Samacheer Kalvi Class 11 Business Maths Solutions Chapter 7 Financial Mathematics Exercise 7.1

Question 1. Find the amount of an ordinary annuity of Rs 3,200 per annum for 12 years at the rate of interest of 10% per year, \( [(1.1)^{12} = 3.1384] \).
Answer:
Given:
Annual payment, \( a = Rs \ 3,200 \)
Number of years, \( n = 12 \)
Interest rate, \( i = \frac{10}{100} = 0.1 \)

The formula for the amount of an ordinary annuity is:
\[ A = \frac{a}{i} [(1+i)^n - 1] \] Substitute the given values into the formula:
\[ A = \frac{3200}{0.1} [(1 + 0.1)^{12} - 1] \] \[ A = 32000 [(1.1)^{12} - 1] \] We are given that \( (1.1)^{12} = 3.1384 \). We will use this value to simplify.
\[ A = 32000 [3.1384 - 1] \] \[ A = 32000 [2.1384] \] \[ A = 68428.8 \] Thus, the total amount of the annuity after 12 years is Rs 68,428.8.
In simple words: If you put Rs 3,200 into an account every year for 12 years and it grows by 10% each year, you would have Rs 68,428.8 in total. This calculation helps you see how your regular savings can grow over time.

🎯 Exam Tip: Remember to correctly identify 'a' (payment per period), 'n' (total number of periods), and 'i' (interest rate per period) from the question before applying the annuity formula.

Question 2. If the payment of Rs 2,000 is made at the end of every quarter for 10 years at the rate of 8% per year, then find the amount of annuity. \( [(1.02)^{40} = 2.2080] \)
Answer:
Given:
Payment per quarter, \( a = Rs \ 2,000 \)
Number of years, \( n = 10 \)
Annual interest rate = 8%
Compounded quarterly, so number of compounding periods per year, \( k = 4 \)

First, calculate the interest rate per quarter:
\[ \text{Rate per quarter} = \frac{\text{Annual Rate}}{k} = \frac{8\%}{4} = 2\% \] So, \( i = \frac{8}{100} = 0.08 \). Interest rate per period is \( \frac{i}{k} = \frac{0.08}{4} = 0.02 \).
Total number of quarters, \( \text{nk} = 10 \times 4 = 40 \).

The formula for the amount of an ordinary annuity with quarterly compounding is:
\[ A = \frac{a}{\frac{i}{k}} \left[ \left(1 + \frac{i}{k}\right)^{nk} - 1 \right] \] Substitute the values:
\[ A = \frac{2000}{0.02} [(1 + 0.02)^{40} - 1] \] \[ A = 100000 [(1.02)^{40} - 1] \] We are given that \( (1.02)^{40} = 2.2080 \). We will use this value.
\[ A = 100000 [2.2080 - 1] \] \[ A = 100000 [1.2080] \] \[ A = 1,20,800 \] Therefore, the amount of the annuity is Rs 1,20,800.
In simple words: If you save Rs 2,000 every three months for 10 years, and your money earns 8% interest each year, then you will have a total of Rs 1,20,800. This shows how regular small savings add up with compound interest.

🎯 Exam Tip: When interest is compounded quarterly, remember to adjust both the interest rate (divide by 4) and the number of periods (multiply by 4) to ensure consistency in your calculations.

Question 3. Find the amount of an ordinary annuity of 12 monthly payments of Rs 1,500 that earns interest at 12% per annum compounded monthly. \( [(1.01)^{12} = 1.1262] \)
Answer:
Given:
Monthly payment, \( a = Rs \ 1,500 \)
Number of payments = 12 (for 1 year)
Annual interest rate = 12%
Compounded monthly, so number of compounding periods per year, \( k = 12 \)

First, calculate the interest rate per month:
\[ \text{Rate per month} = \frac{\text{Annual Rate}}{k} = \frac{12\%}{12} = 1\% \] So, \( i = \frac{12}{100} = 0.12 \). Interest rate per period is \( \frac{i}{k} = \frac{0.12}{12} = 0.01 \).
Total number of months, \( \text{nk} = 1 \times 12 = 12 \).

Using the formula for the amount of an ordinary annuity:
\[ A = \frac{a}{\frac{i}{k}} \left[ \left(1 + \frac{i}{k}\right)^{nk} - 1 \right] \] Substitute the values:
\[ A = \frac{1500}{0.01} [(1 + 0.01)^{12} - 1] \] \[ A = 150000 [(1.01)^{12} - 1] \] We are given that \( (1.01)^{12} = 1.1262 \). Use this value.
\[ A = 150000 [1.1262 - 1] \] \[ A = 150000 [0.1262] \] \[ A = 18930 \] Therefore, the amount of the annuity is Rs 18,930.
In simple words: If you make 12 monthly payments of Rs 1,500, and the money earns 12% interest compounded every month, you will end up with Rs 18,930. This highlights how monthly contributions accumulate into a larger sum.

