21  Mathematical Aptitude (Fraction, Time & Distance, Ratio, Proportion and Percentage, Profit and Loss, Interest and Discounting, Averages etc.)

21.1 What the Syllabus Covers

The syllabus head bundles nine recurring arithmetic areas that NTA Paper-I tests:

  1. Number system & fractions.
  2. Percentages.
  3. Ratio and proportion.
  4. Profit and loss.
  5. Simple and compound interest (with discounting).
  6. Average.
  7. Time, speed and distance (incl. trains, boats-streams).
  8. Time and work (incl. pipes and cisterns).
  9. Permutation, combination and probability.

Each problem rewards formula recognition + clean substitution more than algebra. The candidate trains a small set of formulas and a fast checking procedure.

21.2 Number System and Fractions

TipWorking Categories of Numbers
Type Description Examples
Natural (N) Counting numbers 1, 2, 3 …
Whole (W) Natural + 0 0, 1, 2 …
Integer (Z) All positive, zero, negative …, −2, −1, 0, 1, 2 …
Rational (Q) p/q form, q ≠ 0 1/2, −3/4, 5
Irrational Cannot be p/q √2, π, e
Real (R) Rational + Irrational All non-imaginary
Prime Divisible by 1 and self only 2, 3, 5, 7, 11 …
Composite More than two factors 4, 6, 8, 9, 10 …

21.2.1 Fraction Essentials

TipFraction Quick Facts
  • Proper fraction — numerator < denominator (3/4).
  • Improper fraction — numerator ≥ denominator (5/4).
  • Mixed number — 1¼.
  • To add / subtract: equalise denominators (LCM).
  • To multiply: numerators × numerators, denominators × denominators.
  • To divide: multiply by the reciprocal.
  • Decimal-to-fraction: 0.25 = 25/100 = 1/4.
  • Recurring decimals: 0.333… = 1/3; 0.142857142857… = 1/7.

21.2.2 HCF and LCM

TipHCF and LCM
  • HCF (Highest Common Factor) — largest number dividing all given numbers.
  • LCM (Least Common Multiple) — smallest multiple of all given numbers.
  • Key identity: HCF × LCM = product of the two numbers (for two numbers only).

21.3 Percentages

A percentage is a fraction with denominator 100. x % = x/100.

21.3.1 Core Formulas

TipPercentage Formulas
  • x % of N = (x/100) × N.
  • A is what % of B = (A/B) × 100.
  • Increase from A to B = ((B − A)/A) × 100 %.
  • Decrease from A to B = ((A − B)/A) × 100 %.
  • Successive % change of a% then b% (NET trap): net = a + b + (ab/100).

21.3.2 Useful Conversions

TipFractional Equivalents to Memorise
Fraction Percentage
1/2 50 %
1/3 33⅓ %
1/4 25 %
1/5 20 %
1/6 16⅔ %
1/7 14 ²/₇ %
1/8 12½ %
1/9 11⅑ %
1/10 10 %
1/11 9¹/₁₁ %
1/12 8⅓ %
1/20 5 %

21.3.3 Worked Examples

TipPercentage Worked Examples

Q. 20 % of 150 = (20/100) × 150 = 30. Q. A salary rises from ₹40,000 to ₹44,000. Percentage rise = (4000/40000) × 100 = 10 %. Q. A price first rises 20 % then falls 20 %. Net change = 20 − 20 − (20 × 20)/100 = −4 %. (Not zero!)

21.4 Ratio and Proportion

TipRatio Vocabulary
  • Ratio a : b — comparison; a is antecedent, b is consequent.
  • Proportion a : b :: c : d — two ratios equal; reads a × d = b × c.
  • Direct proportion — when one increases, the other increases.
  • Inverse proportion — when one increases, the other decreases.
  • Compound ratio of (a : b) and (c : d) = ac : bd.
  • Mean proportional of a and b = √(ab).
  • Third proportional of a, b = b²/a.
  • Fourth proportional of a, b, c = bc/a.

21.4.1 Worked Examples

TipRatio Worked Examples

Q. Divide ₹600 among A, B, C in ratio 1 : 2 : 3. Total parts = 6. A = 100, B = 200, C = 300.

Q. If 3 men finish a job in 10 days, how long for 5 men (inverse proportion)? 3 × 10 = 5 × x → x = 6 days.

Q. Mean proportional of 8 and 18 = √(8×18) = √144 = 12.

21.4.2 Partnership

For partners P and Q investing for time periods t₁ and t₂, profit-share ratio = (P × t₁) : (Q × t₂).

