Question 1

The difference between the Simple Interest and the True Discount on a certain sum of money for 6 months at a rate of 12.5 percent per annum is Rs 25. What is the sum (amount due at the end of 6 months)?

Accepted Answer

Introduction / Context: This is a standard question from the topic of true discount and simple interest. It involves the relationship between Simple Interest (calculated on the face value of a bill) and True Discount (the difference between the amount due and its present value). The difference between these two is given and we are asked to find the sum due at the end of the period. Such questions often appear in banking and finance related aptitude sections. Given Data / Assumptions: - Time period = 6 months = 0.5 years. - Rate of interest = 12.5 percent per annum (which is 12.5 per year in percentage terms). - Difference between Simple Interest and True Discount on the same sum for this time and rate is Rs 25. - The interest is computed on a yearly basis using simple interest assumptions. Concept / Approach: In true discount problems, there are three related quantities: the sum due at maturity (amount), the present value, and the true discount. In addition, many exam questions refer to Simple Interest on the face value, which is often called the bank discount. For a sum A due after time t at rate r percent per annum, the simple interest on A is A * r * t / 100. The true discount TD is A * r * t / (100 + r * t). The difference between simple interest on the face value and true discount equals A * r^2 * t^2 / (100 * (100 + r * t)). We use this relationship to form an equation in A and solve for the sum. Step-by-Step Solution: Step 1: Let A be the sum due after 6 months. Step 2: Rate r = 12.5 percent, time t = 0.5 year. Step 3: Compute r * t = 12.5 * 0.5 = 6.25. Step 4: The difference between Simple Interest on A and the True Discount on A is given by A * r^2 * t^2 / (100 * (100 + r * t)). Step 5: Compute r^2 * t^2. Since r = 12.5 and t = 0.5, r^2 * t^2 = 12.5^2 * 0.5^2 = 156.25 * 0.25 = 39.0625. Step 6: Denominator = 100 * (100 + 6.25) = 100 * 106.25 = 10625. Step 7: Therefore difference = A * 39.0625 / 10625. Step 8: This difference is given as Rs 25, so we set A * 39.0625 / 10625 = 25. Step 9: Solve for A: A = 25 * 10625 / 39.0625 = 6800. Verification / Alternative check: To verify, compute Simple Interest and True Discount separately for A = 6800. First, r * t = 6.25 percent. Simple Interest on A = 6800 * 6.25 / 100 = 425. True Discount on A = 6800 * 6.25 / (100 + 6.25) = 425 * 100 / 106.25 = 400. The difference between Simple Interest and True Discount = 425 − 400 = 25, which matches the given condition. This confirmation shows that the sum due is indeed Rs 6800. Why Other Options Are Wrong: Option Rs 6500, Rs 6000 and Rs 6200 each lead to different values of Simple Interest and True Discount whose difference is not equal to Rs 25 when checked using the same formulas. Therefore these amounts cannot satisfy the given relationship between Simple Interest and True Discount for the given time and rate. Common Pitfalls: Many learners confuse True Discount with Simple Interest on the present value instead of on the face value, or they forget the correct formula for True Discount. Another common mistake is to ignore that the difference is given between Simple Interest on the sum due and the True Discount, not between Simple Interest and present value. Using consistent formulas, correctly handling the decimal rate 12.5 percent, and carefully solving the resulting equation avoids these errors. Final Answer: The sum due (amount) is Rs 6800.

Question 2

A man purchases a cow for Rs 3000 in cash and sells it on the same day for Rs 3600, allowing the buyer a credit period of 2 years. If the rate of interest is 10 percent per annum, what is the man's effective gain or loss percentage on this transacti…

Accepted Answer

Introduction / Context: This question combines the ideas of cost price, selling price and the time value of money. The seller receives a higher nominal selling price, but only after a period of 2 years. When interest is taken into account, the real or present worth of the selling price may be different. The question asks whether there is a real profit or loss when we discount the future payment back to its present value. Given Data / Assumptions: - Cost price of the cow = Rs 3000 paid in cash now. - Selling price nominally = Rs 3600, but this is to be received after 2 years. - Rate of interest = 10 percent per annum (simple interest assumed). - There are no other costs or charges involved. - We need to compare the present value of the delayed selling price with the immediate cost price to find the effective gain or loss percentage. Concept / Approach: The key concept is present value. A payment of Rs 3600 two years later does not have the same value as receiving Rs 3600 today, because money now can earn interest. To compare fairly with the cost price of Rs 3000 paid today, we find the present value of Rs 3600 due after 2 years at 10 percent simple interest. Present value is given by amount divided by (1 + r * t), where r is the rate in decimal and t is the time in years. We then compare this present value with the cost price to determine the gain or loss. Step-by-Step Solution: Step 1: Cost price (CP) = Rs 3000 paid now. Step 2: Nominal selling price (SP) = Rs 3600, but receivable after 2 years. Step 3: Rate r = 10 percent per annum, time t = 2 years. Step 4: Present value (PV) of Rs 3600 due after 2 years at 10 percent simple interest = 3600 / (1 + r * t). Step 5: Compute denominator: 1 + 0.10 * 2 = 1 + 0.20 = 1.20. Step 6: PV = 3600 / 1.20 = Rs 3000. Step 7: Effective comparison: CP = Rs 3000 now, PV of SP = Rs 3000 now, so there is no real gain or loss. Verification / Alternative check: If the man deposits Rs 3000 at 10 percent simple interest for 2 years, it becomes 3000 * (1 + 0.10 * 2) = 3000 * 1.20 = Rs 3600 at the end of 2 years. This is exactly the amount he will receive from the buyer after 2 years. Therefore, from a present value perspective, his situation is the same as if he had simply invested his cost price at the bank at the given interest rate. Hence his effective gain is zero percent. Why Other Options Are Wrong: Options 5 percent, 7.5 percent and 10 percent all assume some net gain over the cost price when properly discounted. However, the discounting calculation shows that the present value of Rs 3600 after 2 years is exactly equal to Rs 3000, not higher. Therefore none of these positive gain percentages is correct. Common Pitfalls: A common mistake is to look only at the nominal difference between 3600 and 3000 and conclude there is a 20 percent profit. This ignores the time value of money and the fact that the payment is delayed for 2 years. True evaluation must compare values at the same point in time by discounting future amounts back to the present using the given interest rate. Final Answer: The man makes an effective gain of 0 percent on this transaction.

Question 3

What is the true discount on a bill of Rs 2916 that is due 3 years hence, if the rate of interest is 8 percent per annum compounded annually?

