Arithmetic Word Problems Formula Sheet - IBPS PO Prelims Quant...

Chapter 1: Fundamental Percentages & Net Change

Percentage is the core building block of Quantitative Aptitude in the IBPS PO Prelims exam. Mastery over percentage conversions and formula application enables faster computation in Data Interpretation (DI) and standalone arithmetic word problems.

Core Formulas

Percentage Change: To find the percentage increase or decrease relative to an initial value:

\[\text{\% Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\]

Successive Percentage Changes: When a quantity undergoes two successive percentage changes of \(a\%\) and \(b\%\), the net percentage change is given by:

\[\text{Net Change} = ( a + b + \frac{ab}{100} )\%\]

Note: Use positive values for increase/profit and negative values for decrease/discount.

Percentage Point vs. Percentage: A percentage point refers to the absolute arithmetic difference between two percentage values, whereas percentage change measures the relative change.

Chapter 2: Profit, Loss & Discount Strategies

Profit, Loss, and Discount concepts are heavily tested in IBPS PO Prelims, often combined with ratios or conditional word problems.

Basic Equations

Let \(CP\) be Cost Price, \(SP\) be Selling Price, and \(MP\) be Marked Price.

\[\text{Profit} = SP - CP, \quad \text{Loss} = CP - SP\] \[\text{Profit}\% = ( \frac{SP - CP}{CP} ) \times 100\]

Discount & Marked Price Relations

\[\text{Discount} = MP - SP\] \[\text{Discount}\% = ( \frac{MP - SP}{MP} ) \times 100\]

Combined Formula (Relation between MP and CP):

\[SP = MP \times (1 - \frac{D}{100}) = CP \times (1 + \frac{P}{100})\]

Chapter 3: Ratio, Proportion & Partnership

Ratio concepts govern proportionality problems, ages, and business partnerships in banking examinations.

Proportional Variations

Direct Variation: If \(y \propto x\), then \(y = kx\) or \(\frac{y_1}{x_1} = \frac{y_2}{x_2}\).
Inverse Variation: If \(y \propto \frac{1}{x}\), then \(xy = k\) or \(x_1 y_1 = x_2 y_2\).

Partnership Profit Sharing

The distribution of total profit among partners depends on their capital investments and investment periods:

\[\text{Profit Share} \propto \text{Investment} \times \text{Time}\] \[\frac{\text{Profit}_A}{\text{Profit}_B} = \frac{I_A \times T_A}{I_B \times T_B}\]

Chapter 4: Averages, Mixtures & Alligation Rules

Alligation is a shortcut framework used to solve complex mixture problems and weighted averages in seconds during exam pressure.

Weighted Average Formula

\[\text{Avg}_{avg} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}\]

Rule of Alligation

When two ingredients of prices/qualities \(C_1\) and \(C_2\) (where \(C_1 < \text{Mean} < C_2\)) are mixed to yield a mean price \(\text{Mean}\), the ratio of quantities \(Q_1\) and \(Q_2\) is:

\[\frac{Q_1}{Q_2} = \frac{C_2 - \text{Mean}}{\text{Mean} - C_1}\]

Chapter 5: Time, Speed & Distance (Boats & Streams)

Time, Speed, and Distance (TSD) tests candidate speed and relative motion application.

Core Equations & Average Speed

\[\text{Distance} = \text{Speed} \times \text{Time}\]

When equal distances are covered at speeds \(v_1\) and \(v_2\), the harmonic average speed is:

\[S_{avg} = \frac{2 v_1 v_2}{v_1 + v_2}\]

Relative Speed Concepts

Same Direction: Relative Speed = \(S_1 - S_2\) (where \(S_1 > S_2\))
Opposite Direction: Relative Speed = \(S_1 + S_2\)

Boats and Streams

Let downstream speed be \(u\) and upstream speed be \(v\):

\[\text{Speed of Boat in Still Water} = \frac{u + v}{2}\] \[\text{Speed of Current} = \frac{u - v}{2}\]

Chapter 6: Time & Work, Pipes & Cisterns

Work-rate mechanics form a significant portion of word problems in IBPS PO Prelims.

Work Rate Mechanics

If a person completes a total work in \(n\) days, their 1-day efficiency/work rate is \(\frac{1}{n}\).

If two workers take \(x\) and \(y\) days individually, time taken together is:

\[\text{Time}_{together} = \frac{xy}{x + y}\]

M-D-H Chain Rule Formula

For comparing work outputs with varying manpower, days, hours, and efficiencies:

\[\frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2}\]

Pipes & Cisterns

Inlet pipes perform positive work (filling rate), whereas outlet or leak pipes perform negative work (emptying rate).

Chapter 7: Simple & Compound Interest Foundations

Interest problems test your accuracy with exponential and linear rates over multiple time periods.

Simple Interest (SI)

\[SI = \frac{P \times R \times T}{100}\]

Where \(P\) = Principal, \(R\) = Annual Rate of Interest, and \(T\) = Time in years.

IBPS PO Prelims Exam Pattern & Chapter Weightage Analysis

In the IBPS PO Prelims examination, the Quantitative Aptitude section comprises 35 questions to be answered in 20 minutes. Arithmetic Word Problems contribute approximately 10 to 15 questions directly or form the mathematical base for Arithmetic-based Data Interpretation sets.

Topic Weightage Breakdown

Percentages, Profit & Loss: 3-4 Questions
Ratio, Mixture & Alligation: 2-3 Questions
Time & Work, Pipes & Cisterns: 2 Questions
Time, Speed, Distance & Boats: 2-3 Questions
SI & CI: 1-2 Questions

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Arithmetic Word Problems Formula Sheet - IBPS PO Prelims Quantitative Aptitude