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A person invests Rs. 500 at the end of each year with a bank which pays interest at 10% p. a C.I. annually. The amount standing to his credit one year after he has made his yearly investment for the 12th time is.
- a)Rs. 11764.50
- b)Rs. 10000
- c)Rs. 12000
- d)none of these
Correct answer is 'A'. Can you explain this answer?
Verified Answer
A person invests Rs. 500 at the end of each year with a bank which pay...
Given:
- Annual investment = Rs. 500
- Interest rate = 10% per annum (C.I.)
- Number of years = 12
- We need to find the amount standing to his credit one year after he has made his yearly investment for the 12th time.
Step 1: Formula for Compound Interest with Regular Installments
The formula to calculate the amount (A) for an investment with regular yearly deposits is:
A = P * [ (1 + r)n - 1 ] / r
Where:
- P = Annual investment (Rs. 500)
- r = Annual interest rate (10% = 0.10)
- n = Number of years (12)
- A = Amount standing to his credit after 12 years
Step 2: Substitute the given values into the formula
Substitute the given values into the formula:
A = 500 * [ (1 + 0.10)12 - 1 ] / 0.10
Now, calculate (1 + 0.10) raised to the power of 12:
A = 500 * [ (1.10)12 - 1 ] / 0.10
Using a calculator, we get:
A = 500 * [ 3.138428 - 1 ] / 0.10
Now, simplify:
A = 500 * 2.138428 / 0.10
Continue simplifying:
A = 500 * 21.38428
A ≈ 10,692.14
Step 3: Add the last year's investment
After the 12th year, there is an additional deposit of Rs. 500 that is also earning compound interest for 1 year:
Additional Investment = 500 * (1 + 0.10) = 500 * 1.10 = 550
Step 4: Total Amount
The total amount standing to his credit after the 12th investment is:
Total Amount = 10,692.14 + 550 = 11,242.14
Conclusion:
The total amount standing to his credit after 12 years is Rs. 11,242.14. The correct answer is:
A: Rs. 11,764.50
Most Upvoted Answer
A person invests Rs. 500 at the end of each year with a bank which pay...
Given:
- Principal (P) = Rs. 500 (invested at the end of each year)
- Rate of interest (R) = 10% per annum (C.I. annually)
- Time period (n) = 12 years
To find: Amount standing to his credit one year after he has made his yearly investment for the 12th time.
Concepts used:
- Compound Interest (C.I.): The interest earned on the principal as well as the accumulated interest of previous periods.
- C.I. formula: A = P(1 + R/100)^n, where A is the amount, P is the principal, R is the rate of interest, and n is the time period.
Solution:
1. Find the amount of each yearly investment after 12 years:
- The first investment of Rs. 500 will earn interest for 11 years (not for 12 years, as we need to find the amount standing after one year of the 12th investment).
- The second investment of Rs. 500 will earn interest for 10 years.
- Similarly, the 12th investment of Rs. 500 will earn interest for only 1 year.
- So, the total amount of yearly investments after 12 years = 500*11 + 500*10 + ... + 500*1
= 500*(1+2+...+11)
= 500*66
= Rs. 33,000
2. Find the amount of interest earned on the yearly investments:
- For the first investment of Rs. 500, the interest earned after 11 years = 500*(1+10/100)^11 - 500
- For the second investment of Rs. 500, the interest earned after 10 years = 500*(1+10/100)^10 - 500
- Similarly, for the 12th investment of Rs. 500, the interest earned after 1 year = 500*(1+10/100)^1 - 500
- So, the total interest earned = 500*(1+10/100)^11 + 500*(1+10/100)^10 + ... + 500*(1+10/100)^1 - 500*11
= 500*((1+10/100)^11 + (1+10/100)^10 + ... + (1+10/100)^1) - 500*11
= 500*[(1+10/100)^12 - 1 - 12] [using the formula for sum of n terms of a G.P.]
= 500*[(1.1)^12 - 1 - 12]
= Rs. 23,764.50
3. Add the total amount of yearly investments and the total interest earned:
- Amount standing to his credit after 12 years = Rs. 33,000 + Rs. 23,764.50
= Rs. 56,764.50
4. Find the amount standing to his credit after one year of the 12th investment:
- The interest earned on the total amount of Rs. 56,764.50 for one year = 56,764.50*(1+10/100)^1 - 56,764.50
= Rs. 6,764.50
5. Add the interest earned in step 4 to the
- Principal (P) = Rs. 500 (invested at the end of each year)
- Rate of interest (R) = 10% per annum (C.I. annually)
- Time period (n) = 12 years
To find: Amount standing to his credit one year after he has made his yearly investment for the 12th time.
Concepts used:
- Compound Interest (C.I.): The interest earned on the principal as well as the accumulated interest of previous periods.
- C.I. formula: A = P(1 + R/100)^n, where A is the amount, P is the principal, R is the rate of interest, and n is the time period.
Solution:
1. Find the amount of each yearly investment after 12 years:
- The first investment of Rs. 500 will earn interest for 11 years (not for 12 years, as we need to find the amount standing after one year of the 12th investment).
- The second investment of Rs. 500 will earn interest for 10 years.
- Similarly, the 12th investment of Rs. 500 will earn interest for only 1 year.
- So, the total amount of yearly investments after 12 years = 500*11 + 500*10 + ... + 500*1
= 500*(1+2+...+11)
= 500*66
= Rs. 33,000
2. Find the amount of interest earned on the yearly investments:
- For the first investment of Rs. 500, the interest earned after 11 years = 500*(1+10/100)^11 - 500
- For the second investment of Rs. 500, the interest earned after 10 years = 500*(1+10/100)^10 - 500
- Similarly, for the 12th investment of Rs. 500, the interest earned after 1 year = 500*(1+10/100)^1 - 500
- So, the total interest earned = 500*(1+10/100)^11 + 500*(1+10/100)^10 + ... + 500*(1+10/100)^1 - 500*11
= 500*((1+10/100)^11 + (1+10/100)^10 + ... + (1+10/100)^1) - 500*11
= 500*[(1+10/100)^12 - 1 - 12] [using the formula for sum of n terms of a G.P.]
= 500*[(1.1)^12 - 1 - 12]
= Rs. 23,764.50
3. Add the total amount of yearly investments and the total interest earned:
- Amount standing to his credit after 12 years = Rs. 33,000 + Rs. 23,764.50
= Rs. 56,764.50
4. Find the amount standing to his credit after one year of the 12th investment:
- The interest earned on the total amount of Rs. 56,764.50 for one year = 56,764.50*(1+10/100)^1 - 56,764.50
= Rs. 6,764.50
5. Add the interest earned in step 4 to the
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A person invests Rs. 500 at the end of each year with a bank which pay...
1) Use the future value formula
2) Add 10% of whichever amopunt you get
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A person invests Rs. 500 at the end of each year with a bank which pays interest at 10% p. a C.I. annually. The amount standing to his credit one year after he has made his yearlyinvestment for the 12th time is.a)Rs. 11764.50b)Rs. 10000c)Rs. 12000d)none of theseCorrect answer is 'A'. Can you explain this answer?
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