You can save Rs.5,000 a year for 3 years, and Rs.7,000 a year for 7 years thereafter. What will these savings cumulate to at the end of 10 years, if the rate of interest is 8%?
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Your answer

To solve this problem, we need to calculate the future value of the savings at the end of 10 years. The savings consist of two parts: Rs.5,000 per year for the first 3 years and Rs.7,000 per year for the next 7 years. The interest rate is 8%.

### Step 1: Calculate the future value of Rs.5,000 saved annually for the first 3 years

The future value of an annuity can be calculated using the formula:

\[
FV = A \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- \( A = 5,000 \) Rs.
- \( r = 0.08 \) (8% interest rate)
- \( n = 3 \) years

\[
FV_1 = 5,000 \times \frac{(1 + 0.08)^3 - 1}{0.08} 
\]

\[
FV_1 = 5,000 \times \frac{1.259712 - 1}{0.08} = 5,000 \times 3.2464 \approx 16,232.01 \text{ Rs.}
\]

### Step 2: Calculate the future value of Rs.16,232.01 at the end of 10 years

Since Rs.16,232.01 is accumulated at the end of 3 years, we need to calculate its value at the end of 10 years (which is 7 more years). This is done by compounding it forward:

\[
FV_1 (after 10 years) = 16,232.01 \times (1 + 0.08)^7
\]

\[
FV_1 (after 10 years) = 16,232.01 \times 1.71382 \approx 27,819.49 \text{ Rs.}
\]

### Step 3: Calculate the future value of Rs.7,000 saved annually for the next 7 years

Now, calculate the future value of the Rs.7,000 saved annually for the next 7 years:

\[
FV_2 = 7,000 \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- \( A = 7,000 \) Rs.
- \( r = 0.08 \) (8% interest rate)
- \( n = 7 \) years

\[
FV_2 = 7,000 \times \frac{(1 + 0.08)^7 - 1}{0.08} 
\]

\[
FV_2 = 7,000 \times \frac{1.71382 - 1}{0.08} = 7,000 \times 8.92275 \approx 62,459.25 \text{ Rs.}
\]

### Step 4: Add the future values to get the total savings at the end of 10 years

\[
Total \, Savings = FV_1 (after 10 years) + FV_2 = 27,819.49 \text{ Rs.} + 62,459.25 \text{ Rs.} \approx 90,278.74 \text{ Rs.}
\]

### Final Answer:

The total savings at the end of 10 years will be **approximately Rs.90,278.74**.

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Would you like further details or have any questions?

### Relative Questions:
1. What would be the future value if the interest rate was 10% instead of 8%?
2. How does the time period of savings affect the final accumulated amount?
3. What if you could save Rs.7,000 for all 10 years instead of Rs.5,000 for the first 3 years?
4. How would the savings cumulate if the interest was compounded monthly instead of annually?
5. What would happen if the savings started with Rs.7,000 for the first 7 years and Rs.5,000 for the last 3 years?

### Tip:
When calculating future values of different savings or investments over time, break down the problem into parts and calculate the future value of each part separately before combining them.

Discover how to calculate future savings by making annual contributions and earning compound interest. Use our step-by-step guide to compute the total savings accumulated over 10 years with specific annual contributions and an 8% interest rate.

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