Rupees 4,000 amount to Rs. 5,000 in 8 years, in what time will Rs. 2,100 amount to Rs. 2,800 at the same rate ?

Answer

To solve the problem step by step, we will first determine the rate of interest based on the information provided and then use that rate to find the time required for Rs. 2,100 to amount to Rs. 2,800.

### Step 1: Identify the given values
- Principal (P) = Rs. 4,000
- Amount (A) = Rs. 5,000
- Time (T) = 8 years

### Step 2: Calculate the Interest (I)
Interest (I) can be calculated using the formula:
\[ I = A - P \]
Substituting the values:
\[ I = 5000 - 4000 = 1000 \text{ Rs.} \]

### Step 3: Calculate the Rate of Interest (R)
We can find the rate of interest using the formula:
\[ R = \frac{100 \times I}{P \times T} \]
Substituting the known values:
\[ R = \frac{100 \times 1000}{4000 \times 8} \]
\[ R = \frac{100000}{32000} \]
\[ R = 3.125 \text{ or } \frac{25}{8} \text{ percent} \]

### Step 4: Identify the new values for the second part of the question
- New Principal (P') = Rs. 2,100
- New Amount (A') = Rs. 2,800

### Step 5: Calculate the new Interest (I')
Using the same formula for interest:
\[ I' = A' - P' \]
Substituting the new values:
\[ I' = 2800 - 2100 = 700 \text{ Rs.} \]

### Step 6: Calculate the Time (T') required for Rs. 2,100 to amount to Rs. 2,800
We can rearrange the formula for time:
\[ T' = \frac{100 \times I'}{P' \times R} \]
Substituting the known values:
\[ T' = \frac{100 \times 700}{2100 \times \frac{25}{8}} \]
\[ T' = \frac{70000}{2100 \times \frac{25}{8}} \]
\[ T' = \frac{70000 \times 8}{2100 \times 25} \]
\[ T' = \frac{560000}{52500} \]
\[ T' = 10.6667 \text{ years} \]

### Step 7: Convert the decimal into years and months
The decimal part 0.6667 years can be converted into months:
\[ 0.6667 \times 12 \approx 8 \text{ months} \]
Thus, the time required is approximately:
\[ 10 \text{ years and } 8 \text{ months} \]

### Final Answer
The time required for Rs. 2,100 to amount to Rs. 2,800 at the same rate is **10 years and 8 months**.

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