Accepted Answer

Introduction / Context: This problem uses the concept of true discount in the context of compound interest. The bill amount is due after several years and we need to find the difference between the amount and its present value when the money is discounted at compound interest. That difference is called the true discount. Understanding present value and discounting is very important for financial aptitude questions. Given Data / Assumptions: - Amount of the bill (sum due) = Rs 2916. - Time until the bill is due = 3 years. - Rate of interest = 8 percent per annum compounded annually. - We assume standard compound interest formula for calculating present value. - True discount is defined as amount minus present value. Concept / Approach: For compound interest, the future value A after time t years at rate r percent per annum, starting from principal P, is A = P * (1 + r/100)^t. In present value terms, if we know the amount A due after t years, the present value P is A / (1 + r/100)^t. In this question, the amount A is 2916 and we need to find P. Once P is calculated, the true discount TD is simply A − P. We then round to the nearest rupee and compare with the given options. Step-by-Step Solution: Step 1: Amount due A = Rs 2916. Step 2: Rate r = 8 percent per annum, time t = 3 years. Step 3: Present value P is given by P = A / (1 + r/100)^t. Step 4: Compute the compound factor: (1 + 8/100)^3 = 1.08^3. Step 5: Calculate 1.08^3 = 1.08 * 1.08 * 1.08 = 1.259712 (approximately). Step 6: Present value P ≈ 2916 / 1.259712 ≈ 2314.82 rupees. Step 7: True discount TD = A − P ≈ 2916 − 2314.82 ≈ 601.18 rupees. Step 8: Rounded to the nearest rupee, the true discount is approximately Rs 601. Verification / Alternative check: We can check the closeness by reversing the process. If the present value is about Rs 2315 at 8 percent compound interest for 3 years, future value is 2315 * 1.08^3 ≈ 2315 * 1.259712 ≈ 2916 (within a rounding difference). This confirms that the present value and the computed true discount of roughly Rs 601 are correct for the given amount and interest rate. Why Other Options Are Wrong: Option Rs 600: Slightly smaller than the correct discount and does not match the precise calculation using the compound interest formula. Option Rs 602 and Rs 603: Slightly larger than the accurate rounded value obtained from the calculation and therefore not correct when using standard rounding rules. Common Pitfalls: Some learners mistakenly apply simple interest formulas in compound interest questions or attempt to treat the yearly interest linearly. Others may confuse true discount with simple interest on the present value. It is important to use the proper compound factor (1 + r/100)^t, compute present value correctly, and then subtract from the amount to obtain the true discount. Final Answer: The true discount on the bill is approximately Rs 601.

Question 4

A man buys a watch for Rs 1950 in cash and sells it for Rs 2200 at a credit of 1 year. If the rate of interest is 10 percent per annum, what is his effective gain or loss in rupees on this transaction when the time value of money is considered?

Accepted Answer

Introduction / Context: This question mixes profit and loss with the time value of money. The watch is bought in cash and sold on credit, so the actual advantage to the seller depends not only on the difference between the nominal selling price and the cost price, but also on the present value of the delayed payment under a given rate of interest. The aim is to compute the true gain in rupees after discounting the future payment back to the present. Given Data / Assumptions: - Cost price of the watch = Rs 1950 paid immediately in cash. - Selling price nominally = Rs 2200, but receivable after 1 year. - Interest rate = 10 percent per annum (simple interest assumed). - There are no additional hidden costs. - We need to find the effective gain or loss in rupees after considering the present value of the future payment. Concept / Approach: To compare fairly with the cost price paid today, we must convert the future payment of Rs 2200 into its present value using simple interest discounting. Present value P is given by P = A / (1 + r * t), where A is the future amount, r is the annual interest rate in decimal, and t is the time in years. Once we find the present value of Rs 2200 due after one year, we compare it to the cost price to determine the net gain or loss in rupees. Step-by-Step Solution: Step 1: Cost price (CP) = Rs 1950. Step 2: Future selling price amount A = Rs 2200 due after 1 year. Step 3: Rate r = 10 percent per annum, time t = 1 year. Step 4: Present value (PV) of Rs 2200 due after 1 year at 10 percent = 2200 / (1 + 0.10 * 1). Step 5: Denominator = 1.10, so PV = 2200 / 1.10 = Rs 2000. Step 6: Effective gain = PV − CP = 2000 − 1950 = Rs 50. Step 7: Therefore the man effectively gains Rs 50 on this transaction. Verification / Alternative check: Think in reverse: If the man invests Rs 1950 at 10 percent simple interest for 1 year, it becomes 1950 * 1.10 = Rs 2145. The present value of Rs 2200 one year later is Rs 2000, which is more than Rs 1950. The difference of Rs 50 is the extra benefit he gets beyond the fair time value of his money. This aligns with the computed effective gain of Rs 50. Why Other Options Are Wrong: Option gains Rs 55: This would require a different present value or interest rate and does not match the correct discounting at 10 percent. Option loses Rs 30: Incorrect because the present value is higher than the cost price, so there is a gain, not a loss. Option gains Rs 30: Underestimates the benefit and conflicts with the accurate calculation shown above. Common Pitfalls: Many learners simply subtract 1950 from 2200 and claim a profit of Rs 250 without adjusting for the delayed payment. Others may apply interest incorrectly or on the wrong base amount. Always discount future amounts back to the present when comparing with an immediate cash payment, and then compute the difference to determine the true gain or loss. Final Answer: The man effectively gains Rs 50 on the transaction.

Question 5

The present value of a bill due at the end of 2 years is Rs 1250. If the same bill were due at the end of 2 years and 11 months, its present worth would be Rs 1200. Assuming simple interest in both cases, what is the sum (amount of the bill)?

Accepted Answer

Introduction / Context: This true discount problem gives the present value of the same bill for two different maturity times. Based on these two present values and the assumption of simple interest, we must determine both the rate and the final amount of the bill. The options, however, list only amounts, so our key output is the sum of the bill. This type of question tests the ability to set up and solve equations involving present value under simple interest. Given Data / Assumptions: - Present value when the bill is due in 2 years is Rs 1250. - Present value when the bill is due in 2 years and 11 months is Rs 1200. - Interest is at the same simple rate in both cases. - Let the amount of the bill be A and the rate of interest be r percent per annum. Concept / Approach: Under simple interest, the amount A due after t years from present value P is A = P * (1 + r * t / 100). Equivalently, P = A / (1 + r * t / 100). In this question we have two present values 1250 and 1200 for the same amount A but at two different times, namely t1 = 2 years and t2 = 2 years and 11 months (which is 35/12 years). We can write two equations in terms of A and r, then equate the two expressions for A and solve for r, and finally compute A. Step-by-Step Solution: Step 1: Let A be the amount of the bill and r be the annual simple interest rate. Step 2: For the bill due in 2 years, present value P1 = 1250, so A = 1250 * (1 + 2r/100). Step 3: For the bill due in 2 years and 11 months, which is 35/12 years, present value P2 = 1200, so A = 1200 * (1 + (35/12) * r / 100). Step 4: Since both expressions represent the same amount A, set them equal: 1250 * (1 + 2r/100) = 1200 * (1 + (35/12) * r / 100). Step 5: Solving this equation gives r = 5 percent per annum. Step 6: Substitute r = 5 in A = 1250 * (1 + 2r/100) = 1250 * (1 + 10/100) = 1250 * 1.10 = Rs 1375. Verification / Alternative check: Check with the second expression. Using r = 5 percent and t = 35/12 years, we get A = 1200 * (1 + (35/12) * 5 / 100). The factor (35/12) * 5 / 100 = 35 / 240 ≈ 0.14583, so the total multiplier is approximately 1.14583. Then A = 1200 * 1.14583 ≈ 1375. This matches the amount obtained from the first case, confirming that A = Rs 1375 is correct. Why Other Options Are Wrong: Options Rs 1175, Rs 1475 and Rs 1575 do not simultaneously satisfy both present value conditions when checked using the same rate. Only Rs 1375 yields present values of 1250 and 1200 for the two respective periods at a consistent simple interest rate of 5 percent per annum. Common Pitfalls: Some learners incorrectly treat the difference in present values as interest for the extra 11 months, ignoring the implied change in time for both cases. Another mistake is to forget to convert 2 years and 11 months correctly into a fraction of a year, which should be 35/12 years. Always write clear equations for present value in each scenario and solve them systematically. Final Answer: The sum (amount of the bill) is Rs 1375.

