Basic Numeracy and Data Interpretation

Number system

• Types of number: natural, whole, integers, rational (p/q), irrational (√2, π) and real numbers.
• Prime numbers: 2 is the only even prime. There are 25 primes below 100.
• Divisibility rules:
  • 3 or 9: the sum of the digits is divisible by 3 or 9;
  • 4: the last two digits are divisible by 4;
  • 8: the last three digits are divisible by 8;
  • 11: the difference between the sums of alternate digits is 0 or a multiple of 11.
• HCF and LCM: HCF × LCM = product of two numbers.
• Unit digit cycles: powers of 2 end in 2, 4, 8, 6; of 3 in 3, 9, 7, 1; of 7 in 7, 9, 3, 1; of 8 in 8, 4, 2, 6.
• Orders of magnitude:
  • 1 lakh = 10⁵;
  • 1 crore = 10⁷;
  • 1 million = 10⁶;
  • 1 billion = 10⁹ = 100 crore.

Percentages

• x% of y = xy/100.
• Successive change: net % = a + b + ab/100. Use a minus sign for a decrease. For example, +20% then −20% gives −4%.
• If A is r% more than B, then B is r/(100 + r) × 100 % less than A.
• Common fraction–percentage pairs:
  • ½ = 50%
  • ⅓ = 33⅓%
  • ¼ = 25%
  • ⅕ = 20%
  • ⅙ = 16⅔%
  • ⅛ = 12½%
  • ⅑ = 11.1%
  • ¹⁄₁₁ = 9.09%

Profit and loss, discount

• Profit % = (profit ÷ cost price) × 100.
• Selling price = CP × (100 + profit %)/100.
• Discount is calculated on the marked price.
• Successive discounts of a% and b% = a + b − ab/100.

Ratio, proportion and averages

• If a : b = 3 : 2, then a = 3k and b = 2k.
• Average = sum ÷ count.
• Weighted average = Σ(w·x) ÷ Σw.
• Mixtures (alligation): (cheaper : dearer) = (d − m) : (m − c), where m is the mean price.

Interest

• Simple interest: SI = PRT/100.
• Compound interest: A = P(1 + R/100)ⁿ, and CI = A − P.
• For 2 years: CI − SI = P(R/100)².
• Rule of 72: money doubles in about 72/R years.

Time, work and speed

• Work: if A does a job in a days and B in b days, together they take ab/(a + b) days.
• Pipes and cisterns: inlets are positive, outlets negative.
• Speed = distance ÷ time.
  • To convert km/h to m/s, multiply by ⁵⁄₁₈. To convert m/s to km/h, multiply by ¹⁸⁄₅.
  • Average speed for equal distances = 2xy/(x + y).
• Trains:
  • To pass a pole, distance = length of the train.
  • To pass a platform, distance = train + platform.
  • Relative speed = sum of speeds (opposite directions) or difference (same direction).
• Boats:
  • downstream speed = u + v;
  • upstream speed = u − v.

Basic mensuration and geometry

Shape	Area	Perimeter / volume
Rectangle	l × b	2(l + b)
Square	a²	4a
Triangle	½ × base × height	—
Circle	πr²	2πr
Cube	surface 6a²	volume a³
Cuboid	surface 2(lb + bh + hl)	volume lbh
Cylinder	curved surface 2πrh	volume πr²h
Sphere	surface 4πr²	volume ⁴⁄₃πr³

Pythagoras: a² + b² = c². Common triples: (3, 4, 5), (5, 12, 13), (8, 15, 17).

Probability and counting

• P(event) = favourable outcomes ÷ total outcomes, from 0 to 1.
• Two dice give 36 outcomes. A sum of 7 is the most likely, at ⁶⁄₃₆ = ⅙.
• Permutation: ⁿPᵣ = n!/(n − r)!.
• Combination: ⁿCᵣ = n!/[r!(n − r)!].

Data interpretation

1. Read the title, units and footnotes first (₹ crore? per cent? thousand tonnes?).
2. Approximate. Options are usually far apart, so round to 2 significant figures.
3. Common questions:
   • % change = (new − old)/old × 100;
   • share = part ÷ total × 100;
   • ratio;
   • CAGR ≈ (end/start)^(1/n) − 1;
   • average.
4. Pie chart: angle = share% × 3.6°.
5. Data sufficiency: test each statement alone, then together. Do not solve fully; only decide whether it can be solved.

Worked example (table):

• Production (lakh tonnes): 2021: 40; 2022: 46; 2023: 50.
• Growth 2021–22 = ⁶⁄₄₀ = 15%. Growth 2022–23 = ⁴⁄₄₆ ≈ 8.7%.
• Average production = 136 ÷ 3 ≈ 45.3 lakh tonnes.

Exam tip: UPSC CSAT has about 25–30 maths and DI questions, and BPSC GS has about 10. Learn the formula sheet above and do 15 timed questions a day. Accuracy matters more than speed because of negative marking.