Question 1

The simple interest on Rs. 750 for 2 years is the same as the true discount on Rs. 960 due 2 years hence. If the rate of interest is the same in both cases, what is the annual rate of interest?

Accepted Answer

Introduction / Context: This is a conceptual question connecting simple interest and true discount. The same rate and time apply to two different sums but through different formulas. The simple interest on one sum equals the true discount on another. The task is to find the common rate of interest, which requires equating the algebraic expressions for simple interest and true discount and solving for the rate. Given Data / Assumptions: Principal for simple interest, P1 = Rs. 750. Time for simple interest, t1 = 2 years. Bill amount for true discount, S2 = Rs. 960, due after t2 = 2 years. Same rate of interest r percent per annum in both cases. Simple interest on Rs. 750 for 2 years equals true discount on Rs. 960 for 2 years. Concept / Approach: Simple interest SI is given by: SI = P * r * t / 100 True discount TD for sum S due after time t is: TD = S * r * t / (100 + r * t) We are told: SI on 750 for 2 years = TD on 960 for 2 years. We set up the equations in terms of r and solve for r. Since time is 2 years in both cases, we can simplify the algebra. Step-by-Step Solution: Step 1: Compute SI expression. SI = 750 * r * 2 / 100 = 1500r / 100 = 15r. Step 2: Write TD expression for S2 = 960 and t2 = 2 years. TD = 960 * r * 2 / (100 + 2r) = 1920r / (100 + 2r). Step 3: Equate SI and TD. 15r = 1920r / (100 + 2r). Step 4: Since r is non zero, divide both sides by r. 15 = 1920 / (100 + 2r). Step 5: Cross multiply: 15 * (100 + 2r) = 1920. 1500 + 30r = 1920. Step 6: 30r = 1920 − 1500 = 420. Step 7: r = 420 / 30 = 14. So the annual rate of interest is 14 percent. Verification / Alternative check: Compute SI at 14 percent: SI = 15 * 14 = Rs. 210. Compute TD on Rs. 960 at the same rate and time. Here r * t = 28: TD = 960 * 28 / (100 + 28) = 960 * 28 / 128. Simplify 960 / 128 = 7.5, so TD = 7.5 * 28 = 210. Both SI and TD equal Rs. 210, which confirms the value r = 14 percent. Why Other Options Are Wrong: Rates like 12 percent, 15 percent, 16 percent, or 10 percent do not satisfy the equality condition between SI and TD when substituted into the formulas. They give mismatched values for simple interest and true discount and therefore cannot be correct. Common Pitfalls: A common mistake is to directly treat SI and TD as both equal to P * r * t / 100 and ignore the denominator 100 + r * t in the true discount formula. This leads to a trivial equality and no solution. Students may also forget to divide through by r or mishandle cross multiplication. Carefully writing both expressions and keeping the denominator intact is key to solving such problems correctly. Final Answer: The annual rate of interest is 14%.

Question 2

A has to pay Rs. 220 to B after 1 year. B suggests that A should instead pay Rs. 110 in cash now and defer payment of the remaining Rs. 110 for 2 years. A agrees. If the rate of interest is 10% per annum, what is the net gain or loss for A under thi…

Accepted Answer

Introduction / Context: This question compares two different repayment schedules using the concept of time value of money at 10 percent simple interest per annum. We must determine if changing from a single payment after 1 year to a split payment, with part paid now and part after 2 years, results in any net gain or loss for A. Standard exam convention is to compare the total obligations at a common future date using simple interest. Given Data / Assumptions: Original obligation: pay Rs. 220 after 1 year. New plan: pay Rs. 110 now and Rs. 110 after 2 years. Simple interest rate r = 10 percent per annum. We compare both arrangements by equating values at the end of 2 years. Money can earn or cost 10 percent per year in the market. Concept / Approach: The idea is to transform each payment schedule into an equivalent amount at a common comparison date, namely 2 years from now. Under simple interest, a payment made earlier grows to a larger amount at a later date. A payment made between now and the comparison date can be accumulated, and a payment made later can be taken as it is at the comparison date. If both schemes result in the same total amount at the comparison date, then there is no net gain or loss. Step-by-Step Solution: Step 1: Treat the end of year 2 as the comparison date. Step 2: Under the original scheme, A pays Rs. 220 at the end of year 1. Step 3: If this payment is considered at the end of year 2, it would earn one more year of interest at 10 percent. Amount at end of year 2 = 220 * (1 + 0.10 * 1) = 220 * 1.10 = 242. Step 4: Under the new scheme, A pays Rs. 110 now and Rs. 110 at the end of year 2. Step 5: The Rs. 110 paid now, if hypothetically invested at 10 percent for 2 years, would amount to 110 * (1 + 0.10 * 2) = 110 * 1.20 = 132 at the end of year 2. Step 6: The second payment at the end of year 2 is Rs. 110, so the total equivalent amount at the end of year 2 is 132 + 110 = 242. Step 7: The total amount at the comparison date is the same, 242, under both schemes. Verification / Alternative check: Since both arrangements translate to the same obligation of Rs. 242 at the end of year 2, the creditor and debtor are in equivalent positions when the market rate is 10 percent simple interest. This implies that, under this comparison method, there is no financial advantage to either party. Therefore, A neither gains nor loses. Why Other Options Are Wrong: Options suggesting that A gains or loses Rs. 7.34, Rs. 7.43, or Rs. 11 arise when inconsistent reference dates or incorrect discounting methods are used. If a single comparison date and the same simple interest rate are correctly applied to all cash flows, both arrangements give the same equivalent amount, so such gains or losses should not appear. Common Pitfalls: The main difficulty is choosing and sticking to a single comparison date. Some learners compare cash flows partly at the present and partly at future dates without adjusting using the interest rate, which leads to incorrect differences. Others mix simple interest and true discount formulas incorrectly. Always select a common date, accumulate earlier payments to that date, and then compare totals. Final Answer: There is no net gain or loss for A under the new mode of payment.