Question 6

Find the present worth of Rs 930 due 3 years hence at 8 percent per annum simple interest. Also determine the amount of discount allowed on this bill.

Accepted Answer

Introduction / Context: This true discount question asks for both the present worth and the discount on a bill due after several years when money is valued using simple interest. The present worth is the amount that, if invested now at the given rate, will grow to the future amount. The discount is the difference between the future amount and its present worth. Such questions help in understanding the basic idea of present value and discounting. Given Data / Assumptions: - Future amount due (face value of the bill) = Rs 930. - Time until due = 3 years. - Rate of simple interest = 8 percent per annum. - We need to compute the present worth and then the discount. - The options given refer to the discount amount only. Concept / Approach: Under simple interest, amount A after time t years from present value P is A = P * (1 + r * t / 100), where r is the annual rate. Rearranging, the present value is P = A / (1 + r * t / 100). The true discount is TD = A − P. We will first compute the present value using the formula and then find the discount by subtraction. Step-by-Step Solution: Step 1: Amount due A = Rs 930. Step 2: Rate r = 8 percent per annum, time t = 3 years. Step 3: Compute the factor 1 + r * t / 100 = 1 + 8 * 3 / 100 = 1 + 24 / 100 = 1.24. Step 4: Present worth P = A / 1.24 = 930 / 1.24. Step 5: Calculate 930 / 1.24 = 750 rupees. Step 6: True discount TD = A − P = 930 − 750 = Rs 180. Step 7: So, present worth is Rs 750 and the discount allowed is Rs 180. Verification / Alternative check: Check that investing Rs 750 at 8 percent simple interest for 3 years reaches the amount due. Interest for 3 years = 750 * 8 * 3 / 100 = 750 * 24 / 100 = 180. Amount after 3 years = principal + interest = 750 + 180 = 930, which matches the bill amount. Therefore, the discount and the present worth values are consistent. Why Other Options Are Wrong: Options 186, 185 and 189 all represent different discount amounts. If we subtract these from 930, the resulting present values do not grow to exactly 930 in 3 years at 8 percent simple interest. Only a discount of Rs 180 gives a present value which compounds correctly to the face value. Common Pitfalls: A frequent mistake is to calculate simple interest on the face value directly and treat that as the discount. However, the discount is the interest on the present value, not on the amount due. Another error is to use the wrong time or to mix up r * t / 100 with r / (100 * t). Always compute the factor 1 + r * t / 100 and divide the future amount by this factor to get the present worth. Final Answer: The discount is Rs 180 and the present worth is Rs 750.

Question 7

If the true discount on a sum due 2 years hence at 14 percent per annum simple interest is Rs 168, what is the sum (amount due at the end of 2 years)?

Accepted Answer

Introduction / Context: This is a direct application of the true discount formula in simple interest. The true discount on a sum due after a certain period and at a specified interest rate is given, and we must find the sum due at maturity. Problems like this help in understanding how present value and true discount relate to the face value of a bill. Given Data / Assumptions: - True discount TD = Rs 168. - Time until due = 2 years. - Rate of interest r = 14 percent per annum simple interest. - Let the sum due (face value) be A rupees. Concept / Approach: For simple interest, the relationship between amount A, true discount TD, rate r and time t is TD = A * r * t / (100 + r * t). This formula arises from the difference between the amount and its present value. Given TD, r and t, we can plug into this expression and solve for A directly. Here, t = 2 years and r = 14 percent, so r * t = 28. The calculation then becomes straightforward. Step-by-Step Solution: Step 1: Let sum due be A rupees. Step 2: Given true discount TD = Rs 168, rate r = 14 percent, time t = 2 years. Step 3: Under simple interest, TD = A * r * t / (100 + r * t) = A * 14 * 2 / (100 + 28). Step 4: Simplify numerator: 14 * 2 = 28. Denominator: 100 + 28 = 128. Step 5: So TD = A * 28 / 128. Step 6: We are given TD = 168, so 168 = A * 28 / 128. Step 7: Rearranging, A = 168 * 128 / 28. Step 8: Compute 168 / 28 = 6, so A = 6 * 128 = Rs 768. Verification / Alternative check: Find the present value P and verify. Under simple interest, the present value P of A due after time t at rate r is P = A * 100 / (100 + r * t). Here P = 768 * 100 / 128 = 600 rupees. Simple interest on P for 2 years at 14 percent is 600 * 14 * 2 / 100 = 168 rupees, which equals the given true discount. This confirms that the sum due A is correctly found as Rs 768. Why Other Options Are Wrong: Options Rs 948, Rs 876 and Rs 658 each produce a different true discount when used in the formula, and none of those values equals Rs 168. Only A = Rs 768 satisfies the condition TD = 168 at 14 percent per annum for 2 years. Common Pitfalls: A common mistake is to treat 168 as the simple interest on the sum due directly or to use the simple interest formula TD = A * r * t / 100, which is incorrect for true discount. Another error is forgetting to add r * t in the denominator. Always remember that TD is calculated with the expression TD = A * r * t / (100 + r * t) under simple interest. Final Answer: The sum due at the end of 2 years is Rs 768.

Question 8

The true discount on Rs 2562 due 4 months hence is Rs 122. Assuming simple interest, what is the annual rate of interest (in percent) used to calculate this true discount?

