Averages, Mixtures & Alligation

Averages and the alligation rule turn otherwise messy mixture and cost problems into quick ratios in CDS. The alligation shortcut is the single most time-saving tool here.

Averages

• Average = (sum of observations) / (number of observations).
• If the average of n numbers is A and one number is replaced, the change in sum = n x (change in average).

Alligation (the mixing rule)

• When two ingredients at prices/values c (cheaper) and d (dearer) mix to a mean m:
  (quantity of cheaper) / (quantity of dearer) = (d - m) / (m - c).
• This gives the ratio in which two quantities must be mixed to hit a target average.

Mixture replacement

• If a vessel of volume V has pure liquid and x is removed and replaced with water, repeated n times, remaining pure liquid = V(1 - x/V)^n.

Exam Tricks & Tips

• 🎯 Use alligation for any 'two things mixed to an average' problem — mean price, mean speed, mean age, purity.
• 🎯 The alligation ratio is (dearer - mean) : (mean - cheaper) for cheaper : dearer.
• 🎯 For repeated replacement, remaining fraction = (1 - x/V)^n — apply directly.
• 🎯 Weighted average = (n1 A1 + n2 A2)/(n1 + n2); use it when group sizes differ.
• 🎯 To adjust an average when adding/removing a value, work with total sums, not the averages.
• ❌ Common mistake: taking a plain average of two group averages when the groups have different sizes — you must weight by size.

Expected exam pattern

Average, weighted-average, alligation-ratio and mixture-replacement questions; 6-10 marks, fast with alligation.

Quick recap

Average = sum/count; alligation ratio = (d - m):(m - c); replacement leaves V(1 - x/V)^n; always weight averages by group size.

Cover the answer, recall, then check. 11 cards on averages and mixtures.

Q1. Formula for the average of n numbers?
A1. (sum of the numbers) / n.

Q2. State the alligation rule (cheaper : dearer).
A2. (quantity cheaper)/(quantity dearer) = (dearer - mean)/(mean - cheaper).

Q3. Remaining pure liquid after n replacements of x from volume V?
A3. V(1 - x/V)^n.

Q4. Weighted average of groups (n1, A1) and (n2, A2)?
A4. (n1 A1 + n2 A2)/(n1 + n2).

Q5. Mix rice at 20 and 30 per kg to average 24. Ratio?
A5. (30-24):(24-20) = 6:4 = 3:2 (cheaper:dearer).

Q6. If the average of 5 numbers is 12, their sum is?
A6. 60.

Q7. The average of 11 results is 60; the first 6 average 58 and the last 6 average 63. Find the 6th result.
A7. (6x58 + 6x63) - 11x60 = 726 - 660 = 66.

Q8. From 40 L of pure milk, 4 L is replaced by water twice. Milk left?
A8. 40(1 - 4/40)^2 = 40(0.9)^2 = 32.4 L.

Q9. When can you NOT simply average two averages?
A9. When the two groups have different sizes — you must weight by size.

Q10. Average of the first n natural numbers?
A10. (n + 1)/2.

Q11. What does the alligation mean 'm' represent?
A11. The desired average value of the final mixture.

Averages, mixtures, and the rule of alligation form a connected CDS arithmetic cluster about combining quantities. Alligation in particular is a powerful shortcut that replaces heavy equation-solving with a simple ratio.

Core idea / what this tests

This tests the average (mean) of a set, how averages change when items are added or removed, and how to find the ratio in which two ingredients (or prices, or concentrations) are mixed to reach a desired mean — the rule of alligation.

Deep explanation

Beginner — averages

• Average = (sum of values) ÷ (number of values). So sum = average × count.
• When a new value is added, the average shifts toward it; when the average of n items is A and one item is replaced, the change in total = change in the average × n.
• Weighted average: if group sizes differ, weight each group's average by its size: (n₁A₁ + n₂A₂)/(n₁ + n₂).

Intermediate — the rule of alligation

Alligation finds the ratio of two ingredients needed to obtain a target average (the "mean price"):

Ratio of cheaper : dearer = (dearer price − mean) : (mean − cheaper price).
In words, each quantity is taken in inverse proportion to its distance from the mean. This works for prices, concentrations, speeds — any weighted-average situation.

Advanced — repeated replacement (removal & replacement)

When a vessel of volume V contains pure liquid and you repeatedly remove x litres and replace with water, after n operations the quantity of original liquid left = V(1 − x/V)ⁿ. This models successive dilutions (milk-and-water problems). Combined with alligation, it handles most mixture questions CDS poses.

Worked example

In what ratio must rice at ₹30/kg be mixed with rice at ₹45/kg to get a mixture worth ₹40/kg?

Step 1 — identify: cheaper = 30, dearer = 45, mean = 40.
Step 2 — apply alligation: cheaper : dearer = (dearer − mean) : (mean − cheaper) = (45 − 40) : (40 − 30) = 5 : 10.
Step 3 — simplify the ratio: 5 : 10 = 1 : 2.
Check: taking 1 kg at 30 and 2 kg at 45 → cost = 30 + 90 = 120 for 3 kg → 40/kg ✓.
Answer: mix in the ratio 1 : 2 (cheaper : dearer). Alligation gave the ratio in one line, no equations needed.

CDS relevance

Averages appear directly and inside data interpretation; alligation is a time-saving CDS shortcut for mixture, price, and concentration problems. Recognising a problem as an alligation (any "what ratio to get a target mean") lets you skip algebra and score fast.

Exam tricks & shortcuts

• Average = sum ÷ count; sum = average × count.
• Alligation: cheaper : dearer = (dearer − mean) : (mean − cheaper) — quantities inverse to distance from the mean.
• Alligation applies to any weighted average (prices, speeds, concentrations, percentages).
• Replacement: after n removals/replacements, original left = V(1 − x/V)ⁿ.
• When one number in an average changes, total change = (new avg − old avg) × count.

Key formulas

• Average = sum/count; weighted average = Σwᵢxᵢ/Σwᵢ.
• Alligation: (cheaper:dearer) = (dearer − mean):(mean − cheaper).
• Mean value of a mixture = (Q₁P₁ + Q₂P₂)/(Q₁ + Q₂).
• Repeated replacement: quantity of pure left = V(1 − x/V)ⁿ.
• Average speed (equal distances) = 2uv/(u+v).
• Average of first n naturals = (n+1)/2.

Usage: alligation is just weighted averaging solved for the mixing ratio.

Worked Example

Problem: The average age of 5 members of a club is 20 years. When one new member joins, the average age becomes 21 years. Find the age of the new member.

Solution:
Step 1 — Total age of the original 5 members = average × number = 20 × 5 = 100 years.

Step 2 — After the new member joins there are 6 members, with a new average of 21:
new total age = 21 × 6 = 126 years.

Step 3 — The new member's age = new total − old total:
= 126 − 100 = 26 years.

Quick reasoning check: the new member not only matches the old average (20) but also supplies the "extra" year for each of the 6 members (6 × 1 = 6). So the age = 20 + 6 = 26 ✓, confirming the answer.

Answer: The new member is 26 years old.