Arithmetic, Mensuration, Statistics and Probability

1. Chapter at a Glance

This chapter covers the arithmetic, mensuration, statistics and probability that form the backbone of CDS Elementary Mathematics (60–70 of 100 questions), the whole of AFCAT Numerical Ability (15–20 questions) and about 20–25 questions of NDA Paper I (statistics and probability). The level is Class X for CDS and AFCAT, but questions are time-pressed, so shortcuts and mental calculation matter as much as concepts. Every section below gives the concept, the shortcut and a worked example.

2. Number System, HCF and LCM

• Natural numbers 1, 2, 3...; whole numbers 0, 1, 2...; integers include negatives; rational numbers p/q (q ≠ 0) have terminating or repeating decimals; irrational numbers (√2, π) do not.
• 2 is the only even prime. There are 25 primes below 100.
• Divisibility: by 3 or 9 if the digit sum is divisible by 3 or 9; by 4 if the last two digits form a multiple of 4; by 8 if the last three digits do; by 11 if (sum of odd-place digits − sum of even-place digits) is 0 or a multiple of 11.
• HCF × LCM = product of two numbers. HCF of fractions = HCF of numerators / LCM of denominators; LCM of fractions = LCM of numerators / HCF of denominators.
• Number of factors of N = p^a q^b r^c is (a + 1)(b + 1)(c + 1). Example: 72 = 2³ × 3² has 4 × 3 = 12 factors.
• Unit digit of powers: digits 2, 3, 7, 8 repeat in cycles of 4; 4 and 9 in cycles of 2; 0, 1, 5, 6 never change.
• Converting a recurring decimal: 0.ababab... = ab/99; 0.4777... = (47 − 4)/90 = 43/90.

3. Percentage, Profit and Loss, Discount

• x% of y = y% of x (8% of 50 = 50% of 8 = 4).
• If a value increases by a% and then by b%, the net change is a + b + ab/100 (use negative signs for decreases). A 10% rise followed by a 10% fall gives −1%.
• If the price rises by r%, consumption must fall by r/(100 + r) × 100% to keep expenditure unchanged (a 25% rise needs a 20% cut).
• Profit% = (Profit/CP) × 100; SP = CP × (100 + P%)/100. Discount is always on the marked price: SP = MP × (100 − D%)/100.
• Successive discounts of a% and b% equal a single discount of a + b − ab/100.
• If two articles are sold at the same price, one at x% profit and the other at x% loss, there is always a loss of x²/100 %.
• False weight: a dealer who uses 900 g for 1 kg and sells at cost gains 100/900 = 11.11%.

4. Simple and Compound Interest

• SI = PRT/100. Amount A = P + SI.
• Compound amount A = P(1 + R/100)ᵀ; half-yearly compounding: rate R/2, time 2T.
• Difference between CI and SI: for 2 years = P(R/100)²; for 3 years = P(R/100)²(3 + R/100).
• A sum doubles at SI in T years ⇒ R = 100/T. At CI, if a sum doubles in n years, it becomes 4 times in 2n years and 8 times in 3n years.
• Rule of 72: at CI of R%, money roughly doubles in 72/R years.

5. Ratio, Proportion, Partnership, Mixtures

• If a : b = c : d, then ad = bc. Mean proportional of a and b = √(ab); third proportional = b²/a.
• Partnership: profit is shared in the ratio of (capital × time).
• Alligation: when two items costing c₁ (cheaper) and c₂ (dearer) are mixed to get mean price m, quantity ratio = (c₂ − m) : (m − c₁).
• Repeated replacement: if x litres are removed from a vessel of V litres of pure liquid and replaced by water n times, pure liquid left = V(1 − x/V)ⁿ.

6. Averages and Ages

• Average = sum/number. Average of n consecutive numbers = middle term = (first + last)/2.
• If a new item raises the average of n items by d, the new item = old average + (n + 1)d.
• Average speed for equal distances at speeds x and y = 2xy/(x + y) (not the simple average).
• Age problems: write present ages as variables, translate each sentence into an equation; check the answer against all statements.

7. Time and Work, Pipes and Cisterns

• If A does a job in a days and B in b days, together they take ab/(a + b) days.
• LCM method: take total work = LCM of the days; find each person's per-day work; add or subtract efficiencies.
• M₁D₁H₁/W₁ = M₂D₂H₂/W₂ (men, days, hours, work).
• Pipes: filling pipes are positive, emptying pipes negative. A fills in 10 h, B empties in 15 h ⇒ net = 1/10 − 1/15 = 1/30, i.e. 30 hours.