Question 3

The true discount on a certain sum of money due 3 years hence is Rs. 100 and the simple interest on the same sum for the same time and at the same rate is Rs. 120. What is the rate of interest per annum?

Accepted Answer

Introduction / Context: This question compares true discount and simple interest for the same sum, same period, and same rate. The true discount and simple interest values are given. The target is the rate of interest. This is a standard pattern in aptitude exams and requires understanding how true discount is related to simple interest through the effective rate time product r * t. Given Data / Assumptions: True discount TD = Rs. 100. Simple interest SI = Rs. 120. Time t = 3 years. Rate of interest r percent per annum is unknown. Both TD and SI are calculated on the same sum S and at the same rate r for the same time t. Concept / Approach: For a sum S due after time t at rate r percent, we have: SI = S * r * t / 100 TD = S * r * t / (100 + r * t) If SI and TD for the same sum and time are known, then the ratio SI / TD is: SI / TD = (S * r * t / 100) / (S * r * t / (100 + r * t)) = (100 + r * t) / 100. Thus: SI / TD = (100 + r * t) / 100. We use the given numerical values to find r * t and then r. Step-by-Step Solution: Step 1: Compute SI / TD from the given data. SI / TD = 120 / 100 = 6 / 5. Step 2: Set this equal to (100 + r * t) / 100. 6 / 5 = (100 + r * t) / 100. Step 3: Cross multiply: 6 * 100 = 5 * (100 + r * t). 600 = 500 + 5 * r * t. Step 4: 5 * r * t = 600 − 500 = 100. Step 5: So r * t = 100 / 5 = 20. Step 6: Time t is 3 years, so r = (r * t) / t = 20 / 3. Step 7: 20 / 3 percent is 6 whole and 2 thirds percent, that is (6 + 2/3)%. Verification / Alternative check: We can find the sum S and verify. Using SI = S * r * t / 100: 120 = S * 20 / 100, so S = 120 * 100 / 20 = 600. Now compute TD with S = 600 and r * t = 20: TD = S * r * t / (100 + r * t) = 600 * 20 / 120 = 600 * 1 / 6 = 100. This equals the given true discount, confirming that r = 20 / 3 percent, or (6 + 2/3) percent, is correct. Why Other Options Are Wrong: Rates like (5 + 2/3) percent, (4 + 2/3) percent, or (2 + 2/3) percent correspond to smaller values of r * t and would not satisfy SI / TD = 6 / 5. Substituting them into the relationship leads to a different ratio between SI and TD, which contradicts the given data. Common Pitfalls: One frequent mistake is to treat SI and TD as equal or to ignore the denominator 100 + r * t in the true discount formula, which destroys the key relationship. Others incorrectly cancel variables without considering ratios. Using the ratio SI / TD is a clean and systematic way to avoid dealing with the unknown sum S directly and leads quickly to the effective rate time product and then the rate itself. Final Answer: The annual rate of interest is (6 + 2/3)%.

Question 4

The true discount on a bill due 9 months hence at 12% simple interest per annum is Rs. 540. What is the present worth of the bill?

