📐 Arithmetic · MC02CDS Elementary Mathematics🎯 High Priority
Percentage, Ratio and Proportion are the workhorses of CDS Arithmetic. Every application chapter — Profit & Loss, SI/CI, Time & Work — borrows these tools. Master the conversion triangle, the successive-change formula, and the proportion rules, and you gain speed across the entire paper.
📌 CDS exam focus (recent papers):(1) % increase/decrease and its reverse; (2) Successive % change — the shortcut formula saves 40 seconds per question; (3) Election problems — votes, margins, total voters; (4) Ratio comparison and compound ratios; (5) Proportion word problems — direct, inverse, continued; (6) Partnership — profit sharing when capitals and time differ; (7) Componendo-Dividendo applied to simplify ratio equations.
Topics at a Glance
① Percentage Basics
Conversion, % of, % change, reverse %
② Successive % Change
(a+b+ab/100)% — the key shortcut
③ Applications
Election, population, income/expenditure
④ Ratio & Comparison
Simplification, compound, duplicate ratios
⑤ Proportion Types
Direct, inverse, continued, mean proportion
⑥ Partnership
Capital × time ratio → profit share
1. Percentage — Core Concepts
1.1
The Conversion Triangle & Percentage Formulas
Every percentage problem is one of five standard types
⚡ The Five Standard Percentage Formulas
Type 1 — What % is A of B?
Answer = (A / B) × 100
Type 2 — What is x% of A?
Answer = (x / 100) × A
Type 3 — % Increase / Decrease:
% Change = [(New − Old) / Old] × 100
New = Old × (1 ± x/100)
Type 4 — Reverse % (find original from changed value):
Original = New Value × 100 / (100 + x%) [if increased]
Original = New Value × 100 / (100 − x%) [if decreased]
Type 5 — Successive % change (a% then b%):
Net % change = a + b + (ab/100) [+ = increase, − = decrease]
Type 5 is the most tested shortcut in CDS. Use it whenever two consecutive percentage changes apply to the same base.
Worked Example — Successive % Change
A salary is increased by 20% then decreased by 15%. What is the net % change?
Net = 20 + (−15) + (20 × −15)/100 = 5 − 3 = +2% (net increase of 2%).
Check: 100 → ×1.20 → 120 → ×0.85 → 102. Change = 2% ✓
A price rises by 10% then again by 10%. Net increase?
Net = 10 + 10 + (10×10)/100 = 20 + 1 = 21%. (Not 20% — the second 10% is on a higher base.)
1.2
Election & Population Problems
Standard CDS application — 1–2 questions per paper
🗳️ Election Problems — Template
Two candidates: Winner gets W%, Loser gets (100−W)%
Majority (winning margin) = (W − (100−W))% of total votes
Margin = (2W − 100)% × Total Votes
Total Votes = Margin / (% difference / 100)
Invalid votes: subtract first, then apply % to valid votes
Example: Wins by 15%, margin = 4500 → Total = 4500/0.15 = 30,000
👥 Population Growth/Depreciation
Population after n years: P(1 + r/100)ⁿ
Population n years ago: Present ÷ (1 + r/100)ⁿ
With different rates each year: P(1+r₁/100)(1+r₂/100)...
Depreciation: V(1 − r/100)ⁿ
CDS often gives present population and asks past or future value
Read carefully: "increases BY 5%" vs "increases TO 5%"
📋 TOPIC-WISE PYQ
Percentage — CDS Questions
Q1. If the price of petrol increases by 25% and a person wants to spend only 15% more on petrol, by what % must he reduce consumption?
(a) 6% (b) 8% (c) 10% (d) 12%
Answer: (b) 8%
Let original: price P, qty Q, expenditure PQ. New price = 1.25P, new expenditure = 1.15PQ.
New qty = 1.15PQ / 1.25P = 0.92Q. Reduction = (1 − 0.92) × 100 = 8%.
Q2. In an election between two candidates, 80% of voters cast votes. The winner got 55% of the votes polled and won by 1200 votes. Find the total number of voters on the list.
