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Numerical Ability · AFCAT

NA02 — Percentage, Ratio & Proportion

📈 Numerical Ability – NA02 AFCAT Level ★ High Priority
📌 AFCAT Focus: NA02 is the highest-weightage chapter in Numerical Ability (3–5 questions). Master percentage conversions (fraction↔%↔decimal), successive % change formula, and ratio shortcuts. Also feeds Profit/Loss and SI/CI. Average students who memorise the % fraction table can solve most AFCAT questions in under 30 seconds.

1. Percentage

Fig. 1.1 — Percentage ↔ Fraction ↔ Decimal: Master Conversion Table
PERCENTAGE ↔ FRACTION ↔ DECIMAL CONVERSIONS PERCENTAGE FRACTION DECIMAL 10% 1/10 0.1 20% 1/5 0.2 25% 1/4 0.25 33.33% 1/3 0.333̅ 50% 1/2 0.5 66.67% 2/3 0.667 75% 3/4 0.75 Memorise these! AFCAT uses these fractions directly in questions.
Key Percentage Formulas:
● x% of y = x × y / 100
● % increase = (Increase / Original) × 100
● % decrease = (Decrease / Original) × 100
● If A is x% more than B, then B is x/(100+x) × 100 % less than A
● Successive change: x% then y% → Net % = x + y + xy/100
✎ Worked Example — Successive Percentage Change
A salary is increased by 20% and then decreased by 10%. What is the net % change?
Use: Net = x + y + xy/100 (here x = +20, y = −10)
Net = 20 + (−10) + (20×−10)/100 = 20 − 10 − 2 = +8% (net increase)
✔ Net increase = 8%
RATIO & PROPORTION

2. Ratio & Proportion

Fig. 2.1 — Ratio & Proportion: Concepts & Properties
RATIO (a : b) a : b = a/b    (a = antecedent, b = consequent) Properties: ● Multiply/divide both terms by same number ● a:b = ka:kb (equivalent ratios) Comparison: a/b vs c/d Cross multiply: ad vs bc Types: Compounded ratio: (a:b)×(c:d) = ac:bd Duplicate ratio: a²:b²   Sub-duplicate: √a:√b PROPORTION (a:b = c:d) a, b, c, d in proportion: ad = bc (Product of extremes = Product of means) Key Rules: Componendo: (a+b)/b = (c+d)/d Dividendo: (a−b)/b = (c−d)/d Componendo-Dividendo: (a+b)/(a−b) = (c+d)/(c−d) Types: Continued: a:b:c = a/b = b/c (b = √ac) Direct: y↑ when x↑    Inverse: y↓ when x↑
Topic CPartnership (Profit Sharing)AFCAT Direct
Simple
Equal time: Profit divided in ratio of investments. If A invests Rs 3,000 and B invests Rs 5,000, ratio = 3:5. A gets 3/8 of profit, B gets 5/8.
Compound
Different time: Profit ratio = Capital×Time. If A invests Rs 3,000 for 6 months and B invests Rs 5,000 for 8 months, A:B = (3,000×6):(5,000×8) = 18,000:40,000 = 9:20.
Working Partner
A working partner may receive a salary/commission from profits before the remaining profit is split in the investment ratio.
✎ Worked Example — Ratio & Partnership
A, B and C invest Rs 20,000; Rs 25,000 and Rs 15,000 respectively. Total profit = Rs 9,600. Find each person’s share.
Ratio: 20,000 : 25,000 : 15,000 = 4 : 5 : 3   (divide by 5,000)
Total parts = 4+5+3 = 12
A = (4/12)×9,600 = Rs 3,200
B = (5/12)×9,600 = Rs 4,000
C = (3/12)×9,600 = Rs 2,400
✔ A: Rs 3,200   B: Rs 4,000   C: Rs 2,400
✎ Worked Example — Election Results (% Application)
In an election between two candidates, the winner gets 65% of total votes and wins by 900 votes. Find total votes and votes for each candidate.
Winner: 65%   Loser: 35%   Difference: 30%
30% of total = 900 → Total votes = 900 × 100/30 = 3,000
Winner: 65% of 3,000 = 1,950 votes
Loser: 35% of 3,000 = 1,050 votes
✔ Total = 3,000  |  Winner = 1,950  |  Loser = 1,050
Topic BComponendo, Dividendo, Alternendo & InvertendoAFCAT Theory
Setup
If a/b = c/d (i.e., a:b = c:d), then the following transformations all hold true:
Invertendo
b/a = d/c — flip both fractions. (Invert both sides simultaneously.) E.g. if 3/4 = 6/8 → 4/3 = 8/6 ✓
Alternendo
a/c = b/d — swap the means (b and c swap positions). E.g. if 3/4=6/8 → 3/6=4/8 → 1/2=1/2 ✓
Componendo
(a+b)/b = (c+d)/d — add denominator to numerator on both sides. E.g. a/b=c/d → (a+b)/b=(c+d)/d
Dividendo
(a−b)/b = (c−d)/d — subtract denominator from numerator on both sides.
Componendo-Dividendo
(a+b)/(a−b) = (c+d)/(c−d) — the most used rule in AFCAT. Apply when given a ratio and asked to find (x+y)/(x−y) type expressions.
Example: If x/y = 3/2, find (x+y)/(x−y).
Apply C-D: (3+2)/(3−2) = 5/1 = 5

