Basic numeracy covers percentages, ratios, profit-loss, time-speed-distance, averages, and simple/compound interest. These follow standard formulae — know them cold, practise mental calculations, and learn shortcuts to save time under exam conditions.
Percentages
x% of y = (x × y) / 100 | % change = (Change / Original) × 100
If A is x% more than B: A = B × (1 + x/100)
If A is x% less than B: A = B × (1 − x/100)
Successive changes: +a% then +b% → net change = a + b + ab/100
- Fraction-Percentage equivalents (memorise): 1/8=12.5%, 1/6=16.67%, 1/5=20%, 1/4=25%, 1/3=33.33%, 3/8=37.5%, 1/2=50%, 2/3=66.67%, 3/4=75%
- If price increases by r%, quantity must decrease by r/(100+r) × 100% to keep expenditure constant
Ratios and Proportions
If a:b = m:n, then a = km, b = kn for some k
Componendo: (a+b)/(a−b) = (m+n)/(m−n)
If a/b = c/d, then (a+c)/(b+d) = a/b = c/d (Alternendo/Componendo)
- Mixture problems: use alligation to find ratio of mixing two quantities
- Alligation: |Higher price − Mean price| : |Mean price − Lower price| = parts of lower : parts of higher
Profit and Loss
Profit % = (Profit / CP) × 100 | Loss % = (Loss / CP) × 100
SP = CP × (100 + P%) / 100 | CP = SP × 100 / (100 + P%)
Discount % = (Discount / Marked Price) × 100
If shopkeeper uses false weights (claims W but gives W−e): Profit % = e / (W−e) × 100
Time, Speed and Distance
Distance = Speed × Time | Average Speed = Total Distance / Total Time (NOT average of speeds)
Relative speed: same direction = S₁ − S₂; opposite direction = S₁ + S₂
Train crossing: Time = (Length of train + Length of object) / Speed
Boats: upstream speed = B − R; downstream = B + R (B = boat speed, R = river speed)
- Convert km/hr to m/s: multiply by 5/18 | m/s to km/hr: multiply by 18/5
Simple and Compound Interest
SI = P × R × T / 100 | A = P + SI
CI: A = P × (1 + R/100)ⁿ | CI = A − P
CI − SI (for 2 years) = P × (R/100)²
Effective annual rate for half-yearly compounding: A = P × (1 + R/200)²ⁿ
Averages
Average = Sum of observations / Number of observations
If n numbers, average = A: new number added = x, new average = (nA + x) / (n+1)
Weighted average = (w₁x₁ + w₂x₂ + ...) / (w₁ + w₂ + ...)
Work and Time
If A completes work in a days, A's 1-day work = 1/a
A and B together: 1-day work = 1/a + 1/b; time = ab/(a+b)
Pipes: filling pipe = +; emptying pipe = −; net rate = sum of individual rates
Efficiency ratio = inverse of time ratio (faster worker has higher efficiency)
Key: Average speed = total distance ÷ total time (NOT average of speeds). CI–SI for 2 years = P(R/100)². Alligation for mixture problems. Convert km/hr ↔ m/s using ×5/18 or ×18/5. Successive % change: a + b + ab/100. Work: 1-day work = 1/time; combined = sum of individual rates.