🎯 Exam Tip: For monthly compounding, divide the annual interest rate by 12 and multiply the number of years by 12 to get the correct values for rate per period and total periods.

Question 4. A bank pays 8% per annum interest compounded quarterly. Find equal deposits to be made at the end of each quarter for 10 years to have Rs 30,200? \( [(1.02)^{40} = 2.2080] \)
Answer:
Given:
Future amount of annuity, \( A = Rs \ 30,200 \)
Annual interest rate = 8%
Compounded quarterly, so \( k = 4 \)
Number of years, \( n = 10 \)

Interest rate per quarter, \( \frac{i}{k} = \frac{8\%}{4} = 2\% = 0.02 \).
Total number of quarters, \( nk = 10 \times 4 = 40 \).

We need to find the payment per quarter, \( a \). The formula for the amount of an ordinary annuity is:
\[ A = \frac{a}{\frac{i}{k}} \left[ \left(1 + \frac{i}{k}\right)^{nk} - 1 \right] \] Substitute the known values:
\[ 30200 = \frac{a}{0.02} [(1 + 0.02)^{40} - 1] \] \[ 30200 = \frac{a}{0.02} [(1.02)^{40} - 1] \] We are given \( (1.02)^{40} = 2.2080 \).
\[ 30200 = \frac{a}{0.02} [2.2080 - 1] \] \[ 30200 = \frac{a}{0.02} [1.2080] \] Now, solve for \( a \):
\[ 30200 \times 0.02 = a \times 1.2080 \] \[ 604 = a \times 1.2080 \] \[ a = \frac{604}{1.2080} \] \[ a = 500 \] So, equal deposits of Rs 500 must be made at the end of each quarter.
In simple words: To save Rs 30,200 in 10 years with a bank that gives 8% interest every year (paid every three months), you need to put in Rs 500 at the end of each three-month period. This calculation helps determine how much to save regularly to reach a financial goal.

🎯 Exam Tip: When finding the payment 'a', rearrange the annuity formula carefully. Always double-check your division and multiplication steps, especially with decimals.

Question 5. A person deposits Rs 2,000 from his salary towards his contributory pension scheme. The same amount is credited by his employer also. If an 8% rate of compound interest is paid, then find the maturity amount at end of 20 years of service. \( [(1.0067)^{240} = 4.966] \)
Answer:
Given:
Employee's deposit = Rs 2,000
Employer's contribution = Rs 2,000
Total contribution per period, \( a = Rs \ 2,000 + Rs \ 2,000 = Rs \ 4,000 \).
Service period, \( n = 20 \) years.
Interest rate = 8% per annum, compounded monthly.

Since interest is compounded monthly, \( k = 12 \).
Interest rate per month, \( \frac{i}{k} = \frac{8\%}{12} = \frac{0.08}{12} \approx 0.006666... \approx 0.0067 \).
Total number of months, \( nk = 20 \times 12 = 240 \).

Using the formula for the amount of an ordinary annuity:
\[ A = \frac{a}{\frac{i}{k}} \left[ \left(1 + \frac{i}{k}\right)^{nk} - 1 \right] \] Substitute the values:
\[ A = \frac{4000}{\frac{0.08}{12}} \left[ \left(1 + \frac{0.08}{12}\right)^{240} - 1 \right] \] \[ A = \frac{4000 \times 12}{0.08} [(1 + 0.0067)^{240} - 1] \] \[ A = 600000 [(1.0067)^{240} - 1] \] We are given \( (1.0067)^{240} = 4.966 \).
\[ A = 600000 [4.966 - 1] \] \[ A = 600000 [3.966] \] \[ A = 23,79,600 \] The maturity amount at the end of 20 years is Rs 23,79,600.
In simple words: When a person and their employer both put Rs 2,000 into a pension plan every month for 20 years, and the money grows with 8% interest each year, they will have Rs 23,79,600 at the end. This shows the big benefit of a matching employer contribution and long-term saving.

🎯 Exam Tip: Remember to add both the employee and employer contributions to find the total 'a' (payment per period). Also, convert annual interest rates and periods to match the compounding frequency (e.g., monthly).

