4.1 Arithmetic Foundations: Percentages, Profit & Loss, Ratio & Proportion

Arithmetic Foundations: Percentages, Profit & Loss, Ratio & Proportion

In the Coal India Limited Management Trainee (CIL MT) Paper-I examination, Numerical Ability constitutes 25 vital marks. Speed and numerical agility are paramount: candidates must solve quantitative problems in approximately 45 to 50 seconds per item. Navigating arithmetic through standard algebraic variables often induces unnecessary computational friction. Instead, candidates must master fraction-to-percentage conversions, product constancy relationships, commercial pricing chains, and proportional balancing.

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1. Fraction-to-Percentage Equivalence & Decimal Multipliers

Direct translation between fractions and percentages converts complex division into rapid mental multiplication. In engineering and commercial contexts—such as coal washing yields, stripping ratios, and equipment depreciation—these fractions appear repeatedly.

Standard Conversion Matrix

Multiplying Factors ($MF$)

When a quantity $X$ changes by $r\%$, the new quantity $X'$ is evaluated via the Multiplying Factor ($MF$):

$$X' = X \times MF = X \times \left(1 \pm \frac{r}{100}\right)$$

• Growth ($+r\%$): A $37.5\%$ increase implies $MF = 1 + \frac{3}{8} = \frac{11}{8} = 1.375$.
• Decay ($-r\%$): A $16.67\%$ reduction implies $MF = 1 - \frac{1}{6} = \frac{5}{6} \approx 0.8333$.

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2. Successive Percentage Changes & Product Constancy

Successive Percentage Change Formula

When a base quantity undergoes two successive percentage alterations of $a\%$ and $b\%$ (where positive denotes an increase and negative denotes a decrease), the net percentage change $\Delta_{\text{net}}\%$ is given by:

$$\Delta_{\text{net}}\% = \left(a + b + \frac{a \times b}{100}\right)\%$$

For three successive changes $a\%, b\%, c\%$, apply the formula iteratively or compute the compounded multiplying factor:

$$MF_{\text{net}} = \left(1 + \frac{a}{100}\right)\left(1 + \frac{b}{100}\right)\left(1 + \frac{c}{100}\right)$$ $$\Delta_{\text{net}}\% = (MF_{\text{net}} - 1) \times 100\%$$

Special Case: Equal and Opposite Changes ($ pm x%$) When a quantity is first increased by $x\%$ and subsequently decreased by $x\%$, the net result is always an absolute loss given by: $\Delta_{\text{net}}\% = +x - x + \frac{(+x)(-x)}{100} = -\frac{x^2}{100}\%$

Product Constancy ($A \times B = \text{Constant}$)

In many industrial and economic systems, the product of two variables is fixed:

$$\text{Expenditure} = \text{Price} \times \text{Consumption}$$ $$\text{Total Work} = \text{Efficiency} \times \text{Time}$$ $$\text{Distance} = \text{Speed} \times \text{Time}$$

If variable $A$ increases by a fractional amount $\frac{a}{b}$, variable $B$ must decrease by $\frac{a}{a + b}$ to maintain the invariant product $K$:

$$A' = A\left(1 + \frac{a}{b}\right) = A\left(\frac{a+b}{b}\right) \implies B' = B\left(\frac{b}{a+b}\right) = B\left(1 - \frac{a}{a+b}\right)$$

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3. Commercial Arithmetic: Profit, Loss, Markup, and Discounts

Commercial transactions involve three baseline financial markers: Cost Price ($CP$), Marked Price ($MP$), and Selling Price ($SP$).

Core Equations

1. Profit and Loss Percentages:

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

   Note: Profit and loss percentages are always computed on $CP$ unless explicitly designated on $SP$.

2. Markup and Discount Relationships:

   $$\text{Markup}\% = \frac{MP - CP}{CP} \times 100\% \implies MP = CP\left(1 + \frac{\text{Markup}\%}{100}\right)$$ $$\text{Discount}\% = \frac{MP - SP}{MP} \times 100\% \implies SP = MP\left(1 - \frac{\text{Discount}\%}{100}\right)$$

3. Master Ratio Formula ($CP$ to $MP$): Equating $SP$ expressions yields:

   $$CP\left(1 + \frac{\text{Profit}\%}{100}\right) = MP\left(1 - \frac{\text{Discount}\%}{100}\right)$$ $$\frac{MP}{CP} = \frac{100 + \text{Profit}\%}{100 - \text{Discount}\%}$$

   (For a net loss, substitute $-\text{Loss}\%$ in the numerator).

4. Successive Discounts: Two successive discounts $d_1\%$ and $d_2\%$ produce a single equivalent discount $D_{\text{eq}}\%$:

   $$D_{\text{eq}}\% = \left(d_1 + d_2 - \frac{d_1 \times d_2}{100}\right)\%$$

Dishonest Dealer & Faulty Weighing Systems

In raw material procurement and coal dispatch logistics, dishonest dealer scenarios test faulty scales and adulteration.

• Case 1: Sells at Cost Price but Uses False Weight If a trader claims to sell goods at $CP$ but delivers a false measure $W_{\text{false}}$ instead of the true measure $W_{\text{true}}$:

  $$\text{Gain}\% = \left(\frac{W_{\text{true}} - W_{\text{false}}}{W_{\text{false}}}\right) \times 100\% = \left(\frac{\text{Error}}{\text{True Measure} - \text{Error}}\right) \times 100\%$$

• Case 2: Combined Markup/Discount with False Measure Let a trader mark up goods by $m\%$ (or discount by $d\%$) and simultaneously deliver $W_{\text{false}}$ grams instead of $W_{\text{true}}$ grams. The overall multiplying factor is:

  $$MF_{\text{overall}} = \left(\frac{W_{\text{true}}}{W_{\text{false}}}\right) \times \left(1 + \frac{m}{100}\right) \times \left(1 - \frac{d}{100}\right)$$ $$\text{Net Profit}\% = (MF_{\text{overall}} - 1) \times 100\%$$

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4. Ratio, Proportion, and Partnership Profit Distribution

Properties of Ratios and Proportions

1. Compound Ratio: For ratios $a:b$ and $c:d$, the compound ratio is $(a \times c) : (b \times d)$.
2. Duplicate & Sub-duplicate Ratios:
   • Duplicate ratio of $a:b$ is $a^2 : b^2$.
   • Sub-duplicate ratio of $a:b$ is $\sqrt{a} : \sqrt{b}$.
   • Triplicate ratio of $a:b$ is $a^3 : b^3$.
3. Proportional Variations ($a : b :: c : d$): $$\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c$$
   • Mean Proportional between $a$ and $c$: $b = \sqrt{a \times c}$.
   • Third Proportional to $a$ and $b$: $c = \frac{b^2}{a}$.
   • Fourth Proportional to $a, b,$ and $c$: $d = \frac{b \times c}{a}$.

Partnership Capital-Time Allocations

When multiple entities partner in a joint venture, the total profit is distributed strictly in proportion to the product of their invested capital ($C$) and the duration of investment ($T$):

$$\text{Profit Ratio } (P_1 : P_2 : P_3) = (C_1 \times T_1) : (C_2 \times T_2) : (C_3 \times T_3)$$

If a managing partner receives a dedicated salary or managerial commission ($S\%$) from the gross profit $P_{\text{gross}}$ before distribution:

$$P_{\text{distributable}} = P_{\text{gross}} \times \left(1 - \frac{S}{100}\right)$$

Each partner $i$ then receives their share from $P_{\text{distributable}}$, with the managing partner additionally retaining their salary $S$.