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Numerical Ability · AFCAT

NA05 — Average & Mixture / Alligation

⚖️ Numerical Ability – NA05 AFCAT Level ☆ Low Priority
📌 AFCAT Focus 2022–2026: Average is the simpler half — master finding the missing number when average changes and average speed (use harmonic mean for equal distances). Alligation is a cross-method visual tool — draw the cross, subtract diagonally, get ratio. Most AFCAT mixture questions are solved in under 60 seconds using the alligation cross.

1. Average

Fig. 1.1 — Average: Core Concept & Effect of Adding/Removing Numbers
Average = Sum / Count Sum = Average × Count Adding a New Number New avg > old avg: New number > Old average New avg < old avg: New number < Old average New sum = Old sum + New number Replacing a Number If replaced number > replacement: New average < Old average Change in average formula: ΔAvg = ΔValue / Count New sum = Old avg × n − removed + added
Topic AAverage Speed (Weighted Average)AFCAT Trap
Equal Distance
If A travels equal distances at speeds s₁ and s₂, then:
Average speed = 2s₁s₂ / (s₁ + s₂)   (Harmonic Mean — NOT arithmetic mean!)
e.g., 40 km/h going, 60 km/h returning → Avg speed = 2×40×60/(40+60) = 4800/100 = 48 km/h
Equal Time
If A travels equal time at speeds s₁ and s₂:
Average speed = (s₁ + s₂)/2   (Simple Arithmetic Mean)
e.g., 40 km/h for 1 hour then 60 km/h for 1 hour → Avg = (40+60)/2 = 50 km/h
✎ Worked Example — Missing Number in Average
Average of 5 numbers is 30. When a 6th number is included the average becomes 32. What is the 6th number?
Sum of 5 numbers = 5 × 30 = 150
Sum of 6 numbers = 6 × 32 = 192
6th number = 192 − 150 = 42
✔ 6th number = 42
MIXTURE & ALLIGATION

2. Mixture & Alligation

Fig. 2.1 — Rule of Alligation: The Cross Method
RULE OF ALLIGATION — Cross Method Cost of Cheaper (c) Cost of Dearer (d) Mean Price (m) Qty of Cheaper d − m (dearer − mean) : Qty of Dearer m − c (mean − cheaper) Subtract diagonally
💡 Alligation 3-Step Method:
Step 1: Write cheaper (c) top-left, dearer (d) top-right, mean (m) in centre.
Step 2: Subtract diagonally: Qty of cheaper = (d−m), Qty of dearer = (m−c).
Step 3: Ratio of cheaper:dearer = (d−m):(m−c).
✎ Worked Example — Alligation
In what ratio must rice at Rs 12/kg be mixed with rice at Rs 18/kg to get a mixture worth Rs 14/kg?
c = 12, d = 18, m = 14
Qty of cheaper : Qty of dearer = (d−m) : (m−c) = (18−14) : (14−12) = 4 : 2 = 2 : 1
✔ Ratio = 2 : 1 (cheaper : dearer)
Topic DReplacement ProblemsAFCAT Direct
Formula
A container has V litres of liquid A. Each time x litres is removed and replaced with liquid B:
After n operations: Quantity of A = V × [(V−x)/V]ⁿ
Fraction of A remaining = [(V−x)/V]ⁿ
Example
40L of milk, 4L removed & replaced with water each time. After 2 operations:
Milk remaining = 40×(36/40)² = 40×0.81 = 32.4L. Water = 40−32.4 = 7.6L.

📐 Formula Sheet — NA05

Average
Avg = Sum / Count
Sum = Avg × Count
New number added: new sum = old sum + new no.
ΔAvg = ΔValue / n
Average Speed
Equal distance: 2s₁s₂/(s₁+s₂) (harmonic)
Equal time: (s₁+s₂)/2 (arithmetic)
Total distance / Total time (always safe)
Never average speeds directly for unequal distances!
Alligation Rule
Cheaper(c) & Dearer(d) mixed for Mean(m):
Ratio = (d−m) : (m−c)
c ≤ m ≤ d always
Works for price, %, concentration
Replacement
After n operations: A left = V×[(V−x)/V]ⁿ
Fraction remaining = [(V−x)/V]ⁿ
V = total volume, x = amount replaced
Formula applies to any mixture replacement
Weighted Average
Groups n₁ and n₂ with avgs A₁ and A₂:
Avg = (n₁A₁ + n₂A₂)/(n₁+n₂)
Overall avg between A₁ and A₂
Closer to whichever group is larger
AP Series Average
Avg of AP = (First + Last)/2
= Middle term (if odd count)
1 to n: avg = (n+1)/2
Consecutive evens/odds: avg = (first+last)/2

📝 Topic-Wise PYQs — NA05

Q1. The average of 11 results is 50. If the average of first 6 is 49 and last 6 is 52, what is the 6th result? AFCAT PYQ
(a) 46(b) 56(c) 48(d) 50
✔ Answer: (b) 56
Total sum = 11×50 = 550. First 6 sum = 6×49 = 294. Last 6 sum = 6×52 = 312. 6th result = 294+312−550 = 56. (6th is counted in both groups.)
Q2. A car travels 60 km at 30 km/h and returns at 60 km/h. Average speed for the whole journey is: ⚡ Tricky
(a) 45 km/h(b) 40 km/h(c) 42 km/h(d) 48 km/h
✔ Answer: (b) 40 km/h
Equal distance → harmonic mean: 2×30×60/(30+60) = 3600/90 = 40 km/h. Common trap: (30+60)/2 = 45 is WRONG for equal distances.
Q3. In what ratio must water be added to milk worth Rs 16/litre to get a mixture worth Rs 12/litre? AFCAT PYQ
(a) 1:3(b) 1:4(c) 2:3(d) 3:1
✔ Answer: (a) 1:3
Water cost = 0, Milk = 16, Mean = 12. Alligation: Water:Milk = (16−12):(12−0) = 4:12 = 1:3.
Q4. A vessel has 30 litres of milk. 6 litres is removed and replaced by water. This is done twice. Find the quantity of milk in the vessel now. ⚡ Tricky
(a) 18 L(b) 19.2 L(c) 20 L(d) 16.8 L
✔ Answer: (b) 19.2 L
Milk left = 30×(24/30)² = 30×(4/5)² = 30×16/25 = 19.2 L.
Q5. Average marks of 30 students is 60. Average of top 10 is 80 and bottom 10 is 50. Average of middle 10 is: AFCAT PYQ
(a) 55(b) 60(c) 50(d) 70
✔ Answer: (c) 50
Total = 30×60 = 1800. Top 10 = 800, Bottom 10 = 500. Middle 10 = 1800−800−500 = 500. Average = 500/10 = 50.

🧠 Quick Memory Chart — NA05

📈 Average
  • Avg = Sum/Count
  • Sum = Avg × Count
  • New avg = (old sum ± new no.) / new count
  • ΔAvg = ΔValue / n
  • AP avg: (first+last)/2
📈 Avg Speed
  • Equal distance: 2s₁s₂/(s₁+s₂)
  • Equal time: (s₁+s₂)/2
  • Avg speed < arithmetic mean (for equal dist.)
  • Always use: Total dist / Total time
📈 Alligation
  • Ratio = (d−m):(m−c)
  • c = cheaper, d = dearer, m = mean
  • Draw cross; subtract diagonally
  • Replacement: V×[(V−x)/V]ⁿ
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