21.5 Profit and Loss

TipProfit and Loss Vocabulary
  • Cost Price (CP) — what the seller paid.
  • Selling Price (SP) — what the buyer paid.
  • Marked Price (MP) / List Price — printed price before discount.
  • Profit = SP − CP (if SP > CP).
  • Loss = CP − SP (if SP < CP).
  • Profit % = (Profit / CP) × 100.
  • Loss % = (Loss / CP) × 100.
  • Discount = MP − SP; Discount % = (Discount / MP) × 100.

21.5.1 Useful Identities

TipQuick Identities
  • SP = CP × (100 + Profit %)/100 = CP × (100 − Loss %)/100.
  • CP = SP × 100/(100 + Profit %).
  • If equal SPs, one with x % profit and another with x % loss → overall loss = (x²/100) % (NET trap).
  • Trade discount % = (MP − SP)/MP × 100; given MP, SP = MP × (1 − d/100).

21.5.2 Worked Examples

TipProfit & Loss Worked Examples

Q. CP = ₹500, SP = ₹600. Profit % = (100/500) × 100 = 20 %.

Q. Selling for ₹450 gives 10 % loss. CP = 450 × 100/90 = ₹500.

Q. A trader gives 20 % discount on MP ₹1000. SP = 1000 × 0.80 = ₹800. If his CP was ₹600, profit % = (200/600) × 100 = 33⅓ %.

21.6 Simple and Compound Interest

TipInterest Formulas
  • Principal (P) — borrowed amount.
  • Rate (R %) — annual interest rate.
  • Time (T) — in years.
  • Simple Interest (SI) = P × R × T / 100.
  • Compound Interest (CI) = A − P, where Amount A = P (1 + R/100)^T.
  • CI compounded n times per year: A = P (1 + R/(100n))^(nT).
  • Difference between CI and SI for 2 years = P × (R/100)².

21.6.1 Worked Examples

TipInterest Worked Examples

Q. SI on ₹10,000 at 8 % p.a. for 3 years = 10000 × 8 × 3 / 100 = ₹2,400.

Q. CI on ₹10,000 at 10 % p.a. for 2 years compounded annually = 10000 × (1.1)² − 10000 = 10000 × 1.21 − 10000 = ₹2,100.

Q. Difference between CI and SI on ₹2000 at 10 % for 2 years = 2000 × (10/100)² = ₹20.

21.6.2 Discount (Banker’s vs True)

TipDiscount Concepts
  • Banker’s discount — simple interest on face value.
  • True discount — simple interest on present value.
  • Banker’s discount > True discount.

21.7 Average

TipAverage Formulas
  • Average = sum of all values / number of values.
  • Weighted average = Σ(wᵢ × xᵢ) / Σ wᵢ.
  • If the average of n values is x, sum = n × x.
  • When one value is replaced by another, change in average = (new − old)/n.
  • Average of first n natural numbers = (n + 1)/2.
  • Average of first n even numbers = n + 1.
  • Average of first n odd numbers = n.
  • Average speed for equal distances with speeds a and b = 2ab/(a+b) (harmonic mean).

21.7.1 Worked Examples

TipAverage Worked Examples

Q. Average of 5, 8, 11, 14, 17 = 55/5 = 11.

Q. A class of 30 has average mark 60. With a new student of mark 90, new average = (30 × 60 + 90)/31 = 1890/31 ≈ 61.0.

Q. A car covers 30 km at 60 km/h and 30 km at 40 km/h. Average speed = 2 × 60 × 40 / (60 + 40) = 4800/100 = 48 km/h.

21.8 Time, Speed and Distance

TipTSD Formulas
  • Speed = Distance / Time.
  • Distance = Speed × Time.
  • Time = Distance / Speed.
  • Convert km/h to m/s: multiply by 5/18.
  • Convert m/s to km/h: multiply by 18/5.
  • Relative speed (same direction) = v₁ − v₂.
  • Relative speed (opposite directions) = v₁ + v₂.

21.8.1 Trains

TipTrain Problems
  • Train of length L passing a pole: time = L / speed.
  • Train passing a platform of length P: time = (L + P) / speed.
  • Two trains crossing same direction: time = (L₁ + L₂) / (v₁ − v₂).
  • Two trains crossing opposite directions: time = (L₁ + L₂) / (v₁ + v₂).

