Maharashtra Board Class 12 Maths Part 2 Chapter 2 Insurance and Annuity Miscellaneous Solutions

Exercise 2(I) Choose The Correct Alternative

Question 1. "A contract that pledges payment of an agreed-upon amount to the person (or his/her nominee) on the happening of an event covered against" is technically known as
(a) Death coverage
(b) Saving for future
(c) Life insurance
(d) Provident fund
Answer: (c) Life insurance
In simple words: Life insurance is a contract where the insurer pays a sum of money upon the death of the insured or after a set period, protecting against financial loss from specific events.

🎯 Exam Tip: Understanding the fundamental definition of insurance contracts is crucial for MCQs. Focus on the core purpose of financial protection against specific events.

Question 2. Insurance companies collect a fixed amount from their customers at a fixed interval of time. This amount is called
(a) EMI
(b) Installment
(c) Contribution
(d) Premium
Answer: (d) Premium
In simple words: The regular payment made by a policyholder to an insurance company to maintain their coverage is called a premium.

🎯 Exam Tip: Distinguish between premium and other financial terms. Premium is the cost of insurance coverage.

Question 3. Following are different types of insurance.
I. Life insurance
II. Health insurance
III. Liability insurance
(a) Only I
(b) Only II
(c) Only III
(d) All the three
Answer: (d) All the three
In simple words: Life, health, and liability are all distinct and common categories of insurance designed to protect against different types of financial risks.

🎯 Exam Tip: Be familiar with the broad classifications of insurance policies. This question tests general knowledge of insurance types.

Question 4. By taking insurance, an individual
(a) Reduces the risk of an accident
(b) Reduces the cost of an accident
(c) Transfers the risk to someone else
(d) Converts the possibility of large loss to the certainty of a small one
Answer: (d) Converts the possibility of large loss to the certainty of a small one
In simple words: Insurance doesn't prevent risks, but it converts the uncertain financial impact of a large potential loss into a certain, smaller, regular payment (premium).

🎯 Exam Tip: Understand the fundamental economic principle behind insurance: risk transfer and converting uncertainty into certainty through predictable costs.

Question 5. You get payments of Rs. 8,000 at the beginning of each year for five years ta 6%, what is the value of this annuity?
(a) Rs. 34,720
(b) Rs. 39,320
(c) Rs. 35,720
(d) Rs. 40,000
Answer: (c) Rs. 35,720
In simple words: This question asks for the present value of an annuity due, where payments are made at the beginning of each period, considering a 6% interest rate over five years. The calculation involves using the present value of an annuity due formula.

🎯 Exam Tip: For annuity problems, correctly identify whether it's an ordinary annuity or an annuity due, as this affects the formula. Pay attention to the interest rate and number of periods.

Question 6. In an ordinary annuity, payments or receipts occur at
(a) Beginning of each period
(b) End of each period
(c) Mid of each period
(d) Quarterly basis
Answer: (b) End of each period
In simple words: An ordinary annuity is characterized by payments or receipts occurring at the end of each specified period.

🎯 Exam Tip: Memorize the definition of an ordinary annuity (payments at the end of the period) versus an annuity due (payments at the beginning of the period).

Question 7. The amount of money today which is equal to a series of payments in the future is called
(a) Normal value of the annuity
(b) Sinking value of the annuity
(c) Present value of the annuity
(d) Future value of the annuity
Answer: (c) Present value of the annuity
In simple words: The present value of an annuity is the single lump sum amount today that is equivalent to a series of future payments, discounted back to the present.

🎯 Exam Tip: Understand the difference between present value (value today) and future value (value at a future point) in the context of annuities.

Question 8. Rental payment for an apartment is an example of
(a) Annuity due
(b) Perpetuity
(c) Ordinary annuity
(d) Installment
Answer: (b) Perpetuity
In simple words: Rental payments that continue indefinitely are an example of a perpetuity, a type of annuity where payments are expected to continue forever.

🎯 Exam Tip: Recognize real-world examples of financial concepts. Perpetuities are annuities with an infinite number of payments.

Question 9. - is a series of constant cash flows over a limited period of time.
(a) Perpetuity
(b) Annuity
(c) Present value
(d) Future value
Answer: (b) Annuity
In simple words: An annuity is a sequence of equal payments made at regular intervals over a specific, finite period.

🎯 Exam Tip: A key characteristic of an annuity is that payments are constant and occur over a defined, limited period, unlike a perpetuity which is infinite.

Question 10. A retirement annuity is particularly attractive to someone who has
(a) A severe illness
(b) Risk of low longevity
(c) Large family
(d) Chance of high longevity
Answer: (d) Chance of high longevity
In simple words: A retirement annuity provides a steady income stream for life, making it beneficial for those who expect to live a long time and want to ensure their financial security throughout retirement.

🎯 Exam Tip: Connect the purpose of financial products to individual needs. Annuities mitigate the risk of outliving one's savings.

Exercise 2(II) Fill In The Blanks

Question 1. An installment of money paid for insurance is called -
Answer: premium
In simple words: The regular payment for an insurance policy is known as a premium.

🎯 Exam Tip: Basic terminology is essential. Remember that the cost of insurance is the premium.

Question 2. General insurance covers all risks except -
Answer: life
In simple words: General insurance covers non-life risks like property, health, or liability, excluding risks related to human life which are covered by life insurance.

🎯 Exam Tip: Differentiate between life insurance and general insurance. General insurance typically covers everything but life itself.

Question 3. The value of insured property is called -
Answer: property value
In simple words: The property value is the market worth of the asset being insured.

🎯 Exam Tip: Understand the terms related to insurance policies, distinguishing between property value and policy value.

Question 4. The proportion of property value to insured is called -
Answer: policy value
In simple words: The policy value refers to the specific amount for which the property is insured, often a proportion of its total market value.

