A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.

Question

Accepted Answer

> [!TLDR]
> The machine’s value falls by a fixed percentage each year, so we use the depreciation formula $V = P \left(1 - \frac{r}{100}ight)^n$. With $P = 15625$, $r = 20$, and $n = 5$, the value after 5 years is Rs 5120.

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### Why the Depreciation Formula Works

When an asset loses value by a constant percentage each year, the decrease is **not** a fixed amount — it’s proportional to the current value. That means the value at the end of each year is the previous year’s value multiplied by $(1 - \frac{r}{100})$, where $r$ is the depreciation rate. This is exactly like compound interest, but with a negative rate. So after $n$ years, the formula is:

> [!FORMULA]
> $$V = P \left(1 - \frac{r}{100}ight)^n$$

Here, $P$ is the initial cost, $r$ is the annual depreciation percentage, and $n$ is the number of years.

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### Step-by-Step Solution

1. **Identify the given values.**  
   The machine costs Rs 15625 initially, so $P = 15625$.  
   It depreciates by 20% each year, so $r = 20$.  
   We need the value after 5 years, so $n = 5$.

2. **Write the depreciation factor.**  
   Each year, the machine retains $100\% - 20\% = 80\%$ of its value.  
   As a decimal, that’s $0.80$, or as a fraction, $\frac{80}{100} = \frac{4}{5}$.  
   So the factor is $\left(1 - \frac{20}{100}ight) = \frac{4}{5}$.

3. **Apply the formula.**  
   $$V = 15625 	imes \left(\frac{4}{5}ight)^5$$

4. **Compute $\left(\frac{4}{5}ight)^5$.**  
   $$ \left(\frac{4}{5}ight)^5 = \frac{4^5}{5^5} = \frac{1024}{3125} $$ …

Method	Best for	Key feature
Straight-Line	Buildings, furniture, patents	Equal expense every year
WDV	Vehicles, machinery, electronics	Higher expense early on

Q.A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.

Concept understanding — Depreciation Formula

Depreciation: Why an asset loses value over time

The core idea: two ways to think about it

1. Straight-Line Method (SLM) — equal loss every year

2. Written Down Value Method (WDV) — decreasing loss each year

Which formula to use?

The one formula you must remember for WDV over multiple years

Why the Depreciation Formula Works

Step-by-Step Solution

Q.A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.

Why the Depreciation Formula Works

Step-by-Step Solution

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