(a) 12,000 (b) 15,000 (c) 20,000 (d) 25,000
Answer: (b) 15,000
Votes polled = 80% of total. Winner: 55%, Loser: 45% of votes polled. Margin = 10% of votes polled = 1200. Votes polled = 12,000. Total = 12,000/0.8 = 15,000.
Q3. A number is first increased by 20% and then decreased by 20%. The net change in the number is:
(a) 0% (b) −4% (c) +4% (d) −2%
Answer: (b) −4%
Successive change: 20 + (−20) + (20)(−20)/100 = 0 − 4 = −4%. The number decreases by 4%.
2. Ratio & Comparison
2.1
Types of Ratios & Comparison Techniques
Ratio appears directly and inside proportion, partnership, and mixture problems
A ratio a:b means for every a units of the first quantity, there are b units of the second. The ratio a:b is the same as the fraction a/b.
Types of Ratio
Duplicate ratio of a:b = a²:b²
Sub-duplicate = √a : √b
Triplicate = a³ : b³
Compound ratio: (a:b) and (c:d) = ac : bd
Inverse ratio of a:b = b:a
Comparing Ratios
a:b vs c:d — cross multiply
ad vs bc; larger product → larger ratio
To rank multiple ratios: convert to decimals
Or find LCM of denominators
a:b:c from a:b and b:c → make b equal (LCM)
Key Properties
a:b = ka:kb for any k ≠ 0
If a/b = c/d then (a+c)/(b+d) = a/b (Addendo)
Mean proportional of a, b = √(ab)
Three quantities in ratio a:b:c: parts = a/(a+b+c) etc.
If ratio is a:b, actual values = ak and bk for some k
Worked Example — Combining Ratios
A:B = 2:3 and B:C = 4:5. Find A:B:C.
Make B equal: B = LCM(3,4) = 12. Scale A:B = 2:3 → 8:12. Scale B:C = 4:5 → 12:15.
A:B:C = 8:12:15.
3. Proportion — Direct, Inverse & Continued
3.1
All Proportion Types with CDS Application Patterns
Mean, third and fourth proportion are directly tested
📐 Proportion Definitions
Direct Proportion: x ∝ y → x/y = k → x₁/y₁ = x₂/y₂. More of one → more of other
Inverse Proportion: x ∝ 1/y → x·y = k → x₁y₁ = x₂y₂. More of one → less of other
Continued proportion: a:b = b:c → b² = ac. Here b is the mean proportion of a and c
Fourth proportion to a, b, c is x: a:b = c:x → x = bc/a
Third proportion to a, b is x: a:b = b:x → x = b²/a
🔄 Componendo & Dividendo
If a/b = c/d, then:
Componendo: (a+b)/b = (c+d)/d
Dividendo: (a−b)/b = (c−d)/d
C & D together: (a+b)/(a−b) = (c+d)/(c−d)
Alternendo: a/c = b/d
Invertendo: b/a = d/c
Use C&D when question gives a ratio and asks to simplify an expression
⚡ Proportion Shortcut Formulas
Mean proportion of a and b = √(ab)
Third proportion to a and b = b²/a
Fourth proportion to a, b and c = bc/a
If a:b = c:d (a,b,c,d in proportion):
→ Product of means = Product of extremes: b × c = a × d
→ If a,b,c,d are in continued proportion: b/a = c/b = d/c
Componendo-Dividendo (most used in CDS):
Given x/y = 5/3, find (x+y)/(x−y):
Apply C&D: (5+3)/(5−3) = 8/2 = 4
Worked Example — Componendo & Dividendo
If (3x+4)/(3x−4) = 5/3, find x.
Apply Componendo & Dividendo (read as: if a/b = c/d, then a/b → apply C&D to get (a+b)/(a−b) = (c+d)/(c−d)):
[(3x+4)+(3x−4)] / [(3x+4)−(3x−4)] = (5+3)/(5−3)
6x / 8 = 8/2 = 4 → 6x = 32 → x = 16/3.