🎯 Exam Tip: Policy value is the sum for which an asset is officially covered by the insurance agreement, which can be less than the actual property value.

Question 5. The person who receive annuity is called -
Answer: Annuitant
In simple words: An annuitant is the individual who receives the periodic payments from an annuity.

🎯 Exam Tip: Know the specific terminology for financial roles, such as annuitant for someone receiving annuity payments.

Question 6. The payment of each single annuity is called -
Answer: installment
In simple words: Each individual payment within a series of annuity payments is referred to as an installment.

🎯 Exam Tip: Installment is a common term for any single payment in a series, including annuity payments.

Question 7. The intervening time between payment of two successive installments is called as -
Answer: payment period
In simple words: The payment period is the regular interval between two consecutive payments in an annuity.

🎯 Exam Tip: Clearly define terms like 'payment period' in the context of annuities to avoid confusion with other time-related financial terms.

Question 8. An annuity where payments continue forever is called -
Answer: perpetuity
In simple words: A perpetuity is an annuity that makes payments indefinitely, never ending.

🎯 Exam Tip: The key differentiator for a perpetuity is its infinite duration of payments, unlike a standard annuity with a fixed term.

Question 9. If payments of an annuity fall due at the beginning of every period, the series is called -
Answer: annuity due
In simple words: An annuity due is an annuity where payments are made at the beginning of each period.

🎯 Exam Tip: Differentiate an annuity due (payments at the beginning) from an ordinary annuity (payments at the end) as this impacts calculations.

Question 10. If payments of an annuity fall due at the end of every period, the series is called annuity -
Answer: immediate
In simple words: An annuity immediate, also known as an ordinary annuity, is where payments are made at the end of each period.

🎯 Exam Tip: "Annuity immediate" is another name for an ordinary annuity, where payments are made at the end of each interval.

Exercise 2(III) State Whether Each Of The Following Is True Or False.

Question 1. General insurance covers life, fire, and theft.
Answer: False
In simple words: General insurance covers fire and theft, but life insurance covers life, as these are distinct categories of insurance.

🎯 Exam Tip: Remember the basic distinction: life insurance covers life, while general insurance covers other specific risks like fire, theft, health, etc.

Question 2. The amount of claim cannot exceed the amount of loss.
Answer: True
In simple words: Insurance is based on the principle of indemnity, meaning the insured should only be compensated for the actual loss suffered, not profit from it.

🎯 Exam Tip: The principle of indemnity is fundamental to insurance, ensuring that the claimant is restored to their pre-loss financial position, not a better one.

Question 3. Accident insurance has a period of five years.
Answer: False
In simple words: Accident insurance policies typically have a period of one year and are renewable, not fixed for five years.

🎯 Exam Tip: Most general insurance policies, including accident insurance, are short-term (usually annual) and require renewal.

Question 4. Premium is the amount paid to the insurance company every month.
Answer: True
In simple words: Premiums are indeed regular payments, often monthly, made to the insurer for coverage.

🎯 Exam Tip: While premiums can be paid annually, semi-annually, or quarterly, monthly payments are a very common and valid option, making the statement true in a general sense.

Question 5. Payment of every annuity is called an installment.
Answer: False
In simple words: While annuity payments are installments, not every installment is an annuity payment; an installment is a broader term for any part payment.

🎯 Exam Tip: An installment is a general term for a part of a larger sum; annuity payments are a specific type of installment that occur regularly over a period.

Question 6. Annuity certainly begins on a fixed date and ends when an event happens.
Answer: True
In simple words: A contingent annuity's payments are tied to a specific event, such as a death or a person reaching a certain age, rather than a fixed end date.

🎯 Exam Tip: Differentiate between an annuity certain (fixed start and end dates) and a contingent annuity (start/end tied to an event).

Question 7. Annuity contingent begins and ends on certain fixed dates.
Answer: False
In simple words: A contingent annuity's duration is uncertain as its start or end is dependent on a specific event, not fixed dates.

🎯 Exam Tip: A contingent annuity's defining feature is its dependence on an uncertain event, which contradicts the idea of fixed start and end dates.

Question 8. The present value of an annuity is the sum of the present value of all installments.
Answer: True
In simple words: The present value of an annuity is calculated by discounting each future payment back to the present and summing these individual present values.

🎯 Exam Tip: This statement directly reflects the definition and calculation method of the present value of an annuity, where each future cash flow is individually discounted.

Question 9. The future value of an annuity is the accumulated value of all installments.
Answer: False
In simple words: The future value of an annuity is the sum of the future values of all installments, compounded forward to the end of the annuity period, not just their accumulated value.

🎯 Exam Tip: It's important to understand that future value involves compounding each installment, not simply summing them, to account for interest earned over time.

Question 10. The sinking fund is set aside at the beginning of a business.
Answer: True
In simple words: A sinking fund is an amount saved periodically, typically starting early in a project or business's life, to meet a future financial obligation like replacing an asset or repaying debt.

🎯 Exam Tip: Sinking funds are a form of financial planning for future liabilities, so establishing them early allows for smaller, more manageable periodic contributions due to the power of compounding.

Exercise 2(IV) Solve The Following Problems.

Question 1. A house valued at Rs. 8,00,000 is insured at 75% of its value. If the rate of premium is 0.80%. Find the premium paid by the owner of the house. If the agent's commission is 9% of the premium, find the agent's commission.
Solution:
Property value = Rs. 8,00,000
Policy value = 75% \( \times \) 8,00,000 = Rs. 6,00,000
Rate of Premium = 0.80%
Amount of Premium = 0.80% \( \times \) 6,00,000 = Rs. 4,800
Rate of commission = 9%
Agent commission = 9% \( \times \) 4800 = Rs. 432
In simple words: First, calculate the policy value as 75% of the house's total value. Then, determine the premium by applying the 0.80% rate to this policy value. Finally, calculate the agent's commission as 9% of the premium paid.