21.8.2 Boats and Streams

TipBoats and Streams

Boat speed in still water = u, stream speed = v. - Downstream speed = u + v. - Upstream speed = u − v. - Boat speed = ½ × (downstream + upstream). - Stream speed = ½ × (downstream − upstream).

21.8.3 Worked Examples

TipTSD Worked Examples

Q. A 100 m train at 36 km/h = 10 m/s. Time to pass a pole = 100/10 = 10 sec.

Q. Two trains 100 m and 150 m, speeds 36 and 54 km/h opposite directions. Relative speed = 90 km/h = 25 m/s. Time = (100+150)/25 = 10 sec.

Q. A boat goes 10 km downstream in 2 hr and 5 km upstream in 1 hr. Downstream speed 5 km/h, upstream 5 km/h. Boat speed = (5+5)/2 = 5 km/h. Stream = 0.

21.9 Time and Work

TipTime and Work
  • Work done = 1 (whole); rate = 1 / time taken.
  • If A does work in a days, A’s rate = 1/a per day.
  • Combined rate of A and B = 1/a + 1/b.
  • Together they finish in ab/(a + b) days.
  • Three workers in ab+bc+ca / (abc) days (with the inverse-rate formula).
  • M₁ D₁ H₁ W₂ = M₂ D₂ H₂ W₁ (chain rule).

21.9.1 Pipes and Cisterns

A filling pipe = positive rate; an emptying pipe = negative rate. Net rate = sum.

21.9.2 Worked Examples

TipTime & Work — Worked

Q. A finishes a work in 12 days, B in 18 days. Together they finish in 12 × 18 / (12 + 18) = 216/30 = 7.2 days.

Q. A pipe fills a tank in 6 hrs, another empties it in 12 hrs. Net rate = 1/6 − 1/12 = 1/12 per hr. Time to fill = 12 hrs.

21.10 Permutation, Combination, and Probability

TipCounting Formulas
  • Factorial: n! = n × (n−1) × … × 1; 0! = 1.
  • Permutation: ⁿPᵣ = n! / (n−r)! — ordered.
  • Combination: ⁿCᵣ = n! / (r!(n−r)!) — unordered.
  • Circular permutation of n distinct = (n−1)!.
  • Linear permutation of n distinct = n!.
TipProbability Basics
  • Probability of an event = favourable / total outcomes; lies in [0, 1].
  • P(A or B) = P(A) + P(B) − P(A and B).
  • For mutually exclusive events: P(A and B) = 0.
  • For independent events: P(A and B) = P(A) × P(B).
  • P(not A) = 1 − P(A).

21.10.1 Worked Examples

TipCounting and Probability — Worked

Q. ⁵P₂ = 5! / 3! = 20.

Q. ⁵C₂ = 5! / (2! × 3!) = 10.

Q. Probability of getting a “head” in a fair coin = 1/2. Q. Probability of getting a sum of 7 on two dice = 6/36 = 1/6.

21.11 Calendars, Clocks, and Mensuration — Quick Notes

21.11.1 Calendars

TipDay Counting
  • 365 days = 52 weeks + 1 odd day (year shifts by 1 day next year).
  • 366 days (leap year) = 52 weeks + 2 odd days.
  • Leap year = divisible by 4 (centenary years must be divisible by 400).

21.11.2 Clock Angles

TipClock Angle Formula
  • Hour hand moves 0.5°/min (30°/hr).
  • Minute hand moves 6°/min.
  • Angle between hands = |30H − 5.5M|° at H hours and M minutes.
  • Hands coincide every 65⁵⁄₁₁ minutes (22 times in 12 hours).
  • Hands are perpendicular 44 times in 12 hours; straight (180°) 22 times.