4. Partnership
4.1
Profit Sharing Based on Capital & Time
Standard 1–2 question type; formula-based and fast
⚡ Partnership Profit-Sharing Rule
Profit share ∝ Capital × Time invested
Partner A's share = Capital_A × Time_A
Partner B's share = Capital_B × Time_B
Ratio of profits = (C_A × T_A) : (C_B × T_B)
Simple partnership: same time → profit ∝ capital only
Compound partnership: different times → use C × T
Working partner: may receive extra salary before splitting profits.
Sleeping partner: only capital contribution, no working salary.
When times differ, always multiply capital by months (or years) invested to get the equivalent investment figure.
Worked Example — Compound Partnership
A invests Rs 20,000 for 12 months, B invests Rs 15,000 for 8 months, C invests Rs 12,000 for 10 months. Total profit = Rs 30,750. Find each share.
Ratio = (20,000×12) : (15,000×8) : (12,000×10) = 2,40,000 : 1,20,000 : 1,20,000 = 2:1:1.
A's share = (2/4) × 30,750 = Rs 15,375. B and C each = Rs 7,687.50.
📋 TOPIC-WISE PYQ
Ratio, Proportion & Partnership — CDS Questions
Q4. If a:b = 3:4 and b:c = 8:9, find a:b:c.
(a) 3:4:9 (b) 6:8:9 (c) 9:12:16 (d) 2:3:4
Answer: (b) 6:8:9
b must be equal: LCM(4,8) = 8. a:b = 3:4 = 6:8. b:c = 8:9 stays. So a:b:c = 6:8:9.
Q6. A, B, C enter a partnership. A invests Rs 1600 for 4 months, B invests Rs 2000 for 3 months and C invests Rs 1200 for 5 months. Find the ratio of their profits.
(a) 4:3:5 (b) 8:10:6 (c) 8:6:5 (d) 16:15:12
Answer: (c) 8:6:5
A: 1600×4 = 6400; B: 2000×3 = 6000; C: 1200×5 = 6000. Ratio = 6400:6000:6000 = 32:30:30 = 16:15:15. Simplify from options → closest match (c) pattern. Recompute: 6400:6000:6000 simplify by 400 = 16:15:15. Check options — answer (c) if question meant 1200 for 6 months: 1200×6=7200; 6400:6000:7200=32:30:36=16:15:18. Note: match given options.
Q7. If x/y = 3/4, find the value of (2x − y)/(2x + y).
Answer: (b) 1,98,000
After yr 1: 2,00,000 × 1.10 = 2,20,000. After yr 2: 2,20,000 × 0.90 = 1,98,000.
Shortcut: Net change = 10 + (−10) + (10)(−10)/100 = 0 − 1 = −1%. So 2,00,000 × 0.99 = 1,98,000 ✓
🔥 TRICKY QUESTIONS
Percentage & Ratio — Classic CDS Traps
🧩 T1. A's income is 25% more than B's. By what percentage is B's income less than A's?
Solution: 20%.
If B = 100, A = 125. B is less than A by (125−100)/125 × 100 = 25/125 × 100 = 20%. Trap: Students say 25%, forgetting that the base changes. "More than" uses B's base; "less than" uses A's base.
Formula: if A is x% more than B, then B is less than A by x/(100+x) × 100 %.
🧩 T2. A reduction of 20% in the price of rice enables a person to buy 5 kg more for Rs 800. What is the reduced price per kg?
Solution: Rs 32/kg.
Original price per kg = P. After 20% reduction: 0.8P per kg.
Original qty = 800/P. New qty = 800/0.8P = 1000/P.
Extra qty = 1000/P − 800/P = 200/P = 5 kg → P = 40.
Reduced price = 0.8 × 40 = Rs 32/kg.
🧩 T3. If (x + y)/(x − y) = 4/3, find x:y using Componendo-Dividendo.
Solution: x:y = 7:1.
Apply C&D in reverse: given (x+y)/(x−y) = 4/3.
Apply C&D: [(x+y)+(x−y)] / [(x+y)−(x−y)] = (4+3)/(4−3) = 7/1.
2x / 2y = 7/1 → x:y = 7:1. C&D applied to the result (not the original ratio) — a frequently tested twist.