Question 6. Find the present value of Rs 2,000 per annum for 14 years at the rate of interest of 10% per annum. \( [(1.04)^{-14} = 0.6252] \)
Answer:
Given:
Annual payment, \( a = Rs \ 2,000 \)
Number of years, \( n = 14 \)
Interest rate, \( i = 10\% = 0.1 \)

The formula for the present value of an ordinary annuity is:
\[ P = \frac{a}{i} [1 - (1+i)^{-n}] \] Substitute the given values into the formula:
\[ P = \frac{2000}{0.1} [1 - (1 + 0.1)^{-14}] \] \[ P = 20000 [1 - (1.1)^{-14}] \] The question seems to have a typo in the given value, providing \( (1.04)^{-14} = 0.6252 \) instead of \( (1.1)^{-14} \). Assuming the intent was for \( (1.1)^{-14} \), and if we follow the pattern of similar problems, we would need the value for \( (1.1)^{-14} \). However, since a specific value is provided in the question's hint `[(1.04)-14 = 0.6252]`, it's highly likely this is intended to be used directly or there's a misprint in the base of the power. If we assume the source intended for a slightly different interest rate that yields 1.04, or the question meant to give a direct value for `(1+i)^-n`, let's check the given solution steps. The solution uses `[1 - 0.2632]`, implying `(1.1)^-14` is `0.2632`. This value is consistent with `(1.1)^-14`. Let's use `(1.1)^{-14} = 0.2632` from standard tables/calculation for `1.1^-14`. \[ P = 20000 [1 - 0.2632] \] \[ P = 20000 [0.7368] \] \[ P = 14736 \] Therefore, the present value of the annuity is Rs 14,736.
In simple words: The present value of an annuity tells you how much money you would need right now to be able to make Rs 2,000 payments for 14 years, with a 10% interest rate. In this case, that amount is Rs 14,736. It's like finding the lump sum you'd need today to cover future regular payments.

🎯 Exam Tip: Be very careful with negative exponents in present value calculations. A common mistake is to miscalculate \( (1+i)^{-n} \) or use the future value factor instead.

Question 7. Find the present value of an annuity of Rs 900 payable at the end of 6 months for compounded at 8% per annum. \( [(1.04)^{-12} = 0.6252] \)
Answer:
Given:
Payment per period, \( a = Rs \ 900 \)
Interest rate = 8% per annum, compounded semi-annually (every 6 months).
The total number of periods is not explicitly given, but the hint \( [(1.04)^{-12} = 0.6252] \) suggests \( nk=12 \). Since payments are "at the end of 6 months", the payment period is semi-annual.

Since compounded semi-annually, \( k=2 \).
Interest rate per period, \( \frac{i}{k} = \frac{8\%}{2} = 4\% = 0.04 \).
Total number of periods, \( nk = 12 \). (This implies a total of 6 years because \( 6 \text{ years} \times 2 \text{ periods/year} = 12 \text{ periods} \)).

The formula for the present value of an ordinary annuity is:
\[ P = \frac{a}{\frac{i}{k}} \left[1 - \left(1 + \frac{i}{k}\right)^{-nk}\right] \] Substitute the values:
\[ P = \frac{900}{0.04} [1 - (1 + 0.04)^{-12}] \] \[ P = 22500 [1 - (1.04)^{-12}] \] We are given \( (1.04)^{-12} = 0.6252 \).
\[ P = 22500 [1 - 0.6252] \] \[ P = 22500 [0.3748] \] \[ P = 8433 \] Therefore, the present value of the annuity is Rs 8,433.
In simple words: If someone promises to pay you Rs 900 every six months for 6 years, and the money grows at 8% interest each year, the value of all those future payments today is Rs 8,433. This is what you would pay upfront to receive those future payments.

🎯 Exam Tip: When the payment frequency and compounding frequency match (e.g., both semi-annual), ensure 'a' is the payment per period, and adjust 'i' and 'n' to reflect the period's rate and total number of periods.

Question 8. Find the amount at the end of 12 years of an annuity of Rs 5,000 payable at the beginning of each year, if the money is compounded at 10% per annum.
Answer:
Given:
Annual payment, \( a = Rs \ 5,000 \)
Number of years, \( n = 12 \)
Interest rate, \( i = 10\% = 0.1 \)
Payments are made at the beginning of each year, which means it's an annuity due.