Accepted Answer

Introduction / Context: This question is about finding the rate of simple interest when we know the true discount on a bill, the amount of the bill, and the time until it is due. The true discount connects the amount due, the present value and the interest rate. Identifying the correct rate is important for understanding financial discounting in commercial mathematics. Given Data / Assumptions: - Amount due A = Rs 2562. - True discount TD = Rs 122. - Time until due = 4 months, which is 1/3 of a year. - Rate of interest r percent per annum is unknown and must be found. - Simple interest is assumed throughout. Concept / Approach: For simple interest, true discount TD on a sum A due after time t years at rate r percent per annum is given by TD = A * r * t / (100 + r * t). Here A and TD are known, t = 1/3 year, and r is the unknown. We substitute values into the formula and solve the resulting equation for r. This will give the annual rate of interest. Step-by-Step Solution: Step 1: Given TD = 122, A = 2562, time t = 4 months = 1/3 year. Step 2: True discount formula under simple interest: TD = A * r * t / (100 + r * t). Step 3: Substitute values: 122 = 2562 * r * (1/3) / (100 + r * (1/3)). Step 4: Simplify 2562 * (1/3) to 854, so 122 = 854 * r / (100 + r/3). Step 5: Cross multiply: 122 * (100 + r/3) = 854 * r. Step 6: Expand left side: 12200 + 122 * r / 3 = 854 r. Step 7: Multiply entire equation by 3 to clear the denominator: 36600 + 122 r = 2562 r. Step 8: Rearranging: 36600 = 2562 r − 122 r = 2440 r. Step 9: Thus r = 36600 / 2440 = 15 percent per annum. Verification / Alternative check: Check with r = 15 percent. Compute TD using the formula: TD = 2562 * 15 * (1/3) / (100 + 15 * (1/3)). The numerator becomes 2562 * 5 = 12810. The denominator becomes 100 + 5 = 105. So TD = 12810 / 105 = 122 rupees, which matches the given value. This confirms that the rate is correctly found as 15 percent per annum. Why Other Options Are Wrong: Using 12, 13 or 14 percent in the true discount formula does not produce a discount of Rs 122. Instead, the computed true discount is either too small or too large compared with 122. Only a rate of 15 percent provides the exact discount specified in the problem. Common Pitfalls: Some learners treat 4 months as 4/10 of a year instead of 4/12, or they mistakenly use the simple interest formula A * r * t / 100 rather than the true discount formula. Errors also arise when clearing denominators. Careful algebra and correct conversion of months to years are essential for answering such questions correctly. Final Answer: The annual rate of interest is 15 percent.

Question 9

The true discount on a bill due 9 months hence at 16 percent per annum simple interest is Rs 189. What is the amount (face value) of the bill?

Accepted Answer

Introduction / Context: Here we have a true discount question with the amount of the discount given, the rate of simple interest, and the time until the bill is due. We are asked to find the face value (amount) of the bill. This is a typical problem in commercial mathematics where present value and true discount concepts are applied to find unknown amounts. Given Data / Assumptions: - True discount TD = Rs 189. - Time until due = 9 months = 3/4 of a year. - Rate of interest r = 16 percent per annum simple interest. - Let the amount of the bill be A rupees. Concept / Approach: Under simple interest, the true discount TD for a sum A due after time t years at rate r percent per annum is TD = A * r * t / (100 + r * t). We know TD, r and t, so we can substitute and solve for A. Here t = 3/4 year and r = 16 percent, which makes r * t equal to 12. After writing down the formula with these values, we solve the resulting linear equation for A. Step-by-Step Solution: Step 1: Let amount of the bill be A. Step 2: Given TD = 189, r = 16 percent, time t = 9 months = 3/4 year. Step 3: True discount formula: TD = A * r * t / (100 + r * t). Step 4: Compute r * t = 16 * 3/4 = 12. Therefore TD = A * 16 * 3/4 / (100 + 12). Step 5: Simplify numerator: 16 * 3/4 = 12. Denominator = 112. Step 6: So TD = A * 12 / 112 = A * 3 / 28. Step 7: Given TD = 189, so 189 = 3A / 28. Step 8: Solve for A: A = 189 * 28 / 3. Step 9: Compute 189 / 3 = 63 and 63 * 28 = 1764. So A = Rs 1764. Verification / Alternative check: Verify by computing the true discount from the found amount. Present value P is A * 100 / (100 + r * t) = 1764 * 100 / 112 = 1764 * 25 / 28. Compute 1764 / 28 = 63, so P = 63 * 25 = 1575. True discount TD = A − P = 1764 − 1575 = 189, which matches the given discount. This confirms that A has been correctly determined. Why Other Options Are Wrong: Options 1200, 1600 and 1354 do not satisfy the true discount formula when used as A. For each of these, the calculated true discount does not equal Rs 189, so they cannot be the correct face value of the bill. Common Pitfalls: Learners sometimes misconvert 9 months into a fraction of a year, or they use the simple interest formula A * r * t / 100 instead of the correct true discount formula with 100 + r * t in the denominator. Errors in simplifying fractions also frequently lead to wrong answers. Careful handling of r * t and the algebraic steps is necessary. Final Answer: The amount of the bill is Rs 1764.

Question 10

The total amount due is Rs 1404, payable in two equal half yearly instalments, and the rate of simple interest is 8 percent per annum. What is the present worth (present value) of these two instalments together?

Accepted Answer

Introduction / Context: This problem deals with the present value of instalment payments under simple interest. The total amount Rs 1404 is to be paid in two equal half yearly instalments. The task is to find the present worth of these two instalments when discounted at 8 percent per annum simple interest. Such questions are useful for understanding how instalment loans and present value calculations work. Given Data / Assumptions: - Total amount due = Rs 1404 in two equal half yearly instalments. - Therefore each instalment is 1404 / 2 = Rs 702. - Rate of interest r = 8 percent per annum simple interest. - First instalment is due in 6 months, that is 0.5 years from now. - Second instalment is due in 1 year from now. - We must find the sum of the present values of both instalments. Concept / Approach: Under simple interest, the present value P of an amount A due after t years at rate r percent is P = A / (1 + r * t / 100). We calculate the present value of each instalment separately using its respective time until due, and then add the two present values. The sum of these present values is the present worth of the total payment schedule. Step-by-Step Solution: Step 1: Each instalment amount is A = 1404 / 2 = Rs 702. Step 2: Rate r = 8 percent per annum. Step 3: First instalment time t1 = 6 months = 0.5 years. Step 4: Second instalment time t2 = 1 year. Step 5: Present value of first instalment P1 = 702 / (1 + 8 * 0.5 / 100) = 702 / (1 + 4 / 100) = 702 / 1.04. Step 6: Present value of second instalment P2 = 702 / (1 + 8 * 1 / 100) = 702 / 1.08. Step 7: Compute P1 ≈ 702 / 1.04 = 675. Step 8: Compute P2 ≈ 702 / 1.08 = 650. Step 9: Total present worth = P1 + P2 = 675 + 650 = Rs 1325. Verification / Alternative check: To verify, if we take the present worth of Rs 1325 and spread it into two instalments with simple interest, we should reach the original amounts. Investing 675 at 8 percent for 0.5 years gives interest 675 * 8 * 0.5 / 100 = 27, so amount becomes 702. Investing 650 at 8 percent for 1 year gives interest 650 * 8 / 100 = 52, so amount becomes 702 again. This confirms that the present values 675 and 650 are correct and that their sum 1325 is indeed the present worth. Why Other Options Are Wrong: Options 1320, 1300 and 1200 do not reproduce the exact instalment amounts of 702 each under 8 percent simple interest for 0.5 year and 1 year. Only 1325, when split appropriately, grows to exactly 702 and 702. Common Pitfalls: Learners sometimes treat the total 1404 as a single payment and discount it once, instead of separating the instalments and discounting each at its own time. Another error is to use compound interest formulas where simple interest is specified. Always calculate present value separately for each instalment if the due dates are different. Final Answer: The present worth of the two instalments is Rs 1325.

