The value of a machine decreases every year at the rate of 10% on its value at the beginning of the year. If its present value be Rs. 729, what was its value 3 years ago?

The problem involves calculating the original value of a machine three years ago, given that its value depreciates by 10% annually and its current value is Rs. 729. This is a classic depreciation problem, where the value decreases by a fixed percentage each year.

Step 1: Understand the Depreciation Formula

The value of the machine depreciates by 10% each year, meaning it retains 90% (or 0.9) of its value at the beginning of each year. If the value 3 years ago was $V$, then:

• After 1 year (2 years ago): $V\times 0.9$
• After 2 years (1 year ago): $V\times (0.9{)}^2$
• After 3 years (present value): $V\times (0.9{)}^3=729$

Step 2: Solve for $V$

We need to find the original value $V$ three years ago. Rearranging the equation:

$$V\times (0.9{)}^3=729$$

$$V=\frac{729}{(0.9{)}^3}$$

First, calculate $(0.9{)}^3$:

$$(0.9{)}^3=0.9\times 0.9\times 0.9=0.729$$

Now, substitute back:

$$V=\frac{729}{0.729}=1000$$

Step 3: Verification

To ensure correctness, let’s verify by calculating the depreciated value step-by-step:

1. 3 years ago: Rs. 1000
2. 2 years ago: $1000\times 0.9=900$
3. 1 year ago: $900\times 0.9=810$
4. Present value: $810\times 0.9=729$

The calculations match the given present value, confirming the solution.

Final Answer

The value of the machine 3 years ago was Rs. 1000.