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%

To find out the cumulative savings at the end of 10 years, we need to calculate the future value of the savings for both periods (3 years and 7 years) separately and then sum them up. 

### Step 1: Calculate the future value of savings for the first 3 years
For the first 3 years, you save Rs.5,000 annually. The future value of these savings can be calculated using the future value of an annuity formula:

\[
FV_1 = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- \( P = 5,000 \) (annual savings)
- \( r = 0.08 \) (annual interest rate)
- \( n = 3 \) (number of years)

\[
FV_1 = 5,000 \times \frac{(1 + 0.08)^3 - 1}{0.08} = 5,000 \times \frac{1.2597 - 1}{0.08} = 5,000 \times 3.7468 = 18,734
\]

Now, this amount needs to grow for the remaining 7 years at 8% interest:

\[
FV_1' = FV_1 \times (1 + r)^7 = 18,734 \times (1.08)^7 \approx 18,734 \times 1.7138 = 32,117.44
\]

### Step 2: Calculate the future value of savings for the next 7 years
For the next 7 years, you save Rs.7,000 annually. The future value of these savings is:

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

Where:
- \( P = 7,000 \) (annual savings)
- \( r = 0.08 \)
- \( n = 7 \)

\[
FV_2 = 7,000 \times \frac{(1.08)^7 - 1}{0.08} = 7,000 \times \frac{1.7138 - 1}{0.08} = 7,000 \times 8.9225 = 62,457.5
\]

### Step 3: Calculate the total future value at the end of 10 years
Now, add the future values from both periods:

\[
\text{Total Future Value} = FV_1' + FV_2 = 32,117.44 + 62,457.5 = 94,574.94
\]

### Final Answer:
The total savings cumulated at the end of 10 years will be approximately **Rs. 94,574.94**.

Would you like more details on any of these steps or have any other questions?

### Relative Questions:
1. How does changing the interest rate affect the final savings?
2. What would the total savings be if the savings amount doubled after the first 3 years?
3. How would the result change if the savings were compounded quarterly instead of annually?
4. Can this method be used to calculate retirement savings?
5. How does inflation impact the future value of savings?

**Tip:** When dealing with future value calculations, ensure that the interest rate and the period of compounding match (e.g., annual rate with annual compounding).

Discover how to calculate cumulative savings over 10 years with compound interest. This problem involves saving Rs.5,000 annually for 3 years and Rs.7,000 annually for 7 years, compounded at 8% interest. Learn step-by-step how to compute the future value of annuities and sum them to find the total savings. Suitable for advanced high school students and beyond.

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