🎯 Exam Tip: Carefully follow the sequential steps: calculate policy value, then premium, then commission. Ensure you apply percentages to the correct base amounts at each step.

Question 2. A shopkeeper insures his shop and godown are valued at Rs. 5,00,000 and Rs. 10,00,000 respectively for 80% of their values. If the rate of premium is 8%, find the total annual premium.
Solution:
Property value of shop = Rs. 5,00,000
Policy value = 80% \( \times \) 5,00,000 = Rs. 4,00,000
Rate of Premium = 8%
Amount of premium = 8% \( \times \) 4,00,000 = Rs. 32,000
Property value of Godown = Rs. 10,00,000
Policy value = 80% \( \times \) 10,00,000 = Rs. 8,00,000
Rate of Premium = 8%
Amount of Premium = 8% \( \times \) 8,00,000 = Rs. 64,000
Total annual Premium = 64,000 + 32,000 = Rs. 96,000
In simple words: Calculate the policy value for the shop and the godown separately, which is 80% of their respective property values. Then, calculate the premium for each by applying an 8% rate to their policy values. The total annual premium is the sum of these two individual premiums.

🎯 Exam Tip: When dealing with multiple insured items, treat each calculation (policy value, premium) separately before summing up for the total. Be careful with percentage calculations.

Question 3. A factory building is insured for \( \left(\frac{5}{6}\right)^{\text {th }} \) of its value at a rate of premium of 2.50%. If the agent is paid a commission of Rs. 2,812.50, which is 7.5% of the premium, find the value of the building.
Solution:
Let the Property value be x
Policy value = \( \frac{5x}{6} \)
Rate of premium = 2.50%
Amount of premium = \( \frac{5x}{6} \times 2.50\% = \frac{x}{48} \)
Rate of Agent commission = 7.5%
Agent commission = \( 7.5\% \times \frac{x}{48} \)

\( \implies 2812.50 = \frac{x}{640} \)

\( \implies 2812.50 \times 640 = x \)

\( \implies x = \) Rs. 18,00,000
Value of the building is 18,00,000.
In simple words: Start by using the agent's commission and commission rate to find the total premium amount. Then, knowing the premium rate, calculate the policy value. Finally, use the policy value (which is 5/6ths of the building's value) to determine the full value of the building.

🎯 Exam Tip: Work backward from the known values. First find the total premium from the agent's commission, then the policy value, and finally the property value. Be precise with fractions and percentages.

Question 4. A merchant takes a fire insurance policy to cover 80% of the value of his stock. Stock worth Rs. 80,000 was completely destroyed in a fire. While the rest of the stock was reduced to 20% of its value. If the proportional compensation under the policy was 67,200, find the value of the stock.
Solution:
Let the Property value be x
Policy value 80% \( \times \) x = Rs. \( \frac{4x}{5} \)
Complete loss = Rs. 80,000
Partial loss = 20% \( \times \) (x – 8,00,000) = \( \frac{x-8,00,000}{5} \)
Total loss = 80,000 + \( \frac{x-8,00,000}{5} \) = \( \frac{x-80,000}{5} \) + 64,000
Claim = Rs. 67,200
\[ \frac{\text { Policy value }}{\text { Property value }} \times \text { loss }=67,200 \]
\[ \frac{\frac{4x}{5}}{x} \times \left[80,000+\frac{x-8,00,000}{5}\right]=67,200 \]
\[ \frac{4x}{25}+\frac{4 \times 64,000}{5}=67,200 \]
\[ \frac{4x}{25}+51,200=67,200 \]
\[ \frac{4x}{25}=67,200-51,200 \]
\[ \frac{4x}{25}=16,000 \]
\[ x = \frac{16,000 \times 25}{4} \]

\( \implies x = \) Rs. 1,00,000
The value of the stock is Rs. 1,00,000.
In simple words: Set up an equation for the total loss, considering both completely destroyed and partially damaged stock. Use the proportional compensation formula (Claim = (Policy Value / Property Value) * Total Loss) to solve for the unknown total value of the stock.

🎯 Exam Tip: This problem involves calculating total loss from partial and complete destruction. Ensure the proportional claim formula is applied correctly, and be careful with algebraic manipulation to solve for 'x'.

Question 5. A 35-year old person takes a policy for Rs. 1,00,000 for a period of 20 years. The rate of premium is 76 and the average rate of bonus is 7 per thousand p.a. If he dies after paying 10 annual premiums, what amount will his nominee receive?
Solution:
Policy value = Rs. 1,00,000
Period of Policy = 20 years
Rate of premium = 76 per thousand
Amount of premium = \( \frac{76}{1,000} \times 1,00,000 \) = Rs. 7,600
Total Premium = 7,600 \( \times \) 10 = Rs. 76,000
Rate of Bonus = Rs. 7 per thousand p.a
Total Bonus = \( \frac{7}{1,000} \times 1,00,000 \) = Rs. 7,000
Amount received by Nominee = Policy value + Bonus earned
= 1,00,000 + 7,000
= Rs. 1,07,000
In simple words: Calculate the annual premium based on the policy value and premium rate. Then, determine the total bonus earned over the 10 years by applying the bonus rate to the policy value. The total amount the nominee receives is the sum of the policy value and the accrued bonus.

🎯 Exam Tip: Be careful with the "per thousand" unit for premium and bonus rates. Calculate annual premium, total bonus, and then sum these with the policy value for the final nominee payout. Note that the total premiums paid (Rs. 76,000) are not directly added to the policy value for the nominee's payout, but rather the policy value itself (Rs. 1,00,000) plus the bonus.