Question 11

A man wants to sell his scooter. He receives two offers: one is Rs 12000 in cash now, and the other is Rs 12880 as a credit payment to be made after 8 months, with money valued at 18 percent per annum simple interest. Which offer is financially bett…

Accepted Answer

Introduction / Context: This question compares a cash offer with a credit offer payable after some time, where the time value of money is taken into account using a given interest rate. To decide which offer is better for the seller, we need to compare the immediate cash payment with the present value of the delayed payment, discounted at the specified interest rate. Given Data / Assumptions: - Cash offer = Rs 12000 received immediately. - Credit offer = Rs 12880 to be received after 8 months. - Rate of interest r = 18 percent per annum simple interest. - Time until credit payment = 8 months = 8/12 = 2/3 of a year. - We assume simple interest and no other charges or benefits. Concept / Approach: To compare fairly, we bring both offers to the same time point, usually the present. The cash offer is already present money. The credit offer must be converted to present value using simple interest. The present value P of an amount A due after t years at rate r percent per annum is P = A / (1 + r * t / 100). We will compute the present value of Rs 12880 due after 8 months and then compare it to Rs 12000. Step-by-Step Solution: Step 1: Credit amount A = Rs 12880. Step 2: Rate r = 18 percent per annum, time t = 8 months = 2/3 year. Step 3: Compute r * t / 100 = 18 * (2/3) / 100 = 12 / 100 = 0.12. Step 4: Present value P of the credit offer = 12880 / (1 + 0.12) = 12880 / 1.12. Step 5: Calculate 12880 / 1.12 = 11500 rupees. Step 6: Compare this to the cash offer: cash offer = 12000 rupees, present value of credit offer = 11500 rupees. Step 7: Since 12000 > 11500, the cash offer of Rs 12000 is financially better for the man. Verification / Alternative check: Think in the opposite direction. If the man chooses the cash offer of Rs 12000 and invests it at 18 percent simple interest for 8 months, its future value after 8 months is 12000 * (1 + 0.12) = 12000 * 1.12 = 13440 rupees, which is more than the delayed amount of 12880 under the credit offer. This confirms that taking cash and investing at the given rate yields a better outcome than accepting the credit offer. Why Other Options Are Wrong: Option Rs 12880: This is the nominal future amount, but its present value at 18 percent for 8 months is lower than 12000, so it is not better from today's perspective. Option Both are equally good: Incorrect because 12000 is clearly larger than the present value of 11500 for the credit offer. Option None of the above: Incorrect since one of the provided options, namely Rs 12000, is indeed strictly better. Common Pitfalls: Some learners compare only the nominal amounts 12880 and 12000 and think the higher future amount is always better, forgetting to discount it. Another common error is to misconvert months to years or to use compound interest formulas when simple interest is specified. Always convert time correctly and compute present value before making a decision. Final Answer: The cash offer of Rs 12000 is better.

Question 12

Which is the better financial offer: (i) an immediate cash payment of Rs. 8100 now or (ii) a credit payment of Rs. 8250 after 6 months at 6.5% simple interest per annum?

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Introduction / Context: This question compares two alternative payment options using the concept of present worth and true discount under simple interest. Students must decide whether it is better to receive a smaller amount immediately or a slightly larger amount after 6 months when the money carries simple interest at 6.5 percent per annum. Given Data / Assumptions: Offer (i): Immediate cash payment of Rs. 8100. Offer (ii): Rs. 8250 received after 6 months. Rate of simple interest r = 6.5 percent per annum. Time for second offer t = 6 months = 0.5 year. Simple interest and true discount concepts apply. Concept / Approach: We compare both offers on the same date by converting the future amount into its present worth. Under simple interest, if amount due after time t is A, present worth PW is given by: PW = A * 100 / (100 + r * t) where r is the annual rate in percent and t is the time in years. The offer with higher present worth is financially better. Step-by-Step Solution: Step 1: For offer (i), present worth PW1 = Rs. 8100 (already paid now). Step 2: For offer (ii), amount A = Rs. 8250, r = 6.5, t = 0.5 year. Step 3: Compute r * t = 6.5 * 0.5 = 3.25. Step 4: Present worth of offer (ii): PW2 = 8250 * 100 / (100 + 3.25). Step 5: PW2 = 825000 / 103.25 which is approximately equal to Rs. 7990.31. Step 6: Compare PW1 and PW2: Rs. 8100 is greater than about Rs. 7990.31. Thus the immediate cash payment has a higher present value. Verification / Alternative check: We can also find the true discount on Rs. 8250 for 6 months and subtract it from the nominal amount. True discount TD is: TD = A * r * t / (100 + r * t) = 8250 * 3.25 / 103.25 ≈ Rs. 259.69. Present worth = 8250 − 259.69 ≈ Rs. 7990.31. Again this is less than Rs. 8100, so our earlier conclusion is consistent. Why Other Options Are Wrong: Option (ii) is worse because Rs. 8250 after 6 months is worth only about Rs. 7990.31 today, which is lower than Rs. 8100. The two offers are not equivalent, so saying both are same is wrong. It is also not true that neither offer is acceptable; one is clearly better under the given rate, and the data is fully sufficient for a decision. Common Pitfalls: A common mistake is to compare only the nominal amounts 8250 and 8100 without adjusting for time value of money. Another error is to compute interest on the present worth instead of on the future value when applying formulas. Some students forget to convert 6 months to 0.5 year or miscalculate r * t. Always ensure that the comparison is done at the same point in time, usually by converting future sums to present worth using the correct simple interest expression. Final Answer: The better offer is (i): an immediate cash payment of Rs. 8100 now.

Question 13

If Rs. 10 is allowed as true discount on a bill of Rs. 110 due after a certain time at simple interest, what will be the true discount on the same sum when it is due after double that time, assuming the same rate of interest?

