3.1 Percentages, Profit, Loss & Discount Calculations

Percentages, Profit, Loss & Discount Calculations

Percentages and Profit & Loss form the foundation of Quantitative Aptitude in banking examinations. In the SBI PO exam, direct arithmetic questions and complex Data Interpretation (DI) sets heavily rely on fast percentage computations and profit dynamics. Mastering fraction conversions, successive changes, and dishonest trader concepts is essential to solve questions within tight sectional time limits.

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1. Percentage Fundamentals & Fraction Equivalents

A percentage represents a fraction with a denominator of 100. Converting percentages into simplified fractions enables rapid mental math during calculation-heavy sections.

Essential Fraction-to-Percentage Reference Table

Percentage Increase, Decrease & Product Constancy

1. Percentage Change Formula: $\text{Percentage Change} = \left( \frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}} \right) \times 100$

2. Product Constancy Rule: If $A \times B = C$, and $A$ increases by $\frac{1}{x}$, then to keep $C$ constant, $B$ must decrease by $\frac{1}{x + 1}$.

   • Example: If the price of petrol increases by 25% (which is $\frac{1}{4}$), consumption must decrease by $\frac{1}{4+1} = \frac{1}{5} = 20%$ to keep expenditure unchanged.

3. Successive Percentage Changes: If a value undergoes successive percentage changes of $x%$ and $y%$, the effective net percentage change is: $\text{Net Change \%} = x + y + \frac{xy}{100}$ (Note: Use positive values for increases and negative values for decreases.)

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2. Profit, Loss & Marked Price Core Principles

Understanding financial terminology is critical for solving word problems:

• Cost Price (CP): Price at which an item is purchased, including overhead expenses.
• Selling Price (SP): Price at which an item is sold to a buyer.
• Marked Price (MP): Price printed on the tag (List Price).
• Profit / Loss: Profit occurs when $SP > CP$; Loss occurs when $CP > SP$.
• Discount: Reduction offered on the Marked Price: $\text{Discount} = MP - SP$.

Fundamental Formulas

Golden Rule: Profit and Loss percentages are ALWAYS computed on the Cost Price (CP) unless explicitly instructed otherwise. Discount is ALWAYS computed on the Marked Price (MP).

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3. Step-by-Step Worked Math Examples

Worked Example 1: Net Percentage & Expenditure Adjustments

Problem: The price of sugar rises by 30%. By what percentage should a household reduce its sugar consumption so that total expenditure increases by only 4%?

Step-by-Step Solution:

1. Let initial Price $= P_1 = 100$ and initial Consumption $= C_1 = 100$. Initial Expenditure $= 100 \times 100 = 10,000$.
2. New Price $P_2 = 100 + 30% \text{ of } 100 = 130$.
3. Desired New Expenditure $= 10,000 + 4% \text{ of } 10,000 = 10,400$.
4. Let new consumption be $C_2$. Then $130 \times C_2 = 10,400 \implies C_2 = \frac{10,400}{130} = 80$.
5. Reduction in consumption $= 100 - 80 = 20$.
6. Percentage reduction $= \left(\frac{20}{100}\right) \times 100 = 20%$.

Worked Example 2: Markup, Discount, and CP-MP Ratio

Problem: A trader marks his goods 40% above the cost price and grants a discount of 15% to cash buyers. If he makes a profit of ₹380, find the cost price of the goods.

Step-by-Step Solution:

1. Let Cost Price ($CP$) $= 100x$.
2. Marked Price ($MP$) $= 100x + 40% \text{ of } 100x = 140x$.
3. Discount $= 15% \text{ of } 140x = 0.15 \times 140x = 21x$.
4. Selling Price ($SP$) $= MP - \text{Discount} = 140x - 21x = 119x$.
5. Profit $= SP - CP = 119x - 100x = 19x$.
6. Given Profit $= ₹380 \implies 19x = 380 \implies x = 20$.
7. Therefore, $CP = 100 \times 20 = ₹2,000$.

Worked Example 3: Dishonest Shopkeeper & False Weights

Problem: A merchant claims to sell wheat at a 5% loss on cost price, but uses a false weight that measures 800 grams instead of 1 kilogram. Calculate his net profit percentage.

Step-by-Step Solution:

1. Let the Cost Price of $1,000\text{ g}$ of wheat be ₹$1,000$ (so $1\text{ g} = ₹1$).
2. The merchant claims a 5% loss on CP, so his Selling Price for $1\text{ kg} (1,000\text{ g})$ is ₹$1,000 - 5% \text{ of } 1,000 = ₹950$.
3. However, he actually delivers only $800\text{ g}$ of wheat to the customer.
4. Actual cost incurred by the merchant for $800\text{ g} = ₹800$.
5. Net Profit $= \text{Revenue} - \text{Actual Cost} = 950 - 800 = ₹150$.
6. $\text{Net Profit %} = \left(\frac{150}{800}\right) \times 100 = \frac{150}{8} = 18.75%$.

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4. Advanced SBI PO Exam Strategies & Common Traps

• Selling Two Items at Same SP: If two items are sold at the same selling price, one at a profit of $x%$ and the other at a loss of $x%$, there is ALWAYS an overall net loss given by: $\text{Net Loss \%} = \frac{x^2}{100}$ Trap: Candidates often assume the net effect is 0% profit/loss. Remember that $CP$ values differ, creating an overall loss.

• "Buy X Get Y Free" Offer Discount Formula: $\text{Equivalent Discount \%} = \left( \frac{\text{Free Quantity (Y)}}{\text{Total Quantity (X + Y)}} \right) \times 100$ Example: "Buy 3 Get 2 Free" yields a discount of $\frac{2}{3+2} \times 100 = \frac{2}{5} \times 100 = 40%$.

• Successive Discounts Shortcut: For two successive discounts $d_1%$ and $d_2%$, single equivalent discount is: $\text{Single Equivalent Discount} = d_1 + d_2 - \frac{d_1 d_2}{100}$