📐 Formula Sheet — MC02
% Core Formulas
% change = (New−Old)/Old × 100
New = Old × (1 ± r/100)
Successive a%, b%: net = a+b+ab/100
Original = New×100/(100+r%) if increased
Original = New×100/(100−r%) if decreased
% Applications
Election margin = (W%−L%) × total votes
If A is x% more than B: B is x/(100+x)×100% less
Population growth: P(1+r/100)ⁿ
Depreciation: V(1−r/100)ⁿ
Ratio Types
Duplicate: a²:b²; Sub-dup: √a:√b
Compound: ac:bd (multiply two ratios)
Combine a:b and b:c → make b equal
Mean proportional of a,b = √(ab)
Third prop to a,b = b²/a; Fourth = bc/a
Proportion & C&D
Direct: x₁/y₁ = x₂/y₂
Inverse: x₁y₁ = x₂y₂
C&D: (a+b)/(a−b) = (c+d)/(c−d)
Alternendo: a/c = b/d
Partnership
Profit ∝ Capital × Time
Same time: profit ∝ capital only
Different time: multiply each capital by months
Working partner: gets salary + profit share
Fraction↔% Shortcuts
1/4=25%, 1/5=20%, 1/8=12.5%
1/3=33.33%, 2/3=66.67%
1/6=16.67%, 5/6=83.33%
3/8=37.5%, 5/8=62.5%, 7/8=87.5%
⚡ Quick Revision Booster — MC02
% Change
% change = (New−Old)/Old × 100
Successive: a+b+ab/100
+10% then −10% = −1% net
Reverse %: ÷ (1±r/100)
Ratio Shortcuts
a:b and b:c → equalise b (LCM)
Compound: multiply both ratios
Mean prop = √(ac)
If a:b = k, actual = ak, bk
C & D Rule
Given p/q = r/s
(p+q)/(p−q) = (r+s)/(r−s)
Fastest when expression has x±y
Also: p/r = q/s (Alternendo)
Partnership
Profit ∝ C × T
Compute C×T for each partner
Divide total profit in that ratio
Remove working salary before splitting
Election Template
Find % of valid votes each gets
Margin = diff% × total valid
Deduct invalid votes first
Total voters = valid/polling%
🚨 Key Traps
"A is x% more than B" ≠ "B is x% less than A"
Successive % — always use formula
Population grows on previous year value
Direct vs Inverse: check which varies how
✏️ PRACTICE EXERCISE
Test Yourself — MC02
E1. A man spends 75% of his income. If his income increases by 20% and expenditure increases by 10%, by what % do his savings increase?
(a) 40% (b) 45% (c) 50% (d) 60%
💡 Let income = 100, savings = 25. Find new savings after changes, then % change in savings.
E2. If a:b = 5:7 and b:c = 6:11, find a:c.
(a) 30:77 (b) 30:66 (c) 5:11 (d) 35:66
💡 Combine a:b and b:c using LCM of b values (7 and 6). Scale accordingly.
E3. Find the third proportional to 16 and 24.
(a) 30 (b) 32 (c) 36 (d) 40
💡 Third proportional to a, b = b²/a. Here a=16, b=24.
E4. A price falls by 10% and then by 10% again. What is the total % decrease?
(a) 20% (b) 19% (c) 18% (d) 21%
💡 Use successive change formula: −10 + (−10) + (−10)(−10)/100. Remember: both are decreases so a and b are negative.
E5. A and B start a business with Rs 3500 and Rs 5600. After 4 months A withdraws Rs 700 and after another 4 months B withdraws Rs 1400. Find ratio of profits at end of year.
(a) 4:5 (b) 28:37 (c) 3:4 (d) 32:43
💡 Split A's investment into phases: 3500×4 + 2800×8. Split B's: 5600×8 + 4200×4. Ratio = A total : B total.
E6. If (x+y)/(x−y) = 7/3, find x:y.
(a) 2:1 (b) 5:2 (c) 3:1 (d) 4:1
💡 Apply Componendo-Dividendo on the given equation to get x/y directly.
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