Question 6. 15,000 articles costing 200 per dozen were insured against fire for 1,00,000. If 20% of the articles were burnt completely and 2,400 other articles were damaged to the extent of 80% of their value, find the amount that can be claimed under the policy.
Solution:
Total Articles = 15,000
Property value = \( \frac{15,000}{12} \times 200 \) = 2,50,000
Policy value = Rs. 1,00,000
Complete loss = 20% \( \times \) 2,50,000 = Rs. 50,000
Partial loss = 80% \( \times \frac{2,400}{12} \times 200 \) = Rs. 3,20,000
Total loss = 32,000 + 50,000 = Rs. 82,000
\[ \text { Claim }=\frac{\text { Policy value }}{\text { Property value }} \times \text { Loss } \]
\[ =\frac{1,00,000}{2,50,000} \times 82,000 \]
= Rs. 32,800
In simple words: First, determine the total property value of all articles. Calculate the value of completely lost articles and partially damaged articles separately. Sum these to find the total loss. Finally, use the formula for proportional claim (Claim = (Policy Value / Property Value) \( \times \) Loss) to find the amount that can be claimed.

🎯 Exam Tip: Break down the loss into components: completely destroyed and partially damaged. Ensure you calculate the value of damaged articles correctly (2,400 articles is 200 dozen). Apply the average clause accurately for proportional claims.

Question 7. For what amount should a cargo worth Rs. 25,350 be insured so that in the event of a total loss, its value, as well as the cost of insurance, may be recovered when the rate of premium is 2.5%.
Solution:
Let the policy value be Rs. 100 which includes the cost of insurance and premium
Property value = 100 – 2.50 = Rs. 97.50
If the value of the cargo is 97.50, then the policy value is Rs. 100.
If the value of the cargo is 25,350, then
Policy value = \( \frac{100 \times 25,350}{97.50} \) = Rs. 26,000
In simple words: To recover both the cargo's value and the premium in case of total loss, the insurance policy value must effectively cover 100% of the cargo's value plus the premium itself. Set up a proportion where 97.5% (100% - 2.5% premium) of the policy value equals the cargo's worth, then solve for the total policy value.

🎯 Exam Tip: This is a "grossing up" problem. The insured amount must be high enough to cover the actual loss AND the premium paid. Treat the cargo's value as 100% minus the premium rate, then use cross-multiplication.

Question 8. A cargo of grain is insured at \( \left(\frac{3}{4}\right)\% \) to cover 70% of its value. Rs. 1,008 is the amount of premium paid. If the grain is worth 12 per kg, how many kg of the grain did the cargo contain?
Solution:
Let the Property value be x
policy value = 70% \( \times \) x = Rs. \( \frac{7x}{10} \)
Rate of premium = \( \frac{3}{4}\% \)
Amount of premium = Rate \( \times \) Policy value
\[ 1,008=\frac{7x}{10} \times \frac{3}{4} \times \frac{1}{100} \]
\[ 1,008 = \frac{21x}{4,000} \]
\[ x = \frac{1,008 \times 4,000}{21} \]

\( \implies x = \) Rs. 1,92,000
Rate of Jowar = Rs. 12/kg
Quantity of Jowar = \( \frac{1,92,000}{12} \) = 16,000 kgs
In simple words: First, express the policy value in terms of the total property value 'x'. Use the given premium amount and premium rate to set up an equation to solve for 'x', the total property value. Finally, divide the total property value by the worth per kg to find the total quantity of grain in kgs.

🎯 Exam Tip: Carefully convert percentages and fractions into decimals or simpler fractions for calculation. Ensure all steps are correctly laid out from premium to property value, then to quantity.

Question 9. 4,000 bedsheets worth Rs. 6,40,000 were insured for \( \left(\frac{3}{7}\right)^{\text {th }} \) of their value. Some of the bedsheets were damaged in the rainy season and were reduced to 40% of their value. If the amount recovered against damage was 32,000. Find the number of damaged bedsheets.
Solution:
Property value = Rs. 6,40,000
Policy value = \( 6,40,000 \times \frac{3}{7} = \text { Rs. } \frac{19,20,000}{7} \)
Cost of one Bedsheet = \( \frac{6,40,000}{4,000} \) = Rs. 160
Let 'x' bedsheets be damaged.
Cost of x bedsheets = Rs. 160x
Value of loss = \( 160x \times \frac{40}{100} \) = Rs. 64x
\[ \text { Claim }=\frac{\text { Policy value }}{\text { Property value }} \times \text { Loss } \]
\[ 32,000=\frac{\frac{19,20,000}{7}}{6,40,000} \times 64x \]
\[ 32,000=\frac{19,20,000 \times 64x}{6,40,000 \times 7} \]
\[ 32,000=\frac{192}{7} x \]
\[ x=\frac{32,000 \times 7}{192} \]

\( \implies x = 875 \)
875 Bedsheets damaged.
In simple words: Calculate the value of one bedsheet. If 'x' bedsheets are damaged, determine the total loss for these 'x' bedsheets (reduced to 40% of their value). Using the given claim amount and the average clause formula (Claim = (Policy Value / Property Value) * Loss), solve for 'x', the number of damaged bedsheets.

🎯 Exam Tip: This problem requires calculating the unit cost of an item first. Ensure the loss calculation for the damaged items is correct (value reduced to 40% means 60% loss). Apply the proportional claim formula to find the number of damaged units.

