6.6 Time & Work, Speed-Time-Distance, Simple & Compound Interest

Time & Work, Speed-Time-Distance, Simple & Compound Interest

Quick Answer: Arithmetic aptitude in Paper 1 relies on structured algebraic formulas. Solve Time and Work via the LCM efficiency method (Total Work $= \text{LCM}$, Time $= \text{Work}/\sum \text{Efficiencies}$) and the Chain Rule: $\frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2}$. For Speed-Time-Distance, convert $\text{km/h} \times \frac{5}{18} = \text{m/s}$, apply average speed $\frac{2xy}{x+y}$, and combine train/platform lengths. For Interest, $SI = \frac{PRT}{100}$, $CI = P(1 + \frac{R}{100})^T - P$, and the $2$-year difference between CI and SI is $P(\frac{R}{100})^2$.

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1. Time and Work: The LCM / Efficiency Framework

While the unitary method solves work problems via fractions, the LCM (Total Work) Method avoids fractional addition, accelerating speed and accuracy.

The LCM Efficiency Method Steps

1. Define Total Work: Let Total Work equal the $\text{LCM}$ of the individual completion times.
2. Compute Individual Daily Efficiencies ($E$): $$\text{Efficiency } (E) = \frac{\text{Total Work}}{\text{Time Taken}}$$
3. Combine Efficiencies: Add efficiencies for collaborative work; subtract efficiency for negative work (leaks/drainage).
4. Calculate Time Taken: $$\text{Time Taken} = \frac{\text{Total Work}}{\text{Combined Efficiency}}$$

$$\text{Time for A, B, C Together} = \frac{60\text{ units}}{12\text{ units/day}} = 5\text{ days}$$

Two-Worker Algebraic Shortcut

If $A$ takes $x$ days and $B$ takes $y$ days:

$$\text{Time Together} = \frac{x \times y}{x + y}$$

The Multi-Variable Chain Rule (Man-Days-Hours Formula)

To resolve scenarios involving changing workforce sizes, daily operating hours, differing efficiencies, and varied work quotas:

$$\frac{M_1 \times D_1 \times H_1 \times E_1}{W_1} = \frac{M_2 \times D_2 \times H_2 \times E_2}{W_2}$$

• $M = \text{Number of workers}$
• $D = \text{Number of working days}$
• $H = \text{Working hours per day}$
• $E = \text{Efficiency factor of workers}$
• $W = \text{Quantum of work produced (e.g., metres of wall, units manufactured, wages earned)}$

Pipes and Cisterns

• Inlet Pipe ($A$): Fills tank $\implies$ Positive Efficiency ($+E_A$).
• Outlet / Drainage Pipe / Leak ($B$): Empties tank $\implies$ Negative Efficiency ($-E_B$).
• Net Filling Rate: $E_{\text{net}} = E_{\text{inlet}} - E_{\text{outlet}}$.
• If an inlet fills a tank in $x$ hours and an outlet empties it in $y$ hours ($y > x$): $$\text{Net Time to Fill} = \frac{x \times y}{y - x}$$

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2. Speed, Time, and Distance Mastery

Unit Conversions and Basic Relations

$$\text{Distance } (D) = \text{Speed } (S) \times \text{Time } (T)$$ $$\text{Speed in m/s} = \text{Speed in km/h} \times \frac{5}{18}$$ $$\text{Speed in km/h} = \text{Speed in m/s} \times \frac{18}{5}$$

Average Speed Formulations

• General Definition: $\text{Average Speed} = \frac{\text{Total Distance Travelled}}{\text{Total Time Taken}}$.
• Equal Distance Traveled at Two Different Speeds ($x$ and $y$): $$\text{Average Speed} = \frac{2xy}{x + y}$$
• Equal Distance Traveled at Three Different Speeds ($x, y, z$): $$\text{Average Speed} = \frac{3xyz}{xy + yz + zx}$$

Relative Speed Principles

• Opposite Directions (Moving toward or away from each other): $$\text{Relative Speed} = S_1 + S_2$$
• Same Direction (One pursuing the other): $$\text{Relative Speed} = |S_1 - S_2|$$

Train Physics and Distance Rules

Boats and Streams Vector Mechanics

Let the speed of a boat in still water be $u\text{ km/h}$, and the speed of the water stream/current be $v\text{ km/h}$:

• Downstream Speed ($D$): Moving with the current: $D = u + v$.
• Upstream Speed ($U$): Moving against the current: $U = u - v$.
• Speed of Boat in Still Water ($u$): $$u = \frac{D + U}{2}$$
• Speed of Stream / Current ($v$): $$v = \frac{D - U}{2}$$

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3. Simple and Compound Interest Formulations

Simple Interest (SI)

Simple interest accrues linearly exclusively on the initial principal $P$:

$$SI = \frac{P \times R \times T}{100}$$ $$\text{Total Amount } (A) = P + SI = P\left(1 + \frac{R \times T}{100}\right)$$

• Rule of Doubling in SI: If a sum of money doubles itself in $T$ years, then $R = \frac{100}{T}\%$.
• Rule of $n$-times in SI: If a sum becomes $n$ times itself in $T$ years, $(n - 1) = \frac{R \times T}{100}$.

Compound Interest (CI)

Compound interest accrues on both the principal and previously accumulated interest:

$$\text{Total Amount } (A) = P\left(1 + \frac{R}{100}\right)^T$$ $$CI = A - P = P\left[\left(1 + \frac{R}{100}\right)^T - 1\right]$$

Compounding Frequencies Adjustment Table

Difference Between CI and SI Theorems

For a principal sum $P$ invested at annual rate $R\%$:

1. Difference for 2 Years ($D_2$): $$D_2 = CI_2 - SI_2 = P\left(\frac{R}{100}\right)^2 = \frac{P R^2}{10000}$$
2. Difference for 3 Years ($D_3$): $$D_3 = CI_3 - SI_3 = P\left(\frac{R}{100}\right)^2 \left(3 + \frac{R}{100}\right) = \frac{P R^2 (300 + R)}{10^6}$$

• Worked Example: Find the difference between CI and SI on ₹$10,000$ for $2$ years at $8\%$ per annum. $$D_2 = 10000 \times \left(\frac{8}{100}\right)^2 = 10000 \times \frac{64}{10000} = 64$$ The calculated difference is ₹$64$.