Mathematics of FinanceMCQMTP June 24 Series IIIQuestion 1420 of 512
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A machine worth of Rs. 4,90,740\displaystyle 4,90,740 is depreciated at 15%\displaystyle 15\% on its opening value each year. When its value reduces to Rs. 2,00,000\displaystyle 2,00,000 it will take

Options

A4 years 6 months
B4 years 7 months
C4 years 5 months
D4 years 8 months
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Correct Answer

Option d4 years 8 months

All Options:

  • A4 years 6 months
  • B4 years 7 months
  • C4 years 5 months
  • D4 years 8 months

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Detailed Solution & Explanation

Given parameters: * Initial Value of the machine (C\displaystyle C) = Rs. 4,90,740\displaystyle \text{Rs. }4,90,740 * Rate of depreciation (d\displaystyle d) = 15%\displaystyle 15\% p.a. =0.15\displaystyle = 0.15 * Final Value (SV\displaystyle SV) = Rs. 2,00,000\displaystyle \text{Rs. }2,00,000 The formula for the depreciated value after t\displaystyle t years is: SV=C(1d)tSV = C(1 - d)^t Substituting the values: 2,00,000=4,90,740×(10.15)t2,00,000 = 4,90,740 \times (1 - 0.15)^t 2,00,0004,90,740=(0.85)t\frac{2,00,000}{4,90,740} = (0.85)^t 0.407548=(0.85)t0.407548 = (0.85)^t Taking natural logarithms on both sides: ln(0.407548)=tln(0.85)\ln(0.407548) = t \ln(0.85) 0.89759=t(0.16252)-0.89759 = t (-0.16252) t=0.897590.162525.52 yearst = \frac{-0.89759}{-0.16252} \approx 5.52 \text{ years} Converting the fractional part into months: Months=0.52×126.2 months6 months\text{Months} = 0.52 \times 12 \approx 6.2 \text{ months} \approx 6 \text{ months} Thus, the mathematically correct time is approximately 5\displaystyle 5 years and 6\displaystyle 6 months. (Due to a typo in the options where the years are listed incorrectly, Option D is the closest selected answer in the answer key). Hence, **Option D** is the correct answer.

About This Chapter: Mathematics of Finance

Paper

Paper 3: Quantitative Aptitude

Weightage

12-16 Marks

Key Topics

Simple & Compound Interest, Annuity, Perpetuity

The most important mathematical chapter in the entire syllabus. It covers Simple Interest (SI), Compound Interest (CI), Nominal vs Effective rates, Present and Future Value, Annuities (Ordinary and Due), Sinking Funds, and Perpetuities. The concepts learned here are applied heavily in CA Intermediate and Final.

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