6.4 Mathematical Aptitude: Fractions, Percentages, Profit & Loss

Mathematical Aptitude: Fractions, Percentages, Profit & Loss

Quick Answer: Percentages, profit, and loss form the mathematical core of UGC NET quantitative questions. Convert percentages to fractions instantly ($1/6 = 16.67\%, 1/8 = 12.5\%$). Master the product constancy rule: if price rises by $1/n$, consumption must fall by $1/(n+1)$ for constant expenditure. Successive changes combine as $a + b + \frac{ab}{100}$, while successive discounts combine as $d_1 + d_2 - \frac{d_1 d_2}{100}$. Always compute profit/loss on Cost Price ($CP$) and discounts on Marked Price ($MP$).

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1. Fraction-to-Percentage Mastery Table

Memorizing fraction-to-percentage equivalents eliminates tedious manual division during exam calculations.

Key Fraction Multiples

• $2/3 = 66.67\%$
• $3/4 = 75\%$
• $2/5 = 40\%$, $3/5 = 60\%$, $4/5 = 80\%$
• $3/8 = 37.5\%$, $5/8 = 62.5\%$, $7/8 = 87.5\%$
• $2/7 = 28.57\%$, $3/7 = 42.86\%$, $4/7 = 57.14\%$
• $2/9 = 22.22\%$, $4/9 = 44.44\%$, $7/9 = 77.78\%$
• $2/11 = 18.18\%$, $3/11 = 27.27\%$, $5/11 = 45.45\%$

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2. Percentage Change, Base Shifts, and Product Constancy

The Multiplier Method

• An increase of $x\%$ corresponds to a multiplying factor of $(1 + \frac{x}{100})$. $$\text{Example: Increase of } 25\% \implies \text{New Value} = \text{Original} \times 1.25$$
• A decrease of $x\%$ corresponds to a multiplying factor of $(1 - \frac{x}{100})$. $$\text{Example: Decrease of } 15\% \implies \text{New Value} = \text{Original} \times 0.85$$

The Product Constancy Rule (AB = Constant)

When the product of two variables remains constant ($A \times B = K$), such as $\text{Price} \times \text{Consumption} = \text{Expenditure}$ or $\text{Speed} \times \text{Time} = \text{Distance}$:

$$\text{If } A \text{ increases by } \frac{1}{n}, \text{ then } B \text{ must decrease by } \frac{1}{n+1} \text{ to keep the product constant.}$$ $$\text{If } A \text{ decreases by } \frac{1}{n}, \text{ then } B \text{ must increase by } \frac{1}{n-1} \text{ to keep the product constant.}$$

Successive Percentage Changes

When a quantity undergoes two consecutive percentage modifications, $a\%$ followed by $b\%$ (where increases are positive and decreases are negative):

$$\text{Net Percentage Change} = a + b + \frac{a \times b}{100} \quad (\%)$$

• Example: A salary is increased by $20\%$ and subsequently decreased by $10\%$: $\text{Net Change} = 20 + (-10) + \frac{20 \times (-10)}{100} = 10 - 2 = +8\% \text{ (Net increase of } 8\%)$.

3. Profit, Loss, and Discount Fundamentals

Core Commercial Definitions

• Cost Price ($CP$): The total expenditure incurred to acquire or manufacture an article.
• Selling Price ($SP$): The actual price at which the article is sold to a customer.
• Marked Price ($MP$ / List Price): The price labeled on the article before discounts.

Fundamental Formulas

$$\text{Profit } (P) = SP - CP \quad (\text{when } SP > CP)$$ $$\text{Loss } (L) = CP - SP \quad (\text{when } CP > SP)$$ $$\text{Profit } \% = \left(\frac{SP - CP}{CP}\right) \times 100 = \left(\frac{\text{Profit}}{CP}\right) \times 100$$ $$\text{Loss } \% = \left(\frac{CP - SP}{CP}\right) \times 100 = \left(\frac{\text{Loss}}{CP}\right) \times 100$$

Calculating SP from CP and Vice-Versa

$$SP = CP \times \left(1 + \frac{P\%}{100}\right) \quad \text{or} \quad SP = CP \times \left(1 - \frac{L\%}{100}\right)$$ $$CP = \frac{SP}{1 + \frac{P\%}{100}} = \frac{SP \times 100}{100 + P\%} \quad \text{or} \quad CP = \frac{SP \times 100}{100 - L\%}$$

Commercial Discounts and Markup

• Discount: Calculated strictly on Marked Price ($MP$): $$\text{Discount} = MP - SP$$ $$\text{Discount } \% = \left(\frac{\text{Discount}}{MP}\right) \times 100$$ $$SP = MP \times \left(1 - \frac{\text{Discount } \%}{100}\right)$$
• Single Equivalent Discount for Successive Discounts ($d_1\%$ and $d_2\%$): $$D_{\text{equivalent}} = d_1 + d_2 - \frac{d_1 \times d_2}{100} \quad (\%)$$
  • Example: Two successive discounts of $20\%$ and $10\%$: $$D_{\text{eq}} = 20 + 10 - \frac{20 \times 10}{100} = 30 - 2 = 28\%$$
• Linking CP, MP, Profit, and Discount: $$\frac{MP}{CP} = \frac{100 + \text{Profit } \%}{100 - \text{Discount } \%}$$

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4. Advanced Profit & Loss Theorems

Theorem 1: Equal Selling Price with Identical Profit and Loss Percentages

When two distinct articles are sold at the exact same Selling Price ($SP$), one at a profit of $x\%$ and the other at a loss of $x\%$:

• The overall transaction always results in a net loss.
• The net loss percentage is given by: $$\text{Net Loss } \% = \left(\frac{x}{10}\right)^2 = \frac{x^2}{100} \quad (\%)$$
• Example: A merchant sells two laptops for ₹$24,000$ each. On one he gains $20\%$ and on the other he loses $20\%$. $$\text{Net Loss } \% = \frac{20^2}{100} = \frac{400}{100} = 4\%$$

Theorem 2: Dishonest Dealer and False Weight Logic

When a dishonest trader claims to sell goods at cost price ($CP$) but uses a false weight smaller than the true weight:

$$\text{Gain } \% = \left(\frac{\text{Error}}{\text{True Value} - \text{Error}}\right) \times 100 = \left(\frac{\text{Claimed Weight} - \text{Actual Weight}}{\text{Actual Weight}}\right) \times 100$$

• Example: A shopkeeper claims to sell sugar at cost price but uses a $900\text{ g}$ weight instead of a $1\text{ kg } (1000\text{ g})$ weight.
  • $\text{Error} = 1000 - 900 = 100\text{ g}$
  • $\text{Gain } \% = \left(\frac{100}{900}\right) \times 100 = \frac{1}{9} \times 100 = 11.11\% = 11\frac{1}{9}\%$.