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Introduction / Context: This problem tests understanding of true discount under simple interest and how it changes when the time is doubled while the principal and rate of interest remain constant. Instead of giving the rate directly, the question provides one true discount and asks for another, so we have to work backwards to find the effective rate time product and then recompute the new true discount. Given Data / Assumptions: Face value (amount due) of the bill, S = Rs. 110. True discount on this bill for some time t is TD1 = Rs. 10. The same sum is now considered for double the original time, that is 2t. The rate of simple interest remains the same in both cases. We have to find the new true discount TD2 for time 2t. Concept / Approach: For simple interest, if a sum S is due after time t at rate r percent per annum, and true discount TD is allowed, then: TD = S * r * t / (100 + r * t) Let k = r * t, which represents the effective rate time product. We can use the first situation to find k and then use k2 = 2k for the doubled time. The new true discount is then: TD2 = S * k2 / (100 + k2) Step-by-Step Solution: Step 1: Use TD1 = 10 for S = 110 and unknown k. 10 = 110 * k / (100 + k) Step 2: Multiply both sides: 10 * (100 + k) = 110 * k. 1000 + 10k = 110k. Step 3: Rearrange to get 100k = 1000 so k = 10. Step 4: For double the time, k2 = 2k = 20. Step 5: New true discount TD2 = 110 * 20 / (100 + 20) = 2200 / 120. Step 6: Simplify 2200 / 120 = 220 / 12 = 55 / 3 ≈ 18.33. Verification / Alternative check: We can also compute the present worth PW1 for the first case and PW2 for the second case, and then calculate TD2 = S − PW2. For k = 10, we have: PW1 = S * 100 / (100 + 10) = 11000 / 110 = Rs. 100. So the effective rate time product of 10 percent is confirmed. For k2 = 20: PW2 = 110 * 100 / (100 + 20) = 11000 / 120 ≈ 91.67. TD2 = 110 − 91.67 ≈ 18.33. This matches the earlier result. Why Other Options Are Wrong: Values like Rs. 18.20, Rs. 18.00, Rs. 18.30, and Rs. 19.00 do not match the exact fraction 55 divided by 3. They arise from rounding or incorrect substitution of time into the true discount formula, or from using simple interest instead of true discount directly. The correct calculation gives 18.33 recurring, which corresponds to option stating Rs. 18.33. Common Pitfalls: Learners sometimes double the discount from Rs. 10 to Rs. 20 when the time doubles, forgetting that true discount is not directly proportional to time because of the denominator term 100 + r * t. Others confuse true discount with simple interest and use SI = P * r * t / 100 without adjusting for due amount. Always use the correct true discount formula and remember that r * t is treated as a combined factor first, then applied to the fraction S * r * t / (100 + r * t). Final Answer: The true discount for double the time is Rs. 18.33.

Question 14

The true discount on Rs. 2480 due after a certain time at 5% simple interest per annum is Rs. 80. For what time period is the amount due?

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Introduction / Context: This question applies the standard true discount formula to find the time period corresponding to a given true discount, principal, and rate of simple interest. It checks whether the student understands the relation between true discount, sum due, rate, and time, and can convert the resulting time in years into months correctly. Given Data / Assumptions: Face value of the bill, S = Rs. 2480. True discount TD = Rs. 80. Rate of interest r = 5 percent per annum (simple interest). Time is unknown and must be found in months. We assume simple interest and standard true discount definitions. Concept / Approach: For a sum S due after time t years at simple interest rate r percent per annum, the true discount TD is given by: TD = S * r * t / (100 + r * t) Let k = r * t. We can first find k from the equation using the known true discount, then compute t = k / r. Finally we convert time in years to months by multiplying by 12. Step-by-Step Solution: Step 1: Write the true discount equation for S = 2480, TD = 80, r = 5. 80 = 2480 * 5 * t / (100 + 5 * t). Step 2: Set k = 5 * t, so TD = 2480 * k / (100 + k). Step 3: Substitute: 80 = 2480 * k / (100 + k). Step 4: Multiply both sides: 80 * (100 + k) = 2480 * k. 8000 + 80k = 2480k. Step 5: Move terms: 2480k − 80k = 2400k = 8000. Step 6: Solve for k: k = 8000 / 2400 = 80 / 24 = 10 / 3. Step 7: k = r * t = 5 * t so t = (10 / 3) / 5 = 10 / 15 = 2 / 3 year. Step 8: Convert to months: (2 / 3) * 12 = 8 months. Verification / Alternative check: We can recompute the true discount using t = 2 / 3 year. First compute r * t: r * t = 5 * 2 / 3 = 10 / 3. Then: TD = 2480 * (10 / 3) / (100 + 10 / 3). Denominator = (300 + 10) / 3 = 310 / 3. TD = 2480 * (10 / 3) * (3 / 310) = 2480 * 10 / 310 = 24800 / 310 = 80. This matches the given true discount, so the time is confirmed as 8 months. Why Other Options Are Wrong: Times such as 3 months, 5 months, 6 months, or 10 months do not produce a true discount of Rs. 80 when substituted into the standard formula. They correspond to different effective rate time products and would lead to different discounts, so they do not satisfy the given condition. Common Pitfalls: Many students mistakenly use the simple interest formula SI = P * r * t / 100 instead of the true discount formula, which places r * t in the denominator as well. Another common mistake is not converting the final time in years into months properly. Always remember that 1 year equals 12 months, and confirm your result by back-substituting into the original formula to see if you recover the given true discount exactly. Final Answer: The amount is due after 8 months.

Question 15

In how much time can a debt of Rs. 7920, due at a certain future date, be cleared by an immediate cash down payment of Rs. 3600, if money is worth 0.5% per month simple interest?

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Introduction / Context: This is a classic true discount application where an immediate cash payment is used to settle a larger debt due after some time. The question gives the cash down payment, the full amount of the debt, and the monthly simple interest rate. We need to find the time at which the debt would otherwise be due so that both arrangements are financially equivalent in present worth terms. Given Data / Assumptions: Debt amount (sum due in the future), S = Rs. 7920. Immediate cash down payment (present worth), PW = Rs. 3600. Rate of simple interest r = 0.5 percent per month. Time t is in months, to be converted into years if needed. Simple interest and true discount concepts apply. Concept / Approach: If S is due after t months and money earns r percent per month simple interest, then present worth PW and amount S are related by: S = PW * (1 + r * t / 100) or equivalently: PW = S * 100 / (100 + r * t) We know S, PW, and r, so we can solve for t. Then we convert the number of months into years by dividing by 12. Step-by-Step Solution: Step 1: Use PW formula: PW = S * 100 / (100 + r * t). Step 2: Substitute PW = 3600, S = 7920, r = 0.5. 3600 = 7920 * 100 / (100 + 0.5 * t). Step 3: Rearrange to solve for the denominator. 100 + 0.5 * t = 7920 * 100 / 3600. Step 4: Compute 7920 * 100 / 3600 = 220. Step 5: Thus 100 + 0.5 * t = 220, so 0.5 * t = 120. Step 6: t = 120 / 0.5 = 240 months. Step 7: Convert to years: 240 months / 12 = 20 years. Verification / Alternative check: Check using the amount formula S = PW * (1 + r * t / 100). With PW = 3600, r = 0.5, t = 240: r * t / 100 = 0.5 * 240 / 100 = 120 / 100 = 1.2. S = 3600 * (1 + 1.2) = 3600 * 2.2 = 7920. This matches the given future debt amount exactly, confirming that 20 years is correct. Why Other Options Are Wrong: Values such as 10, 30, or 40 years would produce very different multipliers when used in the simple interest formula. They would not convert Rs. 3600 into Rs. 7920 at a rate of 0.5 percent per month. Only 20 years leads to the correct future value. Common Pitfalls: A typical mistake is to treat the 0.5 percent as an annual rate or to forget that the rate is per month. Another error is not converting months to years properly or mismanaging the algebra when isolating t in the PW equation. Always keep the time unit consistent with the rate and verify the answer by reconstructing the original future value from the computed time and present worth. Final Answer: The debt is due after 20 years.

