Mathematics

Arithmetic — The Highest-Scoring Section

Arithmetic carries the maximum weight in SSC, RRB Group D, NTPC, Banking and Insurance exams — typically 12–18 marks. The seven core sub-topics below (Percentage, Profit & Loss, Ratio, Average, Time & Work, Time-Speed-Distance, and Interest) are the 'rank-makers'. Master the formulas and shortcuts here and your accuracy will improve across the entire paper.

Exam relevance: RRB Group D / NTPC: 8–12 questions out of 25 are pure arithmetic. SSC CGL Tier-I: 6–8 questions. Bank PO/Clerk: 5–10 questions in Quant + Data Sufficiency.

On this page

  1. Percentage
  2. Profit & Loss (CP, SP, Discount, Markup)
  3. Ratio & Proportion (Partnership, Ages)
  4. Average
  5. Time & Work (Pipes & Cisterns)
  6. Time, Speed & Distance (Trains, Boats & Streams)
  7. Simple Interest & Compound Interest

1Percentage

A percentage is a ratio expressed out of 100; x% = x/100.

Convert percentages to fractions for speed: 12.5% = 1/8, 16.67% = 1/6, 20% = 1/5, 25% = 1/4, 33.33% = 1/3, 50% = 1/2, 66.67% = 2/3, 75% = 3/4. Successive percentage change formula: net % = a + b + (ab/100), with positive for increase and negative for decrease. To find the original after a known percentage change: Original = Final × 100 / (100 ± change).

Examples
  • What is 35% of 240? = (35/100) × 240 = 84.
  • If price is increased by 20% and then decreased by 20%, net change = +20 − 20 + (20×−20)/100 = −4% (net loss of 4%).
  • If 60 is 30% of x, then x = 60 × 100/30 = 200.
  • A's salary is 25% more than B's. Then B's salary is 20% less than A's (use 25/(100+25)).
Exam tip: When a value increases by x% and then decreases by x%, the result is always less than the original — by x²/100 percent.

2Profit & Loss (CP, SP, Discount, Markup)

Cost Price (CP) is what the seller pays; Selling Price (SP) is what the customer pays; Marked Price (MP) is the labelled price before discount.

Profit% = (SP − CP)/CP × 100. Loss% = (CP − SP)/CP × 100. Discount% = (MP − SP)/MP × 100. Markup% = (MP − CP)/CP × 100. SP = CP × (100 + Profit%)/100. When two articles are sold at the same SP, one at x% gain and the other at x% loss, there is always a net loss of x²/100 %.

Examples
  • CP = ₹400, SP = ₹500 → Profit% = 100/400 × 100 = 25%.
  • MP = ₹1,200, Discount = 20% → SP = 1200 × 80/100 = ₹960.
  • CP = ₹500, Markup = 40%, Discount = 25% → MP = 700, SP = 700 × 0.75 = ₹525, Profit = 5%.
  • Two pens sold at ₹120 each — one at 20% gain, other at 20% loss → net loss = 4% of total CP.
Exam tip: Always express profit/loss as a percentage of CP, never of SP. A common trap in MCQs is calculating against SP.

3Ratio & Proportion (Partnership, Ages)

A ratio compares two quantities of the same kind; a proportion is an equality of two ratios (a:b = c:d).

Properties: a:b = ka:kb. If a:b and b:c are given, then a:b:c = a×c : b×c : b×b/... use LCM/cross-multiplication carefully. In partnership, profit is shared in the ratio of capital × time. Age problems: 'x years ago' means subtract x from current ages; 'after y years' means add y. Set up one equation per condition and solve.

Examples
  • If a:b = 2:3 and b:c = 4:5, then a:b:c = 8:12:15.
  • A invests ₹6,000 for 8 months and B invests ₹4,000 for 12 months. Profit ratio = (6000×8) : (4000×12) = 48000:48000 = 1:1.
  • Father is 4 times son's age; after 5 years he will be 3 times. Now F = 4S, F+5 = 3(S+5) → S=10, F=40.
  • Divide ₹1,500 in ratio 2:3:5 → shares are ₹300, ₹450, ₹750.
Exam tip: If 'a:b = c:d', cross-multiplying gives a×d = b×c. Use this to solve any one unknown directly.

