4.2 Averages, Simple/Compound Interest, Alligations & Mixtures

Averages, Simple/Compound Interest, Alligations & Mixtures

Industrial operations at Coal India Limited routinely involve statistical averaging of blast-hole depths, equipment utilization rates, financial depreciation, capital project discounting, and alligation blending of varied coal grades. This section focuses on rapid calculation techniques that replace tedious long-form arithmetic with deviation analysis, compound difference identities, and alligation cross matrices.

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1. Statistical Averages & The Assumed Mean Method

Weighted Average

When different groups with sample sizes $n_1, n_2, \dots, n_k$ have respective arithmetic means $\bar{x}_1, \bar{x}_2, \dots, \bar{x}_k$, the combined weighted average $\bar{X}_{w}$ is:

$$\bar{X}_{w} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2 + \dots + n_k \bar{x}_k}{n_1 + n_2 + \dots + n_k} = \frac{\sum w_i x_i}{\sum w_i}$$

Assumed Mean (Deviation Method)

For large data points, avoid brute-force addition by selecting an arbitrary assumed mean $A$ close to the center of the dataset. Calculate the deviation $d_i = x_i - A$ for each point:

$$\bar{X} = A + \frac{\sum_{i=1}^N d_i}{N}$$

Because the algebraic sum of deviations from the true arithmetic mean is always zero ($\sum (x_i - \bar{X}) = 0$), any net deviation $\sum d_i$ simply distributes equally across the $N$ observations.

Dynamic Group Changes (Insertion, Deletion, Replacement)

• Inclusion of a New Member: $$\text{Value of New Entrant} = \text{New Average} + N_{\text{old}} \times (\text{New Average} - \text{Old Average})$$
• Exclusion of a Member: $$\text{Value of Excluded Member} = \text{Old Average} + N_{\text{new}} \times (\text{Old Average} - \text{New Average})$$
• Replacement of an Existing Member: $$\text{Value of New Member} = \text{Value of Replaced Member} + N_{\text{total}} \times (\text{New Average} - \text{Old Average})$$

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2. Simple and Compound Interest Mechanics

Simple Interest ($SI$)

Simple interest accrues uniformly solely on the original principal $P$:

$$SI = \frac{P \times R \times T}{100}, \quad A = P + SI = P\left(1 + \frac{RT}{100}\right)$$

Where:

• $P = \text{Principal sum}$
• $R = \text{Annual interest rate in } \%$
• $T = \text{Time duration in years}$

Compound Interest ($CI$)

Compound interest reinvests accrued interest into the principal base at each compounding cycle:

$$A = P\left(1 + \frac{R}{100 \times k}\right)^{k \times T}, \quad CI = A - P = P\left[\left(1 + \frac{R}{100 \times k}\right)^{k \times T} - 1\right]$$

Where $k$ is the compounding frequency per year:

• Annually: $k = 1$
• Semi-annually (Half-yearly): $k = 2$, rate $= R/2$, periods $= 2T$
• Quarterly: $k = 4$, rate $= R/4$, periods $= 4T$

Difference Between $CI$ and $SI$

Questions regarding the difference $D = CI - SI$ for 2 and 3 years appear frequently in PSU examinations:

1. Two-Year Difference ($D_2$):

   $$D_2 = CI_2 - SI_2 = P\left(\frac{R}{100}\right)^2$$

2. Three-Year Difference ($D_3$):

   $$D_3 = CI_3 - SI_3 = P\left(\frac{R}{100}\right)^2 \left(\frac{300 + R}{100}\right) = D_2 \times \left(3 + \frac{R}{100}\right)$$

Quick Growth Rules of Thumb

• Rule of 72 (Doubling Time): Time required for capital to double under compound interest is approximately $T \approx \frac{72}{R\%}$.
• Rule of 114 (Tripling Time): Time required to triple capital is approximately $T \approx \frac{114}{R\%}$.
• Rule of 144 (Quadrupling Time): Time required to quadruple capital is approximately $T \approx \frac{144}{R\%}$.

3. Alligations & Industrial Mixture Blending

The Alligation Rule

The rule of alligation is a modified cross-computation derived from weighted averages. When two ingredients of price/concentration $c$ (cheaper/lower) and $d$ (dearer/higher) are combined to produce a mixture of mean value $m$:

$$Q_c \times (m - c) = Q_d \times (d - m) \implies \frac{Q_c}{Q_d} = \frac{d - m}{m - c}$$

Applications in Coal Quality Blending

In coal preparation and power dispatch, two grades of coal with different ash percentages ($A_1\%$ and $A_2\%$) or calorific values ($CV_1$ and $CV_2$) are blended to meet strict thermal plant specifications ($A_{\text{target}}$):

$$\frac{\text{Mass of Grade 1}}{\text{Mass of Grade 2}} = \frac{|A_2 - A_{\text{target}}|}{|A_{\text{target}} - A_1|}$$

Three-Component Alligation Framework

When three components with values $c_1 < c_2 < c_3$ are mixed to obtain a target mean $m$ (where $c_1 < m < c_3$), pair the components lying below the mean with those above the mean:

1. Pair $(c_1, c_3)$ with difference weights: $Q_1 = |c_3 - m|$, $Q_{3,a} = |m - c_1|$.
2. If $c_2 < m$, pair $(c_2, c_3)$ giving: $Q_2 = |c_3 - m|$, $Q_{3,b} = |m - c_2|$.
3. Sum the allocated parts for component 3: $Q_3 = Q_{3,a} + Q_{3,b}$.

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4. Repeated Replacement Dilution Formula

When a container initially contains $x$ units of a pure liquid (or material), and $y$ units are drawn off and replaced with an adulterant/water, and this process is repeated $n$ times in succession, the remaining quantity of pure liquid is given by:

$$Q_{\text{pure, remaining}} = x\left(1 - \frac{y}{x}\right)^n$$

The fraction of pure liquid remaining relative to the total container capacity is:

$$\frac{Q_{\text{pure, remaining}}}{\text{Total Capacity}} = \left(1 - \frac{y}{x}\right)^n$$

The ratio of pure liquid to replacement liquid (water) after $n$ operations is:

$$\frac{\text{Pure Liquid}}{\text{Water}} = \frac{\left(1 - \frac{y}{x}\right)^n}{1 - \left(1 - \frac{y}{x}\right)^n}$$

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5. Equal Annual Installments (EMIs) Under SI and CI

Financial appraisal of heavy earthmoving machinery leases and capital loans requires calculating equal annual installment payments:

Installments Under Simple Interest

If a total debt of amount $A$ due after $n$ years is discharged in $n$ equal annual installments of $x$ each at simple interest rate $R\%$:

$$A = n \cdot x + \frac{R \cdot x}{100} \times \frac{(n-1) \cdot n}{2}$$

Solving for the annual installment $x$:

$$x = \frac{100 \cdot A}{100n + \frac{R \cdot n(n-1)}{2}}$$

Installments Under Compound Interest

If a principal loan $P$ borrowed today is repaid in $n$ equal annual installments of $x$ at annual compound interest rate $R\%$:

$$P = \frac{x}{\left(1 + \frac{R}{100}\right)^1} + \frac{x}{\left(1 + \frac{R}{100}\right)^2} + \dots + \frac{x}{\left(1 + \frac{R}{100}\right)^n}$$

For a 2-year repayment cycle, let $v = \frac{1}{1 + R/100}$:

$$P = x \left[ \frac{100}{100+R} + \left(\frac{100}{100+R}\right)^2 \right] \implies x = \frac{P \cdot (1 + R/100)^2}{1 + (1 + R/100)}$$