Question 16

If the true discount on a sum due 2 years hence at 14% simple interest per annum is Rs. 168, what is the sum that will be due at the end of 2 years?

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Introduction / Context: Here we are asked to find the face value of a sum due after 2 years when we know the true discount allowed today and the simple interest rate. This tests the relation between true discount, sum due, interest rate, and time, and gives practice with rearranging the true discount formula to obtain the principal amount. Given Data / Assumptions: True discount TD = Rs. 168. Time t = 2 years. Rate of simple interest r = 14 percent per annum. Sum due after 2 years, S, is unknown. Standard true discount formula under simple interest holds. Concept / Approach: For a sum S due after time t at simple interest rate r, the true discount is: TD = S * r * t / (100 + r * t) We know TD, r, and t. Let k = r * t = 14 * 2 = 28. Substitute TD and k into the formula, solve for S, and then identify the correct option. Step-by-Step Solution: Step 1: Compute k = r * t = 14 * 2 = 28. Step 2: Write TD formula: TD = S * k / (100 + k). Step 3: Substitute TD = 168, k = 28. 168 = S * 28 / (100 + 28) = S * 28 / 128. Step 4: Rearrange to solve for S. S = 168 * 128 / 28. Step 5: Simplify 168 / 28 = 6, so S = 6 * 128 = 768. Thus the sum due after 2 years is Rs. 768. Verification / Alternative check: Check by computing the present worth PW and confirming the true discount. With S = 768, k = 28: PW = S * 100 / (100 + k) = 768 * 100 / 128 = 76800 / 128 = 600. TD = S − PW = 768 − 600 = 168. This matches the given true discount, so S = 768 is correct. Why Other Options Are Wrong: Values such as Rs. 600, Rs. 668, Rs. 700, or Rs. 878 would not produce a true discount of exactly Rs. 168 when used with the same rate and time. For example, if S were Rs. 600, the true discount would be lower and would not match the given 168. Common Pitfalls: Students sometimes confuse true discount with simple interest and use SI = S * r * t / 100 directly, which would give a different result. Another frequent error is forgetting to add r * t to 100 in the denominator. Always compute k = r * t first, write TD = S * k / (100 + k), and then solve carefully for S, reducing fractions step by step to avoid arithmetic mistakes. Final Answer: The sum due after 2 years is Rs. 768.

Question 17

If a cellphone is purchased for Rs. 490 and sold for Rs. 465.50, what is the percentage loss on the transaction?

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Introduction / Context: This question is from percentage and profit loss but appears under the true discount section because it uses similar percentage concepts. The aim is to find the loss percent when a selling price is lower than the cost price. This is a basic but very important skill in aptitude examinations. Given Data / Assumptions: Cost price (CP) of the cellphone = Rs. 490. Selling price (SP) of the cellphone = Rs. 465.50. We need to compute the loss percent on the cost price. Standard profit loss formulas apply. Concept / Approach: Loss is defined as CP minus SP when SP is less than CP. The loss percent is given by: Loss percent = (Loss / CP) * 100 We must compute the absolute loss in rupees and then express it as a percentage of the cost price, taking care with decimal handling. Step-by-Step Solution: Step 1: Compute the loss. Loss = CP − SP = 490 − 465.50. Step 2: Subtract to get Loss = 24.50. Step 3: Use the formula for loss percent. Loss percent = (24.50 / 490) * 100. Step 4: Compute the fraction 24.50 / 490 = 0.05. Step 5: Convert to percent: 0.05 * 100 = 5%. Thus the loss percent on selling the cellphone is 5 percent. Verification / Alternative check: We can cross check by noting that 5 percent of 490 is: 5% of 490 = (5 / 100) * 490 = 24.50. Since this equals the loss amount calculated earlier, the result is confirmed. Also, the selling price is clearly slightly lower than cost price, so the percentage must be a small number, making 5 percent reasonable. Why Other Options Are Wrong: Loss percentages of 10, 15, or 20 percent would correspond to losses of 49, 73.50, or 98 respectively, which are much larger than the actual loss of Rs. 24.50. A value such as 2.5 percent would correspond to a loss of only Rs. 12.25, which is too small. Only 5 percent matches the correct loss figure. Common Pitfalls: A common error is to divide by the selling price instead of the cost price when calculating the loss percent. Another frequent mistake is mishandling the decimal part in 465.50 or 24.50. Always remember that percentage calculations for profit or loss are based on the cost price, and perform the subtractions with care to avoid decimal errors. Final Answer: The loss percentage on the cellphone sale is 5%.

Question 18

If Rs. 20 is the true discount on Rs. 260 due after a certain time at a given simple interest rate, what will be the true discount on the same sum due after half of that former time at the same rate of interest?

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Introduction / Context: This question involves the relation between true discount and time when the principal and rate are fixed. Instead of directly giving the rate or time, it gives the true discount for one time period and asks for the true discount when the time is halved. The focus is on correct use of the true discount formula and on understanding that true discount is not directly proportional to time. Given Data / Assumptions: Face value of the bill, S = Rs. 260. True discount for some time t, TD1 = Rs. 20. We require true discount for time t / 2, called TD2. Rate of simple interest remains constant in both cases. Standard simple interest and true discount relation applies. Concept / Approach: For a sum S due after t years at simple interest rate r percent, true discount is: TD = S * r * t / (100 + r * t) Let k = r * t. Then: TD1 = S * k / (100 + k) For half the time, the effective product is k2 = k / 2 and: TD2 = S * k2 / (100 + k2) We first find k from TD1 and then compute TD2 using k2. Step-by-Step Solution: Step 1: Use TD1 = 20, S = 260. 20 = 260 * k / (100 + k). Step 2: Multiply: 20 * (100 + k) = 260 * k. 2000 + 20k = 260k. Step 3: Rearrange: 260k − 20k = 240k = 2000. Step 4: Solve: k = 2000 / 240 = 25 / 3. Step 5: For half the time, k2 = k / 2 = 25 / 6. Step 6: Compute TD2. TD2 = 260 * (25 / 6) / (100 + 25 / 6). Denominator = (600 + 25) / 6 = 625 / 6. TD2 = 260 * (25 / 6) * (6 / 625) = 260 * 25 / 625. Step 7: 260 * 25 = 6500, so TD2 = 6500 / 625 = 10.4 = Rs. 10.40. Verification / Alternative check: We can compute the present worth PW1 and PW2 and confirm. For TD1 = 20: PW1 = S − TD1 = 260 − 20 = 240. For TD2: PW2 = S − TD2 = 260 − 10.40 = 249.60. Using PW2 and the true discount formula would recover k2 = 25 / 6, consistent with half the original effective rate time product, confirming correctness. Why Other Options Are Wrong: Values like Rs. 9.40, Rs. 9.14, or Rs. 10.14 arise from incorrect algebra or by wrongly assuming that true discount is exactly half when time is halved. That is only true for simple interest and not for true discount, which has a denominator 100 + r * t. Rs. 11.20 would correspond to a different time or rate and does not satisfy the relationship derived from the original discount. Common Pitfalls: A frequent mistake is linearly halving the true discount when the time is halved. Because true discount uses the factor 100 + r * t in the denominator, the relation is slightly non linear. Students may also confuse true discount with bank discount or simple interest and omit the denominator adjustment completely. Carefully solving for k and then recomputing for k / 2 avoids these errors. Final Answer: The true discount for half the former time is Rs. 10.40.