Question 10. A property valued at Rs. 7,00,000 is insured to the extent of Rs. 5,60,000 at \( \left(\frac{5}{8}\right)\% \) less 20%. Calculate the saving made in the premium. Find the amount of loss that the owner must bear, including premium, if the property is damaged to the extent of 40% of its value.
Solution:
Property value = Rs. 7,00,000
Policy value = Rs. 5,60,000
Rate of premium = \( \frac{5}{8}\% \)
Amount of premium = \( \frac{5}{8}\% \times 5,60,000 \) = Rs. 3,500
New rate of premium = \( \frac{5}{8}\% \) less 20%
\[ =\frac{5}{8}\left[1-20 \% \times \frac{5}{8}\right] \]
\[ =\frac{5}{8}\left[1-\frac{1}{4}\right] \]
\[ =\frac{5}{8} \times \frac{3}{4} = \frac{15}{32}\% \]
Amount of premium = \( \frac{15}{32}\% \times 5,60,000 \) = Rs. 2,800
Saving made in premium = 3,500 – 2,800 = Rs. 700
Loss = 7,00,000 \( \times \) 40% = 2,80,000
\[ \text { Claim }=\frac{\text { Policy value }}{\text { Property value }} \times \text { Loss } \]
\[ =\frac{5,60,000}{7,00,000} \times 2,80,000 \]
= Rs. 2,24,000
Loss bear by owner = loss – claim + premium
= 2,80,000 – 2,24,000 + 2,800
= Rs. 58,800
In simple words: First, calculate the original premium based on the initial rate. Then, calculate the new premium using the reduced rate (20% less). The difference is the saving. For the owner's loss, calculate the total damage (40% of property value), then determine the claim amount using the average clause. The owner's total burden is the total loss minus the claim received, plus the premium paid.

🎯 Exam Tip: This multi-part problem requires careful calculation of two premiums (original and new) to find the saving. For the owner's bearable loss, remember to include the premium paid, which is a cost to the owner. Be precise with percentages and fractions.

Question 11. Stocks in a shop and godown worth Rs. 75,000 and Rs. 1,30,000 respectively were insured through an agent who receive 15% of the premium as commission. If the shop was insured for 80% and godown for 60% of the value, find the amount of agent's commission when the premium was 0.80% less 20%. If the entire stock in the shop and 20% stock in the godown is destroyed by fire, find the amount that can be claimed under the policy.
Solution:
Rate of premium = 0.80% less 20%
= 0.80 – 20% \( \times \) 0.80
= 0.80 -0.16
= 0.64%
For Shop
Property value = Rs. 75,000
Policy value = 80% \( \times \) 75,000 = Rs. 60,000
Premium = 0.64% \( \times \) 60,000 = Rs. 384
Loss = Rs. 75,000
\[ \text { Claim }=\frac{\text { Policy value }}{\text { Property value }} \times \text { Loss } \]
\[ =\frac{60,000}{75,000} \times 75,000 \]
= Rs. 60,000
For Godown
Property value = Rs. 1,30,000
Policy value = 60% \( \times \) 1,30,000 = Rs. 78,000
Premium = 0.64% \( \times \) 78,000 = Rs. 499.2
Loss = 20% \( \times \) 1,30,000 = Rs. 26,000
\[ \text { Claim }=\frac{\text { Policy value }}{\text { Property value }} \times \text { Loss } \]
\[ =\frac{78,000}{1,30,000} \times 26,000 \]
= Rs. 15,600
Total claim = 15,600 + 60,000 = Rs. 75,600
Rate of commission = 15%
Agent commission = 15% \( \times \) [384 + 499.2]
= 15% \( \times \) 883.2
= Rs. 132.48
In simple words: First, calculate the effective premium rate after the "less 20%" discount. Then, for both the shop and godown separately, determine the policy value, premium paid, and claim amount for their respective losses (full loss for shop, 20% loss for godown). Sum the individual premiums to find the total premium and calculate the agent's 15% commission. Finally, sum the individual claim amounts to find the total claim.

🎯 Exam Tip: This is a complex problem with multiple parts. Systematically calculate the premium rate first. Then, for each asset (shop and godown), calculate policy value, premium, and claim individually. Finally, combine premiums for agent commission and combine claims for the total amount claimed.

Question 12. A person holding a life policy of Rs. 1,20,000 for a term of 25 years wants to discontinue after paying a premium for 8 years at the rate of Rs. 58 per thousand p.a. Find the amount of paid-up value he will receive on the policy. Find the amount he will receive if the surrender value granted is 35% of the premium paid, excluding the first year's premium.
Solution:
Policy value = Rs. 1,20,000
Rate of premium = Rs. 58 per thousand p.a.
Premium for 8 years = \( \frac{8 \times 58}{1000} \times 1,20,000 \) = Rs. 55,680
Amount of 1st premium = \( \frac{55,680}{8} \) = Rs. 6,960
\[ \text { Paid-up value of policy }=\frac{\text { No of Premium paid }}{\text { Terms of policy }} \times \text { Policy value } \]
\[ =\frac{8}{25} \times 1,20,000 \]
= Rs. 38,400
Surrender value = 35% \( \times \) [Total premium – 1st year premium]
= 35% \( \times \) [55,680 – 6,960]
= 35% \( \times \) 48,720
= Rs. 17,052
In simple words: First, calculate the total premium paid over 8 years and the amount of the first year's premium. Then, compute the paid-up value using the ratio of premiums paid to the total policy term, multiplied by the policy value. Separately, calculate the surrender value by taking 35% of the total premiums paid, excluding the first year's premium.

🎯 Exam Tip: This problem has two distinct parts: calculating the paid-up value and calculating the surrender value. Ensure you apply the correct formulas for each and pay close attention to the exclusion of the first year's premium for the surrender value calculation.