8. Time, Speed and Distance

• Distance = speed × time. 1 km/h = 5/18 m/s; 1 m/s = 18/5 km/h.
• Relative speed: opposite directions add, same direction subtract.
• A train crossing a pole covers its own length; crossing a platform or another train covers the sum of the lengths.
• Boats: downstream speed = u + v, upstream = u − v; boat speed u = (D + U)/2, stream v = (D − U)/2.
• If a person travels at 3/4 of the usual speed and is late by t minutes, the usual time = 3t (generally, usual time = t × new speed/(usual speed − new speed)).

9. Mensuration

10. Elementary Geometry (CDS)

• Sum of interior angles of an n-gon = (n − 2) × 180°; each exterior angle of a regular n-gon = 360°/n.
• Angle in a semicircle is 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°.
• Tangent ⊥ radius at the point of contact; tangents from an external point are equal.
• Pythagorean triplets: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25).
• Similar triangles: ratio of areas = square of the ratio of corresponding sides.
• In a triangle, the centroid divides each median 2 : 1; the circumcentre of a right triangle is the midpoint of the hypotenuse.

11. Statistics

• Mean x̄ = Σx/n (grouped: Σfx/Σf). Median = middle value of ordered data; mode = most frequent value.
• Empirical relation: Mode = 3 Median − 2 Mean.
• Variance σ² = Σ(x − x̄)²/n = Σx²/n − x̄²; standard deviation σ = √variance.
• Mean and SD of first n natural numbers: (n + 1)/2 and √[(n² − 1)/12].
• Effect of change: adding a constant to every observation changes the mean but not the variance or SD; multiplying by k multiplies the mean and SD by k and the variance by k².
• Coefficient of variation = (σ/x̄) × 100 — the series with the smaller CV is more consistent.
• Graphs: histogram (continuous classes), bar chart (discrete), ogive (cumulative frequency, used to find median), pie chart (each sector = value/total × 360°).

12. Probability

• P(E) = favourable outcomes / total outcomes; 0 ≤ P(E) ≤ 1; P(E′) = 1 − P(E).
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Mutually exclusive: P(A ∩ B) = 0. Independent: P(A ∩ B) = P(A)P(B).
• Conditional probability P(A|B) = P(A ∩ B)/P(B).
• Two dice: 36 outcomes; most likely sum is 7 (6 ways). A deck has 52 cards: 4 suits × 13, 12 face cards, 4 aces.
• Binomial distribution: P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ; mean np, variance npq.
• Odds in favour of E = P(E) : P(E′).

13. Data Interpretation and Quick Approximation

CDS Elementary Mathematics and the AFCAT Numerical Ability section regularly give a small table, bar chart or pie chart and ask three or four questions on it. These questions use only percentages, ratios and averages, but they reward a calm method. Read the title, the units (thousands, lakhs, crores) and the base year before touching any number.

Worked table. The number of candidates (in thousands) who appeared for a defence entrance test in four years is: 2022 — 400, 2023 — 450, 2024 — 540, 2025 — 600.

Approximation. In a timed paper, round numbers sensibly before calculating and then pick the nearest option. For example, 19.98% of 501 ≈ 20% of 500 = 100. Also, 4,987 ÷ 49.6 ≈ 5,000 ÷ 50 = 100, and √1,230 ≈ 35 because 35² = 1,225. Round one number up and the other down when multiplying so that the errors cancel. Approximation is safe only when the options are spread far apart; if two options are close, calculate exactly.

14. Master Formula Table

15. Master Summary Table

16. Mind Map

17. Tips and Tricks

18. Important One-Liners

• 1 is neither prime nor composite; 2 is the smallest and only even prime.
• The product of two numbers equals the product of their HCF and LCM.
• A number is divisible by 6 if it is divisible by both 2 and 3.
• Discount is always calculated on the marked price, profit on the cost price.
• Simple interest is the same every year; compound interest grows every year.
• 1 km/h = 5/18 m/s.
• Relative speed is the sum of speeds for opposite directions.
• Average speed for equal distances is the harmonic mean of the speeds.
• The volume of a cone is one-third that of a cylinder with the same base and height.
• The angle in a semicircle is a right angle.
• Each interior angle of a regular hexagon is 120°.
• The median is not affected by extreme values; the mean is.
• Adding a constant to all observations does not change the standard deviation.
• Probability of an impossible event is 0 and of a sure event is 1.
• For two dice, the probability of getting a sum of 7 is 1/6.
• Mean of a binomial distribution is np.

19. PYQ-Based Questions with Detailed Answers

20. Quick Revision Summary