📐 Formula Sheet — NA02

Core % Formulas
x% of y = xy/100
% increase = (increase/original)×100
Successive: x+y+xy/100
A more than B by x%: B less than A by x/(100+x)%
Ratio Key Facts
a:b::c:d → ad = bc
Compounded: ac:bd
Duplicate: a²:b²
Componendo-Dividendo: (a+b)/(a−b) = (c+d)/(c−d)
Proportion Types
Mean proportion of a,b: √(ab)
Third proportion to a,b: b²/a
Fourth proportion to a,b,c: bc/a
Direct: y = kx    Inverse: y = k/x
% Fraction Table (Key)
1/8 = 12.5%   1/6 = 16.67%
2/5 = 40%   3/8 = 37.5%
5/8 = 62.5%   7/8 = 87.5%
1/9 = 11.11%   1/11 = 9.09%
Partnership
Simple: Profit ∝ Capital
Compound: Profit ∝ Capital × Time
Working partner gets salary first, then split
Population / Growth
After n years: P×(1 + r/100)ⁿ
% change on original: use net formula
If population falls: P×(1 − r/100)ⁿ

📝 Topic-Wise PYQs — NA02

Q1. 35% of 280 is: AFCAT PYQ
(a) 91(b) 96(c) 98(d) 105
✔ Answer: (c) 98
35% of 280 = (35/100)×280 = 35×2.8 = 98. Shortcut: 35% = 25%+10% = 70+28 = 98.
Q2. A number is increased by 20% and then decreased by 20%. The net change is: AFCAT PYQ
(a) 0% change(b) 4% decrease(c) 4% increase(d) 2% decrease
✔ Answer: (b) 4% decrease
Net = x+y+xy/100 = 20+(−20)+(20×−20)/100 = 0−4 = −4%. A 4% decrease. This is a classic AFCAT trap — students assume 0% change.
Q3. If A:B = 2:3 and B:C = 4:5, then A:B:C = ? AFCAT PYQ
(a) 8:12:15(b) 2:3:5(c) 6:8:10(d) 4:6:10
✔ Answer: (a) 8:12:15
Make B common: A:B = 2:3 = 8:12; B:C = 4:5 = 12:15. So A:B:C = 8:12:15.
Q4. In an election, candidate A gets 55% of total votes. If he wins by 2,400 votes, find total votes: ⚡ Tricky
(a) 20,000(b) 24,000(c) 30,000(d) 12,000
✔ Answer: (b) 24,000
A gets 55%, B gets 45%. Difference = 10%. 10% of total = 2,400 → Total = 24,000.
Q5. A and B invest Rs 4,000 and Rs 6,000. If total profit = Rs 5,500, B’s share is: AFCAT PYQ
(a) Rs 2,000(b) Rs 3,000(c) Rs 3,300(d) Rs 2,500
✔ Answer: (c) Rs 3,300
Ratio = 4:6 = 2:3. B's share = (3/5)×5,500 = Rs 3,300.

🧠 Quick Memory Chart — NA02

📈 % Conversions
  • 10% = 1/10
  • 25% = 1/4
  • 33.33% = 1/3
  • 50% = 1/2
  • 12.5% = 1/8
  • 16.67% = 1/6
📈 % Formulas
  • Successive: x+y+xy/100
  • Both same sign: net increases
  • Opposite signs: may decrease
  • +r% then −r% = −r²/100 %
  • A 20% more than B → B is 16.67% less
📈 Ratio & Partnership
  • ad = bc (proportion)
  • C-D: (a+b)/(a−b) = (c+d)/(c−d)
  • Mean proportion: √(ab)
  • Simple partnership: Capital ratio
  • Compound: Capital × Time
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