4Average

Average = Sum of observations ÷ Number of observations.

If average of n items is A, total = nA. When a new item is added, the new average shifts by (new − old average)/(n+1). Average speed for equal distances at speeds u and v = 2uv/(u+v) (harmonic mean). Average of consecutive numbers from a to b = (a + b)/2. Replacing/adding/removing a member: change in total = change in count × average shift.

Examples
  • Average of 5 numbers is 30 → sum = 150. If a 6th number 42 is added, new average = 192/6 = 32.
  • Average of first 50 natural numbers = (1+50)/2 = 25.5.
  • A car covers 60 km at 30 km/h and another 60 km at 60 km/h → average speed = 2(30)(60)/(30+60) = 40 km/h.
  • Average age of 4 boys = 16. A 5th boy joins and the average becomes 17 → his age = 17×5 − 16×4 = 21.
Exam tip: For average speed of unequal distances at known speeds, use Total Distance / Total Time, not the harmonic-mean shortcut.

5Time & Work (Pipes & Cisterns)

Work done = Rate × Time. If A finishes a job in n days, A's 1-day work = 1/n.

Combined rate: if A and B finish a job in a and b days, together they finish in ab/(a+b) days. For pipes & cisterns, an inlet pipe is positive work and an outlet pipe is negative work. Use the LCM-of-days method: assume total work = LCM(individual times) units, compute units/day for each worker, then add. Efficiency ∝ 1/Time.

Examples
  • A does a job in 12 days, B in 18 days. Together: ab/(a+b) = 216/30 = 7.2 days.
  • LCM method: total = 36 units. A = 3 u/day, B = 2 u/day → together 5 u/day → 36/5 = 7.2 days.
  • Pipe A fills tank in 6 h, pipe B empties in 12 h. Net rate = 1/6 − 1/12 = 1/12 per hour → 12 hours to fill.
  • If 6 men or 8 women complete a work in 10 days, 3 men + 4 women take 10 days too (same composite rate).
Exam tip: When efficiencies are given as a ratio, treat the ratio as units of work per day directly — no need to convert to days first.

6Time, Speed & Distance (Trains, Boats & Streams)

Distance = Speed × Time. Convert km/h to m/s by multiplying by 5/18.

Relative speed: same direction → |u − v|; opposite direction → u + v. Train crossing a stationary object (pole, tree, man): time = length/speed. Train crossing a platform/bridge: time = (train length + platform length)/speed. Trains crossing each other: time = sum of lengths / relative speed. Boats & streams: downstream speed = (boat + stream); upstream = (boat − stream); boat speed = (down+up)/2; stream speed = (down−up)/2.

Examples
  • A 200 m train at 72 km/h (= 20 m/s) crosses a 300 m platform in (200+300)/20 = 25 s.
  • Two trains 120 m & 150 m long, speeds 54 & 36 km/h opposite direction → relative = 90 km/h = 25 m/s; cross time = 270/25 = 10.8 s.
  • Boat speed in still water = 8 km/h, stream = 2 km/h → downstream 10 km/h, upstream 6 km/h.
  • If a man rows 24 km downstream in 3 h and 24 km upstream in 6 h → boat = (8+4)/2 = 6 km/h, stream = (8−4)/2 = 2 km/h.
Exam tip: Always convert to consistent units (m/s with metres-and-seconds, km/h with km-and-hours) before applying formulas.

7Simple Interest & Compound Interest

Simple Interest is calculated on the original principal only. Compound Interest is calculated on the principal plus accumulated interest.

