Arithmetic — Study Notes for IBPS Clerk (Prelims)
Overview
These questions test your ability to apply fundamental mathematical concepts to real-world scenarios—calculating interest, splitting profits, finding ages, or measuring speeds.
Unlike Data Interpretation, arithmetic questions are standalone and can be solved quickly if your basics are strong. The Clerk-level arithmetic is moderate in difficulty—calculations are straightforward, and most problems can be solved within 45–60 seconds using shortcuts. Mastering this topic is non-negotiable for clearing the sectional cutoff.
The key sub-topics you must prepare are: Percentage, Profit & Loss, Simple & Compound Interest, Ratio & Proportion, Average, Time & Work, Time Speed & Distance, Partnership, Ages, and Mixture & Alligation. Pattern recognition and formula recall are your best friends here.
Key Concepts
- Percentage is the foundation: Almost every arithmetic topic—profit/loss, interest, partnership—uses percentage calculations. Master fraction-to-percentage conversions first.
- Ratio links quantities: Ratios compare two quantities; proportions state that two ratios are equal. Always reduce ratios to simplest form before solving.
- Average = Sum ÷ Count: The weighted average formula is essential when groups have different sizes.
- Profit/Loss is always on Cost Price: Unless stated otherwise, profit% and loss% are calculated on CP, not SP.
- Interest problems differentiate SI and CI: SI grows linearly; CI grows exponentially. For 2-year problems, CI – SI = P × (R/100)².
- Time and Work uses the "1/n" concept: If A completes work in n days, A's one-day work = 1/n. Use LCM method for faster calculations.
- Speed, Time, Distance are inversely/directly related: Distance = Speed × Time. Relative speed applies when objects move toward or away from each other.
- Alligation gives the mixing ratio: It finds the ratio in which two items at different values must be mixed to get a desired average value.
Formulas / Key Facts
Percentage
- x% of y = y% of x = (x × y)/100
- % change = (Change / Original) × 100
- Successive % change: Net = a + b + (ab/100)
Profit and Loss
- Profit% = (Profit / CP) × 100
- SP = CP × (100 + Profit%) / 100
- Discount% = (Discount / MP) × 100
- SP = MP × (100 – Discount%) / 100
Simple Interest
- SI = (P × R × T) / 100
- Amount = P + SI
Compound Interest
- A = P × (1 + R/100)^T
- CI = A – P
- For 2 years: CI – SI = P × (R/100)²
Ratio and Proportion
- If a:b = c:d, then ad = bc (cross multiplication)
- Componendo: (a+b)/b = (c+d)/d
Average
- Average = Sum of observations / Number of observations
- Weighted Average = (Σ weight × value) / Σ weight
Time and Work
- Combined work rate: 1/A + 1/B = 1/Total days together
- LCM Method: Take LCM of days as total work units
Time, Speed and Distance
- Speed = Distance / Time
- Average speed (two equal distances) = 2ab / (a+b)
- Relative speed (same direction) = |S1 – S2|
- Relative speed (opposite direction) = S1 + S2
Partnership
- Profit share ∝ Capital × Time
- A's share : B's share = (C1 × T1) : (C2 × T2)
Problems on Ages
- Set up linear equations; past = subtract years, future = add years
Mixture and Alligation
- Ratio = (Higher value – Mean) : (Mean – Lower value)
Worked Examples
Example 1: Profit and Loss
A shopkeeper marks an article 25% above CP and gives 10% discount. Find profit%.
Step 1: Let CP = 100 Step 2: MP = 100 + 25% of 100 = 125 Step 3: SP = 125 – 10% of 125 = 125 – 12.5 = 112.5 Step 4: Profit = 112.5 – 100 = 12.5 Step 5: Profit% = (12.5/100) × 100 = 12.5%
Example 2: Time and Work (LCM Method)
A can do a work in 12 days, B in 15 days. In how many days can they complete it together?
Step 1: LCM of 12 and 15 = 60 (total work units) Step 2: A's efficiency = 60/12 = 5 units/day Step 3: B's efficiency = 60/15 = 4 units/day Step 4: Combined efficiency = 5 + 4 = 9 units/day Step 5: Time = 60/9 = 6⅔ days or 6 days 16 hours
Example 3: Mixture and Alligation
Milk at ₹40/litre is mixed with water (free) to get a mixture worth ₹32/litre. Find the ratio of milk to water.
Using Alligation:
- Milk price = 40, Water price = 0, Mean = 32
- Ratio = (40 – 32) : (32 – 0) = 8 : 32 = 1 : 4
Wait—this gives water more than milk. Let's verify: Ratio of Milk : Water = (32 – 0) : (40 – 32) = 32 : 8 = 4 : 1 ✓
Common Mistakes
- Calculating profit% on SP instead of CP → Always use CP as the base unless the question explicitly says otherwise.
- Confusing SI and CI formulas for 2+ years → SI uses simple multiplication (P×R×T/100); CI requires the power formula. Don't interchange.
- Forgetting to convert units → Speed in km/hr and time in minutes will give wrong answers. Convert to consistent units first.
- Using wrong relative speed formula → Same direction: subtract speeds. Opposite direction: add speeds. Visualize the scenario.
- Ignoring negative age solutions → When solving age equations, reject negative values—ages cannot be negative.
- Mixing up MP, CP, and SP in discount problems → Remember: Discount is on MP, Profit/Loss is on CP.
Quick Reference
- Successive % change: a + b + ab/100 — memorize this for two consecutive changes.
- 1/6 = 16.67%, 1/7 ≈ 14.28%, 1/8 = 12.5% — learn fraction-percentage pairs up to 1/15.
- CI – SI for 2 years = P × (R/100)² — instant formula, no derivation needed.
- Average speed for equal distances = 2S1S2 / (S1 + S2) — not the simple average.
- LCM method beats equation method in Time & Work — always assign total work = LCM of days.
- Alligation cross method: Write values in X pattern, subtract diagonally, ratio emerges directly.