Accepted Answer

Introduction / Context: In this question the true discount on a bill is known, along with the time to maturity and the rate of simple interest. We are asked for the present worth of the bill. This involves first finding the face value of the bill using the true discount formula and then subtracting the true discount to get the present worth. Such problems are common in banking related aptitude topics. Given Data / Assumptions: True discount TD = Rs. 540. Time to maturity t = 9 months = 9 / 12 year = 3 / 4 year. Rate of simple interest r = 12 percent per annum. Face value (amount of the bill at maturity), S, is unknown. Present worth PW = S − TD must be found. Concept / Approach: True discount TD for sum S due after t years at rate r percent is: TD = S * r * t / (100 + r * t) We first find r * t, then solve for S using the given true discount. Finally, we calculate the present worth PW = S − TD. The option list clearly suggests that 6000, 6500, 7000, or 1600 are likely candidates, so accurate calculation is important. Step-by-Step Solution: Step 1: Convert time to years: t = 9 months = 9 / 12 = 3 / 4 year. Step 2: Compute r * t = 12 * 3 / 4 = 36 / 4 = 9. Step 3: Use TD formula: 540 = S * 9 / (100 + 9) = S * 9 / 109. Step 4: Rearrange: S = 540 * 109 / 9. Step 5: Simplify 540 / 9 = 60, so S = 60 * 109 = 6540. Step 6: Present worth PW is S − TD = 6540 − 540 = 6000. Verification / Alternative check: We can recompute TD from S = 6540 to verify. With r * t = 9: TD = 6540 * 9 / (100 + 9) = 6540 * 9 / 109. Compute: 6540 * 9 = 58860, and 58860 / 109 = 540. This matches the given true discount. Then the present worth PW = 6540 − 540 = 6000, confirming the correctness of the answer. Why Other Options Are Wrong: Values such as Rs. 6500 or Rs. 7000 would be closer to the face value S and not to the present worth. Rs. 1600 and Rs. 5400 are much too small to be the present worth when the discount is only Rs. 540 for less than a year at 12 percent. Only Rs. 6000 is consistent with the derived face value and discount. Common Pitfalls: Students often forget to convert months to years correctly and may use t = 9 instead of 3 / 4, causing very large or very small answers. Another common mistake is to misinterpret Rs. 540 as the present worth rather than the true discount. Always distinguish between S (amount due), TD (true discount), and PW (present worth), and apply the formula TD = S * r * t / (100 + r * t) systematically. Final Answer: The present worth of the bill is Rs. 6000.

Question 5

The true discount on a certain sum of money due 3 years hence is Rs. 250, and the simple interest on the same sum for the same time and at the same rate is Rs. 375. What is the annual rate of interest?

Accepted Answer

Introduction / Context: This question is similar in pattern to an earlier one but with larger numbers. Both true discount and simple interest are given for the same sum, same time, and same rate. The relationship between SI and TD helps us find the effective rate time product and then the annual rate. It strengthens understanding of the connection between discounted values and interest over time. Given Data / Assumptions: True discount TD = Rs. 250. Simple interest SI = Rs. 375. Time t = 3 years. Rate of interest r percent per annum is unknown. Both TD and SI are calculated on the same sum S. Concept / Approach: For sum S, time t, and rate r, the formulas are: SI = S * r * t / 100 TD = S * r * t / (100 + r * t) The ratio SI / TD simplifies to: SI / TD = (100 + r * t) / 100. We can therefore compute r * t from the known numerical ratio and then divide by t = 3 to get r. Step-by-Step Solution: Step 1: Compute SI / TD from the data. SI / TD = 375 / 250 = 3 / 2 = 1.5. Step 2: Set this equal to (100 + r * t) / 100. 3 / 2 = (100 + r * t) / 100. Step 3: Cross multiply: 3 * 100 = 2 * (100 + r * t). 300 = 200 + 2 * r * t. Step 4: 2 * r * t = 100, so r * t = 50. Step 5: Time t = 3 years, so r = (r * t) / t = 50 / 3. Step 6: 50 / 3 = 16 whole and 2 thirds percent, which is (16 + 2/3)%. Verification / Alternative check: We may find S and check. Using SI = S * r * t / 100: 375 = S * 50 / 100, so S = 375 * 100 / 50 = 750. Now compute TD with S = 750 and r * t = 50: TD = S * r * t / (100 + r * t) = 750 * 50 / 150 = 750 * 1 / 3 = 250. This matches the given true discount, confirming that r = 50 / 3 percent per year, that is (16 + 2/3)% is correct. Why Other Options Are Wrong: Rates such as (16 + 1/3)% or (16 + 4/3)% correspond to r * t values less than or greater than 50 respectively. They would not preserve the ratio SI / TD = 3 / 2 and therefore cannot match the problem data. The 20 percent option is also incorrect, as it gives r * t = 60, not 50. Common Pitfalls: Mistakes often occur when simplifying the ratio 375 / 250 or when solving the cross multiplication step. Some students also forget that both TD and SI refer to the same sum, which is why the ratio eliminates S and simplifies the problem. Carefully handling the algebra and keeping track of the effective product r * t avoids these errors. Final Answer: The annual rate of interest is (16 + 2/3)%.