Simple Interest (SI) = P × R × T / 100. Amount = P + SI. Compound Interest (CI), compounded annually: A = P × (1 + R/100)ᵀ; CI = A − P. For half-yearly: rate becomes R/2, time 2T. For quarterly: R/4, 4T. Difference between CI and SI for 2 years = P × (R/100)². For 3 years = P × (R/100)² × (3 + R/100).

Examples
  • P = ₹10,000, R = 10%, T = 2 yrs → SI = 10000×10×2/100 = ₹2,000.
  • Same data, CI compounded annually: A = 10000 × 1.21 = ₹12,100; CI = ₹2,100.
  • Difference (CI − SI) for 2 yrs = 10000 × (10/100)² = ₹100. ✓
  • Sum doubles in 8 years at SI → R = 100/8 = 12.5%.
Exam tip: For SI, money doubles in (100/R) years. For CI at R%, use Rule of 72: doubling time ≈ 72/R years.

Short Tricks & Shortcuts

Use these speed tricks in the exam. Each trick is followed by a worked example so you can verify the shortcut yourself.

Trick 1Memorise percentage-to-fraction conversions

Convert percentages to fractions for instant calculation: 12.5%=1/8, 16.66%=1/6, 20%=1/5, 25%=1/4, 33.33%=1/3, 50%=1/2, 66.67%=2/3, 75%=3/4.

Example: What is 12.5% of 480? = 480/8 = 60. (Skip the multiplication entirely.)
Trick 2Net % change for two successive changes

Net % = a + b + (a × b)/100, with + for increase and − for decrease.

Example: Price up 20% then down 10% → 20 + (−10) + (20×−10)/100 = 8% net increase.
Trick 3Same SP, x% gain and x% loss → always loss

When two articles are sold at the same SP, one at x% profit and the other at x% loss, the net loss = x²/100 % of the total CP.

Example: Two pens at ₹120 each, +20% and −20% → net loss = 400/100 = 4%.
Trick 4Time & Work LCM-units shortcut

Set total work = LCM of given days. Compute units/day for each worker. Add the units/day to find combined rate. Total ÷ combined rate = days.

Example: A=12 days, B=18 days → total = 36 units; A=3, B=2 u/day; combined=5 → 36/5 = 7.2 days.
Trick 5CI − SI difference shortcuts

For 2 years: difference = P × (R/100)². For 3 years: P × (R/100)² × (3 + R/100). Skip computing CI and SI separately.

Example: P=₹10,000, R=10%, 2 yrs → CI − SI = 10000 × (0.1)² = ₹100.
Trick 6Average speed for equal distances

If equal distances are covered at speeds u and v, average speed = 2uv/(u+v) (harmonic mean) — NEVER (u+v)/2.

Example: 60 km at 30 km/h, 60 km at 60 km/h → avg = 2(30)(60)/(30+60) = 40 km/h.
Trick 7Train + platform crossing time

Time to cross a platform = (train length + platform length) / speed. Convert km/h to m/s with × 5/18 first.

Example: 200 m train at 72 km/h (=20 m/s) crossing 300 m platform → (200+300)/20 = 25 s.
Trick 8Boat & stream — boat-and-stream split

Boat speed in still water = (downstream + upstream)/2; Stream speed = (downstream − upstream)/2.

Example: Down 10 km/h, Up 6 km/h → Boat = 8 km/h, Stream = 2 km/h.

Quick Revision Facts

  • x% of y = y% of x. Use this to flip awkward calculations.
  • Successive discounts of a% and b% = a + b − ab/100 (always less than a+b).
  • Profit% on SP can never exceed 100%; on CP it can.
  • Average speed = Total distance / Total time — always.

Frequently Asked Questions

Because the profit is calculated on a smaller CP and the loss on a larger CP, the absolute loss exceeds the absolute profit. Net effect = x²/100 % loss on the total CP.

Memorise SI and CI formulas — they appear in 90% of interest problems unchanged. Only the difference formulas (CI − SI for 2 and 3 years) save real time and are worth knowing by heart.
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