Question 13. A godown valued at Rs. 80,000 contained stock worth Rs. 4,80,000. Both were insured against fire. Godown for Rs. 50,000 and stock for 80% of its value. A part of stock worth 60,000 was completely destroyed and the rest was reduced to 60% of its value. The amount of damage to the godown is 40,000. Find the amount that can be claimed under the policy.
Solution:
For Godown
Property value = Rs. 80,000
Policy value = Rs. 50,000
Loss = Rs. 40,000
\[ \text { Claim }=\frac{\text { Policy value }}{\text { Property value }} \times \text { Loss } \]
\[ =\frac{50,000}{80,000} \times 40,000 \]
= Rs. 25,000
For stock
Property value = 4,80,000
Policy value = 80% \( \times \) 4,80,000 = Rs. 3,84,000
Complete loss = Rs. 60,000
Partial loss = (100 – 60)% \( \times \) [4,80,000 – 60,000]
= 40% \( \times \) 4,20,000
= Rs. 1,68,000
Total loss = 1,68,000 + 60000 = Rs. 2,28,000
\[ \text { Claim }=\frac{\text { Policy value }}{\text { Property value }} \times \text { Loss } \]
\[ =\frac{3,84,000}{4,80,000} \times 2,28,000 \]
= Rs. 1,82,400
Total claim = 25,000 + 1,82,400 = Rs. 2,07,400
In simple words: Calculate the claim for the godown using the average clause based on its property value, policy value, and damage. Then, for the stock, calculate the policy value (80% of its worth), determine the total loss from both complete and partial damage, and calculate the claim using the average clause. The final total claim is the sum of claims for the godown and the stock.

🎯 Exam Tip: Treat the godown and stock as separate insurance calculations. For the stock, carefully distinguish between complete loss and partial loss (60% value remaining means 40% loss on that portion). Apply the average clause formula correctly for both components.

Question 14. Find the amount of an ordinary annuity if a payment of Rs. 500 is made at the end of every quarter for 5 years at the rate of 12% per annum compounded quarterly. [Given: (1.03)20 = 1.8061]
Solution:
C = Rs. 500
r = 12% p.a. compounded quarterly,
r = \( \frac{12}{4} \) = 3%
n = 5 years
But, payment is made quarterly
n = 5 \( \times \) 4 = 20
i = \( \frac{r}{100} = \frac{3}{100} \) = 0.03
\[ A=\frac{C}{i}\left[(1+i)^{n}-1\right] \]
\[ A=\frac{500}{0.03}\left[(1+0.03)^{20}-1\right] \]
\[ A=\frac{500}{0.03}[1.8061-1] \]
\[ A=\frac{500}{0.03} \times 0.8061 \]
\[ A = 16,666.67 \times 0.8061 \]
\( \implies A \) = Rs. 13,435
In simple words: First, adjust the annual interest rate and the number of years to quarterly periods. Then, use the future value formula for an ordinary annuity, substituting the quarterly payment (C), quarterly interest rate (i), and total number of quarters (n) to calculate the final amount.

🎯 Exam Tip: It's critical to convert the annual interest rate and term to match the compounding frequency (quarterly in this case). Use the given value for \( (1+i)^n \) directly to simplify calculations for the annuity's future value.

Question 15. Find the amount a company should set aside at the end of every year if it wants to buy a machine expected to cost Rs. 1,00,000 at the end of 4 years and interest rate is 5% p.a. compounded annually.
Answer:
Solution:
\(\because\) A = Rs. 1,00,000
\(\therefore\) r = 5% p.a.
\(\therefore\) i = \( \frac{r}{100} = \frac{5}{100} = 0.05 \)
\(\therefore\) n = 4 years
\(\because\) A = \( \frac{C}{i} [(1 + i)^n - 1] \)
\(\therefore\) 1,00,000 = \( \frac{C}{0.05} [(1 + 0.05)^4 - 1] \)
\(\therefore\) 1,00,000 \(\times\) 0.05 = C [(1.05)^4 - 1]
\(\therefore\) 5,000 = C(1.2155 - 1)
\(\therefore\) 5,000 = C \(\times\) 0.2155
\(\therefore\) \( \frac{5,000}{0.2155} \) = C
\(\therefore\) C = Rs. 23,201.86
In simple words: To reach a target amount of Rs. 1,00,000 in 4 years at a 5% annual interest rate, the company needs to save Rs. 23,201.86 each year. This is calculated using the future value of an annuity formula.

🎯 Exam Tip: Pay close attention to the compounding frequency and the correct formula (future value of annuity) for sinking fund problems.

Question 16. Find the least number of years for which an annuity of Rs. 3,000 per annum must run in order that its amount exceeds Rs. 60,000 at 10%compounded annually. [Given: (1.1)11 = 2,8531, (1.1)12 = 3.1384]
Answer:
Solution:
\(\because\) A = Rs. 60,000
\(\therefore\) C = Rs. 3,000
\(\therefore\) r = 10% p.a.
\(\therefore\) i = \( \frac{r}{100} = \frac{10}{100} = 0.1 \)
\(\because\) A = \( \frac{C}{i} [(1 + i)^n - 1] \)
\(\therefore\) 60,000 = \( \frac{3,000}{0.1} [(1 + 0.1)^n - 1] \)
\(\therefore\) 60,000 = 30,000[(1.1)\(^{n}\) - 1]
\(\therefore\) \( \frac{60,000}{30,000} \) + 1 = (1.1)\(^{n}\)
\(\therefore\) 2 + 1 = (1.1)\(^{n}\)
\(\therefore\) 3 = (1.1)\(^{n}\)
Taking log
\(\therefore\) log 3 = log (1.1)\(^{n}\)
\(\therefore\) log 3 = n log(1.1)
\(\therefore\) \( \frac{\log 3}{\log 1.1} \) = n
\(\therefore\) n = \( \frac{0.4771}{0.0414} \) = 11.52 ~ 12 years
In simple words: To accumulate over Rs. 60,000 with an annual annuity of Rs. 3,000 at a 10% interest rate, it will take approximately 12 years. This is found by solving the future value of an annuity formula for 'n' using logarithms.

🎯 Exam Tip: Remember to use logarithms for solving 'n' (number of periods) in annuity problems and round up to the nearest whole number when looking for the "least number of years" to exceed a certain amount.

