Arithmetic – IBPS PO Prelims Study Notes

Overview

Arithmetic forms the backbone of the Quantitative Aptitude section in IBPS PO Prelims. Roughly 10–15 questions out of 35 in this section are direct arithmetic applications or appear embedded within Data Interpretation sets. Mastery here is non-negotiable—these topics recur in every banking exam and offer quick marks if your fundamentals are solid.

The arithmetic umbrella covers Percentage, Ratio and Proportion, Average, Profit-Loss-Discount, Simple and Compound Interest, Time and Work, Time-Speed-Distance, Mixture and Alligation, Partnership, and Problems on Ages. These topics are deeply interconnected: a single question might combine percentage with ratio, or profit-loss with partnership. Your goal is fluency—solving standard problems in under 60 seconds through shortcuts and mental math.

Speed comes from recognizing problem patterns instantly and applying the right formula without hesitation. This guide distills the must-know concepts, formulas, and traps for each arithmetic sub-topic.

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Key Concepts

• **Percentage is a ratio out of 100.** Every percentage problem converts to a fraction or decimal for calculation. Know fraction equivalents cold: 25% = 1/4, 33.33% = 1/3, 12.5% = 1/8.

• **Ratio compares quantities; proportion equates two ratios.** When a ratio is given as a : b, actual values are ax and bx where x is the multiplier you must find.

• **Average = Sum ÷ Count.** Weighted average applies when groups have different sizes. The average of an arithmetic progression equals the middle term.

• **Profit and Loss are always calculated on Cost Price (CP).** Discount is always on Marked Price (MP). Mixing these up is the #1 error.

• **Simple Interest grows linearly; Compound Interest grows exponentially.** For 2-year problems, CI − SI = P × (R/100)².

• **Work problems are rate problems.** If A completes work in 10 days, A's rate = 1/10 work per day. Add rates when working together.

• **Speed = Distance ÷ Time.** Relative speed adds when objects move toward each other, subtracts when moving in the same direction.

• **Alligation links the mean price to component prices.** The criss-cross rule gives the ratio of quantities directly.

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Formulas / Key Facts

### Percentage

• Percentage change = (Change / Original) × 100
• If a value increases by R%, multiply by (1 + R/100); if decreases, multiply by (1 − R/100)
• Successive changes: Net effect of a% then b% = a + b + (ab/100)

### Ratio and Proportion

• If a : b = c : d, then ad = bc (cross-multiplication)
• Componendo-Dividendo: If a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d)

### Average

• Average = Sum / n
• Weighted Average = (w₁×x₁ + w₂×x₂ + ...) / (w₁ + w₂ + ...)
• New average when one item added/removed: Adjust sum accordingly, then divide

### Profit, Loss and Discount

• Profit % = (Profit / CP) × 100
• SP = CP × (1 + Profit%/100) or CP × (1 − Loss%/100)
• Discount % = (Discount / MP) × 100
• SP = MP × (1 − Discount%/100)
• If two successive discounts are d₁% and d₂%, effective discount = d₁ + d₂ − (d₁×d₂/100)

### Simple and Compound Interest

• SI = P × R × T / 100
• CI = P × (1 + R/100)^T − P
• For 2 years: CI − SI = P × (R/100)²
• For 3 years: CI − SI = P × (R/100)² × (3 + R/100)

### Time and Work

• Total work = Rate × Time (use LCM of days as total work units)
• If A and B together finish in T days: 1/T = 1/A + 1/B
• Pipes and Cisterns: Inlet adds, Outlet subtracts

### Time, Speed and Distance

• Speed = Distance / Time
• Relative speed (opposite directions) = S₁ + S₂
• Relative speed (same direction) = |S₁ − S₂|
• Train crossing a pole: Distance = Length of train
• Train crossing a platform: Distance = Length of train + Length of platform

### Mixture and Alligation

• Alligation rule: (Cheaper quantity) : (Dearer quantity) = (Dearer price − Mean) : (Mean − Cheaper price)
• Replacement formula: After n operations, remaining = Initial × (1 − R/V)ⁿ, where R = quantity replaced, V = total volume

### Partnership

• Profit share ∝ Capital × Time
• If A invests C₁ for T₁ months and B invests C₂ for T₂ months: A's share : B's share = C₁T₁ : C₂T₂

### Problems on Ages

• Set up linear equations using the conditions given (present, past, future ages)
• Ratios of ages change over time—track carefully

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Worked Examples

**Example 1 – Percentage** *A shopkeeper marks goods 25% above CP and gives a 10% discount. Find profit %.*

SP = CP × 1.25 × 0.90 = CP × 1.125 Profit % = (1.125 − 1) × 100 = **12.5%**

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**Example 2 – Time and Work** *A can do a job in 12 days, B in 15 days. They work together for 4 days; A leaves. How many more days does B take?*

Total work = LCM(12, 15) = 60 units A's rate = 5 units/day, B's rate = 4 units/day Together in 4 days = (5 + 4) × 4 = 36 units Remaining = 60 − 36 = 24 units B alone = 24 / 4 = **6 days**

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**Example 3 – Mixture and Alligation** *Milk at ₹60/litre is mixed with water (free) to get a mixture worth ₹45/litre. Find the ratio of milk to water.*

Using alligation: Milk : Water = (45 − 0) : (60 − 45) = 45 : 15 = **3 : 1**

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Common Mistakes

• **Confusing base for percentage calculations** → Profit/Loss is always on CP; Discount is always on MP. Write the formula before substituting.

• **Adding percentages directly for successive changes** → Two successive 10% increases ≠ 20%. Use the formula a + b + ab/100 = 10 + 10 + 1 = 21%.

• **Forgetting to equalize time in partnership problems** → Always convert to same unit (months). A 6-month investment and a 12-month investment are not comparable directly.

• **Ignoring the direction in relative speed** → Same direction = subtract speeds; Opposite direction = add. Visualize the motion before calculating.

• **Mixing up SI and CI formulas for short durations** → For 2-year CI, don't just double the SI. CI = SI + extra interest on first year's interest.

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Quick Reference

• Profit % = (SP − CP) / CP × 100 — always on CP
• SI = PRT/100; CI − SI (2 yrs) = PR²/10000
• Combined work rate = Sum of individual rates
• Alligation gives quantity ratio directly via criss-cross
• Train problems: Add lengths when crossing another train or platform
• Successive % change: a + b + ab/100