The formula for the amount of an annuity due is:
\[ A_{\text{due}} = \frac{a}{i} [(1+i)^n - 1] (1+i) \] Substitute the values:
\[ A_{\text{due}} = \frac{5000}{0.1} [(1 + 0.1)^{12} - 1] (1 + 0.1) \] \[ A_{\text{due}} = 50000 [(1.1)^{12} - 1] (1.1) \] Using the value for \( (1.1)^{12} = 3.1384 \) (from Question 1, as it's a common factor in these problems).
\[ A_{\text{due}} = 50000 [3.1384 - 1] (1.1) \] \[ A_{\text{due}} = 50000 [2.1384] (1.1) \] \[ A_{\text{due}} = 106920 \times 1.1 \] \[ A_{\text{due}} = 1,17,612 \] Thus, the amount of the annuity due at the end of 12 years is Rs 1,17,612.
In simple words: If you start saving Rs 5,000 at the beginning of each year for 12 years, and your money earns 10% interest annually, you will have Rs 1,17,612 by the end. This is higher than a regular annuity because the money starts earning interest earlier each year.

🎯 Exam Tip: For an annuity due, payments are made at the beginning of each period. This means the formula for an ordinary annuity's amount needs to be multiplied by \( (1+i) \) to account for the extra period of interest earned on each payment.

Question 9. Find the present value of an annuity due of Rs 1,500 for 16 years at 8% per annum? \( [(1.08)^{15} = 3.172] \)
Answer:
Given:
Payment per period, \( a = Rs \ 1,500 \)
Number of years, \( n = 16 \)
Interest rate, \( i = 8\% = 0.08 \)
This is an annuity due, meaning payments are made at the beginning of each period.

The formula for the present value of an annuity due is:
\[ P_{\text{due}} = \frac{a}{i} [1 - (1+i)^{-n}] (1+i) \] This can also be written as:
\[ P_{\text{due}} = a \left[ \frac{1-(1+i)^{-(n-1)}}{i} + 1 \right] \] Let's use the first form and substitute the values:
\[ P_{\text{due}} = \frac{1500}{0.08} [1 - (1 + 0.08)^{-16}] (1 + 0.08) \] \[ P_{\text{due}} = 18750 [1 - (1.08)^{-16}] (1.08) \] We know that \( (1.08)^{-16} = (1.08)^{-15} \times (1.08)^{-1} \).
We are given \( (1.08)^{15} = 3.172 \), so \( (1.08)^{-15} = \frac{1}{3.172} \approx 0.31525 \).
The provided solution steps simplify to:
\[ P_{\text{due}} = 18750 \times 1.08 \times [1 - (1.08)^{-16}] \] \[ P_{\text{due}} = 20250 [1 - (1.08)^{-16}] \] Alternatively, using the exact steps from the solution given:
\[ P_{\text{due}} = 18750 \left[1.08 - \frac{1}{(1.08)^{15}}\right] \] We are given \( (1.08)^{15} = 3.172 \). So, \( \frac{1}{(1.08)^{15}} = \frac{1}{3.172} \approx 0.31524 \).
\[ P_{\text{due}} = 18750 [1.08 - 0.31524] \] \[ P_{\text{due}} = 18750 [0.76476] \] \[ P_{\text{due}} \approx 14339.25 \] Rounding to the nearest whole number as in the original solution.
\[ P_{\text{due}} = 14340 \] Therefore, the present value of the annuity due is Rs 14,340.
In simple words: If someone is going to pay you Rs 1,500 at the start of each year for 16 years, and the money could grow at 8% annual interest, then the total value of all those future payments right now is Rs 14,340. This is the lump sum that has the same value as all those future payments.

🎯 Exam Tip: For present value of an annuity due, the payments start immediately. Remember to use the \( (1+i) \) factor or adjust the number of periods in the formula correctly, typically using \( (n-1) \) for the discount factor.

Question 10. What is the amount of perpetual annuity of Rs 50 at 5% compound interest per year?
Answer:
Given:
Annual payment, \( a = Rs \ 50 \)
Interest rate, \( i = 5\% = 0.05 \)
This is a perpetual annuity, also known as a perpetuity, meaning payments continue indefinitely.

The formula for the present value of a perpetuity is:
\[ P = \frac{a}{i} \] Substitute the given values:
\[ P = \frac{50}{0.05} \] \[ P = \frac{50}{\frac{5}{100}} \] \[ P = \frac{50 \times 100}{5} \] \[ P = 10 \times 100 \] \[ P = 1000 \] The present value of the perpetual annuity is Rs 1,000.
In simple words: A perpetual annuity means you receive a fixed payment forever. To find out what that endless stream of Rs 50 payments is worth today, if the interest rate is 5%, you would need Rs 1,000. This is the amount of money you would need to invest once to generate those Rs 50 payments endlessly.

🎯 Exam Tip: The present value of a perpetuity is one of the simplest annuity formulas, but students often confuse it with ordinary annuities. Remember, there's no 'n' (number of periods) in the perpetuity formula because payments go on forever.