Question 17. Find the rate of interest compounded annually if an ordinary annuity of Rs. 20,000 per year amounts to Rs. 41,000 in 2 years.
Answer:
Solution:
\(\because\) C = Rs. 20,000
\(\because\) A = Rs. 41,000
\(\therefore\) n = 2 years
\(\therefore\) A = \( \frac{C}{i} [(1+i)^n -1] \)
\(\therefore\) 41,000 = \( \frac{20,000}{i} [(1+i)^2 -1] \)
\(\therefore\) \( \frac{41,000i}{20,000} \) = 1 + 2i + i\(^2\) -1
\(\therefore\) \( \frac{41i}{20} \) = i(2+i)
\(\therefore\) 41 = 40 + 20i
\(\therefore\) 41-40 = 20i
\(\therefore\) \( \frac{1}{20} \) = i
\(\therefore\) \( \frac{1}{20} = \frac{r}{100} \)
\(\therefore\) \( \frac{100}{20} \) = r
\(\therefore\) r = 5% p.a.
In simple words: If an annual payment of Rs. 20,000 grows to Rs. 41,000 in two years, the annual compound interest rate is 5%. This is calculated by substituting the given values into the future value of an ordinary annuity formula and solving for 'i' (interest rate).

🎯 Exam Tip: For problems requiring the interest rate, setting up the annuity formula correctly and simplifying the algebraic expression is crucial for accurate calculation.

Question 18. A person purchases a television by paying Rs. 20,000 in cash and promising to pay Rs. 1,000 at the end of every month for the next 2 years. If money is worth 12% p.a., converted monthly. Find the cash price of the television. [Given: (1.01)-24 = 0.7880]
Answer:
Solution:
Down payment = Rs. 20,000
\(\therefore\) n = 2 years
But, EMI Payable monthly
\(\therefore\) n = 2 \(\times\) 12 = 24
\(\therefore\) r = 12% p.a. compounded monthly
\(\therefore\) r = \( \frac{12}{12} \) = 1%
\(\therefore\) i = \( \frac{r}{100} = \frac{1}{100} = 0.01 \)
\(\because\) P = \( \frac{C}{i} [1-(1+i)^{-n}] \)
\(\therefore\) P = \( \frac{1000}{0.01} [1-(1+0.01)^{-24}] \)
\(\therefore\) P = 1,00,00 [1 -0.7880]
\(\therefore\) P = 1,00,00 \(\times\) 0.2120
\(\therefore\) P = Rs. 21,200
Cash price = Present value + Down payment
= 21,200 + 20,000
= Rs. 41,200
In simple words: The total cash price of the television is Rs. 41,200. This is calculated by adding the initial down payment to the present value of the monthly installments, discounted at a 1% monthly interest rate over 24 months.

🎯 Exam Tip: When dealing with monthly payments and annual interest rates, always convert the interest rate to a monthly rate and the number of years to months to ensure consistency in calculations.

Question 19. Find the present value of an annuity immediate of Rs. 20,000 per annum for 3 years at 10% p.a. compounded annually. [Given: (1.1)-3 = 0.7513]
Answer:
Solution:
\(\because\) C = Rs. 20,000
\(\therefore\) n = 3 years
\(\therefore\) r = 10% p.a.
\(\therefore\) i = \( \frac{r}{100} = \frac{10}{100} = 0.1 \)
\(\therefore\) P = \( \frac{C}{i} [1-(1+i)^{-n}] \)
\(\therefore\) P = \( \frac{20,000}{0.1} [1-(1+0.1)^{-3}] \)
\(\therefore\) P = 2,00,000 [1 – 0.7513]
\(\therefore\) P = 2,00,000 [0.2487]
\(\therefore\) P = Rs. 49,740
In simple words: The present value of an annuity paying Rs. 20,000 annually for 3 years at a 10% annual interest rate is Rs. 49,740. This means that Rs. 49,740 invested today at 10% annual interest could generate those future payments.

🎯 Exam Tip: Clearly distinguish between present value and future value formulas. For present value, ensure the negative exponent for 'n' in the formula is correctly applied.

Question 20. A man borrowed some money and paid it back in 3 equal installments of Rs. 2,160 each. What amount did he borrow if the rate of interest was 20% per annum compounded annually? Also, find the total interest charged. [Given: (1.2)-3 = 0.5788]
Answer:
Solution:
\(\because\) C = Rs. 2,160
\(\therefore\) n = 3
\(\therefore\) r = 20% p.a.
\(\therefore\) i = \( \frac{r}{100} = \frac{20}{100} = 0.2 \)
\(\because\) P = \( \frac{C}{i} [1-(1+i)^{-n}] \)
\(\therefore\) P = \( \frac{2,160}{0.2} [1-(1+0.2)^{-3}] \)
\(\therefore\) P = \( \frac{21,600}{2} (1-0.5788) \)
\(\therefore\) P = 10,800 \(\times\) 0.42112
\(\therefore\) P = 6,251.04
\(\therefore\) Total amount paid = 2,160 \(\times\) 3 = Rs. 6,480
\(\therefore\) Interest = 6,480 – 6,251.04 = Rs. 228.96
In simple words: The man borrowed Rs. 6,251.04 and paid back a total of Rs. 6,480 through three installments, resulting in an interest charge of Rs. 228.96. The borrowed amount is the present value of the annuity payments.

🎯 Exam Tip: Remember that the "amount borrowed" is the present value of the annuity, and the total interest charged is the difference between the sum of all payments and the present value.

