the see-saw shortcut

To mix two things (cheaper C and dearer D) to reach mean price M, the ratio of quantities is:

$\frac{\text{Qty of cheaper}}{\text{Qty of dearer}}=\frac{D-M}{M-C}$

Draw the cross: put C and D on top corners, M in the middle; each quantity's share is the opposite difference.

• Mix rice at ₹30 and ₹40 to get ₹34/kg: cheaper:dearer = (40−34):(34−30) = 6:4 = 3:2.
• Works for any average: prices, concentrations, speeds, percentages, interest rates — anything that averages linearly.

The number nearer the mean is needed in greater quantity. Sanity-check: if M is close to C, you need mostly the cheaper item.

5. Replacement (repeated removal)

When x units are removed from a V-unit vessel and replaced with another liquid, n times, the fraction of the original liquid left is:

${\left(1-\frac{x}{V}\right)}^n\times V$

• 8 L removed from 40 L of milk and replaced with water, twice ⇒ milk left = 40 × (1 − 8/40)² = 40 × (4/5)² = 25.6 L.
• This is compound-decay math — the same (1 − r)^n form as depreciation.

7. Common traps

• Averaging the averages without weighting — a 20-person and 30-person group don't average 50/50; weight by size.
• Getting the alligation ratio inverted — the quantity is the opposite difference; the item nearer the mean is larger.
• Applying replacement to the wrong liquid — the (1−x/V)^n formula tracks the original liquid remaining.
• Percentage as the "price" — alligation works on any linearly-averaged quantity, including % and rates.

8. The protocol

1. Averages: convert to total sum (sum = avg × count); track the sum through changes.
2. Replacement/adjustment: average change = total change ÷ count.
3. Two-group blend: use the alligation cross — quantity ratio = opposite differences.
4. Sanity-check the ratio: the group nearer the mean is the larger share.
5. Repeated replacement: original left = V·(1 − x/V)^n.