21.11.3 Mensuration — Standard Formulas

TipMensuration Quick Reference
Shape Area Perimeter / Volume
Square (side a) a² 4a
Rectangle (l × b) l × b 2(l + b)
Triangle ½ × base × height a + b + c
Circle (radius r) π r² 2πr
Sphere 4πr² (surface) (4/3) π r³
Cylinder 2πrh + 2πr² π r²h
Cone πr (l + r) ⅓ πr²h
Cube (side a) 6a² a³
Cuboid (l × b × h) 2(lb + bh + lh) l × b × h

21.12 Theory Anchors

TipNames and Concepts Worth Knowing
Concept Origin
Percentage Latin per centum; popularised in 17th-c. commerce
Compound interest formula Jacob Bernoulli (1683) → led to e
Pythagorean theorem Pythagoras / Baudhāyana sūtra (India, 8th c. BCE)
Place-value decimal system Indian — used by Aryabhata, Brahmagupta
Zero as numeral Brahmagupta (628 CE)
Algebra (al-jabr) Al-Khwarizmi (9th c.) — Baghdad
Permutation-combination notation Leibniz, 17th c.
Probability theory Pascal & Fermat correspondence, 1654
Factorial notation (n!) Christian Kramp, 1808
HCF / GCD algorithm Euclid’s algorithm (~300 BCE)

21.13 Practice Questions

Q 01 Percentage Easy

What is 35 % of 240?

  • A72
  • B78
  • C84
  • D96
View solution
Correct Option: C
35/100 × 240 = 84.
Q 02 Successive % Hard

A price is first increased by 20 % and then decreased by 20 %. The net change is:

  • A0 %
  • B+4 %
  • C−4 %
  • D−8 %
View solution
Correct Option: C
Net = a + b + ab/100 = 20 + (−20) + (20×−20)/100 = −4 %. The classic NET trap.
Q 03 Ratio Medium

Divide ₹1500 in ratio 2 : 3 : 5. The largest share is:

  • A₹500
  • B₹600
  • C₹750
  • D₹900
View solution
Correct Option: C
Total parts = 10. Largest = 5/10 × 1500 = ₹750.
Q 04 Mean Proportional Medium

The mean proportional of 9 and 16 is:

  • A10
  • B12
  • C14
  • D16
View solution
Correct Option: B
√(9 × 16) = √144 = 12.
Q 05 Profit % Easy

CP = ₹400, SP = ₹500. Profit % is:

  • A20 %
  • B25 %
  • C30 %
  • D40 %
View solution
Correct Option: B
Profit = 100; % = 100/400 × 100 = 25 %.
Q 06 P&L Trap Hard

A shopkeeper sells two items for ₹500 each. On one he gains 25 %, on the other he loses 25 %. His overall:

  • ANo profit, no loss
  • B6.25 % loss
  • C6.25 % profit
  • D10 % loss
View solution
Correct Option: B
Equal-SP rule: loss = (x²/100)% = (25²/100) = 6.25 % loss. (Always a loss.)
Q 07 SI Easy

Simple interest on ₹5000 at 12 % p.a. for 2.5 years is:

  • A₹1200
  • B₹1400
  • C₹1500
  • D₹1600
View solution
Correct Option: C
SI = 5000 × 12 × 2.5 / 100 = ₹1500.
Q 08 CI Medium

Compound interest on ₹10,000 at 10 % p.a. for 2 years (annual compounding) is:

  • A₹2000
  • B₹2100
  • C₹2200
  • D₹2500
View solution
Correct Option: B
A = 10000 × (1.1)² = 12100; CI = ₹2100.
Q 09 CI-SI Diff Hard

The difference between CI and SI on ₹4000 at 10 % p.a. for 2 years is:

  • A₹30
  • B₹40
  • C₹60
  • D₹80
View solution
Correct Option: B
For 2 years: difference = P × (R/100)² = 4000 × (0.1)² = ₹40.
Q 10 Average Medium

The average of first 10 natural numbers is:

  • A5.0
  • B5.5
  • C6.0
  • D6.5
View solution
Correct Option: B
(n+1)/2 = 11/2 = 5.5.
Q 11 Avg Speed Hard

A car covers equal distances at 60 km/h and 40 km/h. Average speed is:

  • A48 km/h
  • B50 km/h
  • C52 km/h
  • D55 km/h
View solution
Correct Option: A
Harmonic mean: 2 × 60 × 40 / (60 + 40) = 4800/100 = 48 km/h. NOT the simple average (50).
Q 12 TSD Medium

A train 120 m long crosses a pole in 6 seconds. Its speed in km/h is:

  • A60
  • B66
  • C72
  • D80
View solution
Correct Option: C
Speed = 120/6 = 20 m/s = 20 × 18/5 = 72 km/h.
Q 13 Trains Hard

Two trains of length 100 m and 150 m at speeds 36 km/h and 54 km/h move in opposite directions. The time they take to cross each other is:

  • A5 sec
  • B8 sec
  • C10 sec
  • D12 sec
View solution
Correct Option: C
Relative speed = 36+54 = 90 km/h = 25 m/s. Time = (100+150)/25 = 10 sec.
Q 14 Boat-Stream Medium

A boat travels at 8 km/h downstream and 6 km/h upstream. The speed of the stream is:

  • A1 km/h
  • B2 km/h
  • C3 km/h
  • D4 km/h
View solution
Correct Option: A
Stream speed = (8 − 6)/2 = 1 km/h.
Q 15 Time-Work Medium

A finishes a task in 10 days, B in 15 days. Together they finish in:

  • A5 days
  • B6 days
  • C7 days
  • D8 days
View solution
Correct Option: B
10 × 15 / (10 + 15) = 150/25 = 6 days.
Q 16 Pipes Hard

Tap A fills a tank in 6 hr; tap B empties it in 12 hr. Both open. Time to fill the tank:

  • A8 hr
  • B10 hr
  • C12 hr
  • D18 hr
View solution
Correct Option: C
Net rate = 1/6 − 1/12 = 1/12. Time = 12 hr.
Q 17 Permutation Medium

The number of arrangements of 5 books on a shelf is:

  • A25
  • B60
  • C100
  • D120
View solution
Correct Option: D
5! = 120.
Q 18 Combination Medium

In how many ways can a committee of 2 be chosen from 5 people?

  • A5
  • B10
  • C15
  • D20
View solution
Correct Option: B
⁵C₂ = 5!/(2!×3!) = 10.
Q 19 Probability Medium

The probability of getting an odd number on a single roll of a fair die is:

  • A1/6
  • B1/3
  • C1/2
  • D2/3
View solution
Correct Option: C
Odd numbers on die: 1, 3, 5 → 3/6 = 1/2.
Q 20 Clock Hard

The angle between the hour and minute hands at 3:00 is:

  • A60°
  • B75°
  • C90°
  • D120°
View solution
Correct Option: C
Formula |30H − 5.5M| = |30×3 − 0| = 90°.

21.14 Quick Recall

ImportantQuick recall
  • Number system: N · W · Z · Q · Irrational · R · Prime · Composite.
  • HCF × LCM = product of two numbers.
  • Percentage: x % = x/100. Successive a%, b% → net = a + b + ab/100.
  • Fraction-percent equivalents: 1/2=50% · 1/3=33⅓% · 1/4=25% · 1/5=20% · 1/6=16⅔% · 1/8=12½% · 1/10=10%.
  • Ratio & Proportion: a×d = b×c. Mean prop = √(ab); Third prop = b²/a; Fourth prop = bc/a.
  • Direct prop: y ↑ as x ↑. Inverse prop: xy = constant.
  • Partnership profit ratio = (P × t) per partner.
  • Profit/Loss: P% = (P/CP)×100. Equal-SP trap: loss = (x²/100)%.
  • SP = CP(100±x)/100. CP = SP×100/(100+x).
  • SI = PRT/100. CI = P(1+R/100)^T − P.
  • CI−SI for 2 years = P(R/100)².
  • Banker’s discount > True discount.
  • Average: sum/n. Avg of first n naturals = (n+1)/2.
  • Equal-distance avg speed (harmonic mean) = 2ab/(a+b).
  • TSD conversion: km/h × 5/18 = m/s.
  • Relative speed: opposite = sum; same direction = difference.
  • Train + pole: L/v. Train + platform: (L+P)/v.
  • Boat: boat = (down+up)/2; stream = (down−up)/2.
  • Time-work: combined = ab/(a+b). Pipes: net = sum (negative for empties).
  • Permutation ⁿPᵣ = n!/(n−r)!. Combination ⁿCᵣ = n!/[r!(n−r)!]. Circular = (n−1)!.
  • Probability: P(A or B) = P(A)+P(B)−P(A&B). Independent: P(A&B) = P(A)P(B).
  • Leap year: divisible by 4; centenary divisible by 400.
  • Clock angle = |30H − 5.5M|°. Hands coincide 22 times in 12 hrs; perpendicular 44 times.
  • Mensuration: Square a², Rectangle l×b, Triangle ½bh, Circle πr², Sphere 4πr² / (4/3)πr³, Cylinder πr²h, Cone (1/3)πr²h, Cube a³.