Question 21. A company decides to set aside a certain amount at the end of every year to create a sinking fund that should amount to Rs. 9,28,200 in 4 years at 10% p.a. Find the amount to be set aside every year. [Given: (1.1)4 = 1.4641]
Answer:
Solution:
\(\therefore\) A = Rs. 9,28,200
\(\therefore\) n = 4 years
\(\therefore\) r = 10% p.a.
\(\therefore\) i = \( \frac{r}{100} = \frac{10}{100} = 0.1 \)
\(\because\) A = \( \frac{C}{i} [(1 + i)^n - 1] \)
\(\therefore\) 9,28,200 = \( \frac{C}{0.1} [(1+0.1)^4 -1] \)
\(\therefore\) 9,28,200 \(\times\) 0.1 = C[1.4641 – 1]
\(\therefore\) 92,820 = C \(\times\) 0.4641
\(\therefore\) \( \frac{92,820}{0.4641} \) = C
\(\therefore\) C = Rs. 2,00,000
In simple words: To accumulate Rs. 9,28,200 in a sinking fund over 4 years at a 10% annual interest rate, the company must set aside Rs. 2,00,000 at the end of each year. This is calculated using the future value of an ordinary annuity formula.

🎯 Exam Tip: Sinking fund problems are applications of the future value of an ordinary annuity. Ensure you correctly identify 'A' as the target amount and 'C' as the annual payment.

Question 22. Find the future value after 2 years if an amount of Rs. 12,000 is invested at the end of every half-year at 12% p.a. compounded half-yearly. [Given: (1.06)4 = 1.2625]
Answer:
Solution:
\(\therefore\) n = 2 years
Payable half yearly, n = 2 \(\times\) 2 = 4
\(\because\) C = Rs. 12,000
\(\therefore\) r = 12% p.a. Compounded half yearly
\(\therefore\) r = \( \frac{12}{2} \) = 6%
\(\therefore\) i = \( \frac{r}{100} = \frac{6}{100} = 0.06 \)
\(\therefore\) A = \( \frac{C}{i} [(1 + i)^n - 1] \)
\(\therefore\) A = \( \frac{12,000}{0.06} [(1+0.06)^4 -1] \)
\(\therefore\) A = \( \frac{12,00,000}{12} [(1.06)^4 -1] \)
\(\therefore\) A = 1,00,000 [1.2625 – 1]
\(\therefore\) A = 1,00,000 \(\times\) 0.2625
\(\therefore\) A = Rs. 26,250
In simple words: If Rs. 12,000 is invested every six months for two years at a 12% annual interest rate compounded half-yearly, the future value will be Rs. 26,250. This accounts for the regular investments and the compounded interest over four half-year periods.

🎯 Exam Tip: For half-yearly compounding, ensure both the interest rate 'r' and the number of periods 'n' are correctly adjusted (r/2 and n*2, respectively) before applying the annuity formula.

Question 23. After how many years would an annuity due of Rs. 3,000 p.a. accumulated Rs. 19,324.80 at 20% p.a. compounded annually? [Given: (1.2)4 = 2.0736]
Answer:
Solution:
\(\because\) C = Rs. 3,000
\(\because\) A = Rs. 19,324.80
\(\therefore\) r = 20% p.a.
\(\therefore\) i = \( \frac{r}{100} = \frac{20}{100} = 0.2 \)
\(\therefore\) A = \( \frac{C(1+i)}{i} [(1+i)^n -1] \)
\(\therefore\) 19,324.80 = \( \frac{3,000(1+0.2)}{0.2} [(1+0.2)^n -1] \)
\(\therefore\) 19,324.80 = 15,000 \(\times\) 1.2[(1.2)\(^{n}\) - 1]
\(\therefore\) 19,324.80 = 18,000[(1.2)\(^{n}\) - 1]
\(\therefore\) \( \frac{19,324.80}{18,000} \) + 1 = (1.2)\(^{n}\)
\(\therefore\) 1.0736 + 1 = (1.2)\(^{n}\)
\(\therefore\) 2.0736 = (1.2)\(^{n}\)
\(\therefore\) (1.2)\(^{4}\) = (1.2)\(^{n}\)
\(\therefore\) n = 4 years
In simple words: An annuity due of Rs. 3,000 per year will accumulate to Rs. 19,324.80 in 4 years at a 20% annual compound interest rate. This is determined by solving the future value of an annuity due formula for 'n' using the given values.

🎯 Exam Tip: For an annuity due, the formula includes an extra (1+i) factor compared to an ordinary annuity. Ensure this factor is applied correctly, especially when solving for 'n'.

Question 24. Some machinery is expected to cost 25% more over its present cost of Rs. 6,96,000 after 20 yeas. The scrap value of the machinery will realize Rs. 1,50,000. What amount should be set aside at the end of every year at 5% p.a. compound interest for 20 years to replace the machinery? [Given: (1.05)20 = 2655]
Answer:
Solution:
Present cost = Rs. 6,96,000
Expected cost = 25% \(\times\) 6,96,000 + 6,96,000
= 1,74,000 + 6,96,000
= Rs. 8,70,000
\(\therefore\) Scrap value = Rs. 1,50,000
\(\therefore\) Sinking fund = 8,70,000 – 1,50,000 = Rs. 7,20,000
\(\therefore\) A = Rs. 7,20,000, n = 20 years, r = 5% p.a.
\(\therefore\) i = \( \frac{r}{100} = \frac{5}{100} = 0.05 \)
\(\because\) A = \( \frac{C}{i} [(1 + i)^n - 1] \)
\(\therefore\) 7,20,000 = \( \frac{C}{0.05} [(1+0.05)^{20}-1] \)
\(\therefore\) 7,20,000 \(\times\) 0.05 = C[(1.05)20 – 1]
\(\therefore\) 36,000 = C[2.655 – 1]
\(\therefore\) 36,000 = C \(\times\) 1.655
\(\therefore\) \( \frac{36,000}{1.655} \) = C
\(\therefore\) C = Rs. 21,752.27
In simple words: To replace machinery costing Rs. 8,70,000 (after 20 years, adjusted for 25% increase) and considering a scrap value of Rs. 1,50,000, a company needs to set aside Rs. 21,752.27 each year into a sinking fund earning 5% interest. This covers the net replacement cost over 20 years.

🎯 Exam Tip: Sinking fund calculations must account for both the future expected cost of the asset and any residual (scrap) value to determine the net amount to be funded.