What is the repeated-replacement formula?

If a container has x litres and you remove y litres and refill with another liquid, repeated n times, the original liquid left = x × (1 − y/x)ⁿ. This is the classic 'milk and water' replacement problem.

Does alligation give amounts or only the ratio?

Alligation gives only the ratio in which to mix the two ingredients. To find actual quantities, apply that ratio to the given total. For example, a 2:3 ratio for a 50 kg mixture means 20 kg and 30 kg.

Where is mixture and alligation used in exams?

Beyond literal mixtures, alligation solves weighted-average problems — average age, average marks, average speed, and profit/loss blends. It is a high-frequency shortcut in SSC, bank, and TNPSC quant.

Mixture and Alligation — Mean Price, Ratio & Replacement | TNPSC, SSC, Bank

Quick Concept Summary

Mixture and alligation is the exam-friendly form of the weighted average. The alligation (cross-difference) rule instantly gives the ratio in which two ingredients must be combined to reach a target mean — saving you from setting up algebra. The same tool solves average-age, average-marks, and average-speed problems, not just literal mixtures.

The Alligation Rule

To mix a cheaper item and a dearer item to a mean value:

Cheaper (c)        Dearer (d)
        \          /
          Mean (m)
        /          \
 (d − m)            (m − c)

Quantity of cheaper : quantity of dearer = (d − m) : (m − c)

This ratio tells you how to mix; multiply by the total to get actual amounts.

Repeated Replacement (Milk & Water)

If a vessel of x litres has some liquid removed (y litres) and refilled, repeated n times:

Liquid remaining = x × (1 − y/x)ⁿ

Worked Examples

Example 1: In what ratio must rice at ₹30/kg and ₹45/kg be mixed to cost ₹35/kg? cheaper:dearer = (45 − 35) : (35 − 30) = 10 : 5 = 2 : 1.

Example 2: A 40-litre vessel is full of milk. 8 litres are removed and replaced with water, done twice. Milk left? 40 × (1 − 8/40)² = 40 × (4/5)² = 40 × 16/25 = 25.6 litres.

Example 3: Average age of a class is 15. Boys average 16, girls average 14. Ratio of boys to girls? boys:girls = (15 − 14) : (16 − 15) = 1 : 1 → 1 : 1.

Example 4: Mix 60 kg of two sugars (₹20 and ₹28) to sell at ₹25/kg. Quantities? ratio = (28 − 25):(25 − 20) = 3:5 → of 60 kg → 22.5 kg and 37.5 kg.

Common Mistakes

Treating the alligation result as amounts instead of a ratio.
Swapping the cross-differences (cheaper gets d − m, not m − c).
In replacement, using subtraction instead of the (1 − y/x)ⁿ power.

Exam Tips

Use alligation for any weighted-average question — it is faster than algebra.
Keep the orientation fixed: cheaper-top-left pairs with the bottom-right difference.
For replacement, remember the answer is always less than the original — a good sanity check.

Frequently Asked Questions

When two ingredients of different prices/strengths are mixed to a mean value, the ratio of their quantities is (dearer − mean) : (mean − cheaper). Place cheaper and dearer on top, the mean in the middle, and cross-subtract. The result is the mixing ratio, not the actual amounts.