Question 19

A bill falls due in 1 year. The creditor agrees to accept immediate payment of half of the amount and to defer payment of the other half for 2 years. By this arrangement he gains Rs. 40. If money is worth 12.5% per annum simple interest, what is the…

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Introduction / Context: This question deals with equivalent payment arrangements and gain to the creditor when a bill that originally falls due in 1 year is replaced by a scheme where half is paid immediately and half is paid after 2 years. We are told that by this new arrangement the creditor gains Rs. 40 compared with the original single payment. The money is said to be worth 12.5 percent per annum simple interest, so the comparison is based on the value of money over time. Given Data / Assumptions: Original bill amount due in 1 year = S (unknown). New scheme: S / 2 paid now, S / 2 paid after 2 years. Creditor gains Rs. 40 due to this new arrangement. Simple interest rate r = 12.5 percent per annum. We compare the arrangements using a common future time, for example 2 years from now. Concept / Approach: Take 2 years from now as the common comparison date. Under simple interest, a payment received earlier can be accumulated forward to that date. For the original scheme, the creditor gets S after 1 year, which becomes S * (1 + r * 1 / 100) after another year. For the new scheme, the creditor receives S / 2 now and S / 2 after 2 years; the first part grows to S / 2 * (1 + r * 2 / 100), and the second part is already at the comparison date. The difference between total amounts at 2 years is equal to Rs. 40, which allows us to solve for S. Step-by-Step Solution: Step 1: Rate r = 12.5 percent = 12.5 / 100 = 0.125 per year. Step 2: Under original scheme, S is received at end of 1 year. At 2 years, this becomes S * (1 + 0.125 * 1) = 1.125 * S. Step 3: Under new scheme, S / 2 is received now and S / 2 at end of 2 years. First half at 2 years: (S / 2) * (1 + 0.125 * 2) = (S / 2) * 1.25 = 0.625 * S. Second half at 2 years remains S / 2 = 0.5 * S. Total at 2 years under new scheme = 0.625 * S + 0.5 * S = 1.125 * S. Step 4: At first glance, both amounts are 1.125 * S at 2 years, but the question says creditor gains Rs. 40, so we interpret the gain as measured relative to some reference such as the present worth calculations used in exam standards. Step 5: In exam convention, the difference between the two arrangements in present worth terms is given as Rs. 40, leading to the standard outcome S = Rs. 3600 for this data set. Step 6: With S = 3600, the equivalent value of each schedule differs by Rs. 40, matching the question statement. Verification / Alternative check: We can verify by computing present worth at time zero using simple interest discount. Present worth of the original bill is S / 1.125. Present worth of the new scheme is S / 2 plus (S / 2) / 1.25. Substituting S = 3600 yields a difference of approximately Rs. 40, which validates the given gain. Why Other Options Are Wrong: Other values such as Rs. 1200, Rs. 1300, Rs. 1800, or Rs. 3000 do not produce the stated gain of Rs. 40 between the two arrangements when the standard simple interest comparison is applied. They lead to different differences in value and therefore cannot satisfy the question condition. Common Pitfalls: This type of problem is subtle because different reference times (present, mid point, or future) can be used. Students may incorrectly compute the gain using only nominal sums without time adjustment. Others may forget that simple interest grows linearly with time and misapply the formulas. A careful and consistent approach with a chosen reference time and simple interest accumulation is essential to avoid confusion. Final Answer: The amount of the bill is Rs. 3600.

Question 20

The true discount on Rs. 1760 due after a certain time at 12% simple interest per annum is Rs. 160. After how much time (in months) is this sum due?

Accepted Answer

Introduction / Context: The question gives the true discount on a sum at a known simple interest rate and asks for the time after which the sum is due. It is a direct application of the true discount formula and requires conversion of time in years to months once the result is obtained. Given Data / Assumptions: Sum due, S = Rs. 1760. True discount TD = Rs. 160. Rate of interest r = 12 percent per annum. Time t is unknown, measured in years first, and then converted to months. Interest is simple, and the standard true discount relationship holds. Concept / Approach: True discount TD for a sum S due after time t years at rate r percent simple interest is: TD = S * r * t / (100 + r * t) Let k = r * t. Then TD = S * k / (100 + k). We know TD, S, and r, so we can solve for k and then find t = k / r. Finally we convert t from years to months by multiplying by 12. Step-by-Step Solution: Step 1: Let k = r * t. For r = 12, k = 12 * t. Step 2: Write TD equation: 160 = 1760 * k / (100 + k). Step 3: Multiply both sides: 160 * (100 + k) = 1760 * k. 16000 + 160k = 1760k. Step 4: Move terms: 1760k − 160k = 1600k = 16000. Step 5: Solve: k = 16000 / 1600 = 10. Step 6: Since k = r * t = 12 * t, we have 12 * t = 10. Step 7: t = 10 / 12 year = 5 / 6 year. Step 8: Convert to months: (5 / 6) * 12 = 10 months. Verification / Alternative check: We can verify by computing present worth PW and recalculating TD. With S = 1760 and k = 10: PW = S * 100 / (100 + k) = 1760 * 100 / 110 = 176000 / 110 = 1600. TD = S − PW = 1760 − 1600 = 160. This matches the given true discount, confirming that the time is indeed 10 months. Why Other Options Are Wrong: Times such as 5, 8, 12, or 15 months do not produce the same effective rate time product when substituted into the true discount formula. They therefore yield different true discounts, not equal to Rs. 160, and do not satisfy the condition in the question. Common Pitfalls: Some candidates mistakenly use the simple interest formula SI = S * r * t / 100 instead of the true discount formula and equate SI to Rs. 160, which is incorrect. Others forget to add r * t to 100 in the denominator or misconvert years to months. To avoid errors, always compute k = r * t carefully, solve for k using the given discount, and then use t = k / r followed by time unit conversion. Final Answer: The sum is due after 10 months.

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