Question 6

What is the present worth of Rs. 2310 due 2.5 years (2 1/2 years) hence, if the rate of simple interest is 15% per annum?

Accepted Answer

Introduction / Context: This problem asks for the present worth of a sum due after a certain time when the rate of simple interest is known. Present worth is the amount which, if invested now at the given rate, will accumulate to the future sum. In true discount terminology, the difference between the future sum and present worth is the true discount. Given Data / Assumptions: Amount due in the future, S = Rs. 2310. Time t = 2.5 years = 2 and 1 / 2 years. Rate of simple interest r = 15 percent per annum. Present worth PW is unknown and must be found. Simple interest calculations are used throughout. Concept / Approach: If PW is the present worth of S due after time t at rate r, then: S = PW * (1 + r * t / 100). Equivalently: PW = S * 100 / (100 + r * t). We first compute the effective product r * t, then directly compute PW using the second formula. This is standard in discount and present value problems. Step-by-Step Solution: Step 1: Compute r * t. r = 15, t = 2.5 years, so r * t = 15 * 2.5 = 37.5. Step 2: Use PW formula: PW = S * 100 / (100 + r * t). PW = 2310 * 100 / (100 + 37.5) = 231000 / 137.5. Step 3: Compute 231000 / 137.5 to get PW = 1680. Thus the present worth is Rs. 1680. Verification / Alternative check: Confirm by forward calculation using simple interest. If PW = 1680 is invested at 15 percent for 2.5 years: Interest I = PW * r * t / 100 = 1680 * 15 * 2.5 / 100. First calculate 15 * 2.5 = 37.5. I = 1680 * 37.5 / 100 = 1680 * 0.375 = 630. Amount after 2.5 years = PW + I = 1680 + 630 = 2310. This matches the given future amount, confirming that the present worth is correctly found. Why Other Options Are Wrong: Values like Rs. 1440, Rs. 1600, Rs. 1750, or Rs. 1840 either produce too small or too large an amount when accumulated at 15 percent for 2.5 years. Only Rs. 1680 grows exactly to Rs. 2310, so only that satisfies the condition. Common Pitfalls: Errors usually arise from miscalculating r * t (for example, treating 2.5 years as 2 years or 25 years) or from mishandling decimal multiplication when computing the interest. Another issue is confusing present worth with the true discount; here we are asked for the present worth directly, not the discount. Always double check by recomputing the amount from the present worth to see if it matches the future sum. Final Answer: The present worth of Rs. 2310 due after 2.5 years at 15% per annum is Rs. 1680.

Question 7

If the true discount on Rs. 2562 due 4 months hence is Rs. 122, what is the annual rate of simple interest?

Accepted Answer

Introduction / Context: This problem gives the true discount on a sum due after 4 months and asks for the annual rate of interest. It requires careful handling of time in years and the true discount formula. The effective rate time product must be inferred using algebra, then converted to an annual percentage rate. Given Data / Assumptions: Sum due, S = Rs. 2562. True discount TD = Rs. 122. Time t = 4 months = 4 / 12 year = 1 / 3 year. Rate of simple interest r percent per annum is unknown. True discount is calculated under simple interest assumptions. Concept / Approach: We use the true discount formula: TD = S * r * t / (100 + r * t) Let r be the annual rate. Time t is 1 / 3 year, so r * t = r / 3. Substituting the known values leads to an equation in r, which can be solved algebraically. After finding r, we interpret it as an annual percentage rate and choose the matching option. Step-by-Step Solution: Step 1: Substitute S = 2562, TD = 122, t = 1 / 3 into the formula. 122 = 2562 * r * (1 / 3) / (100 + r / 3). Step 2: Simplify numerator: 2562 * r / 3. So 122 = (2562 * r / 3) / (100 + r / 3). Step 3: Multiply both sides by (100 + r / 3). 122 * (100 + r / 3) = 2562 * r / 3. Step 4: Multiply both sides by 3 to clear the denominator. 366 * (100 + r / 3) = 2562 * r. Step 5: Expand left side: 36600 + 366 * r / 3. 366 * r / 3 = 122r. So 36600 + 122r = 2562r. Step 6: Move terms: 2562r − 122r = 2440r = 36600. Step 7: Solve r = 36600 / 2440 = 15. Thus r = 15 percent per annum. Verification / Alternative check: Check by recomputing true discount with r = 15 percent and t = 1 / 3 year. Then r * t = 15 / 3 = 5: TD = 2562 * 5 / (100 + 5) = 2562 * 5 / 105. Multiply: 2562 * 5 = 12810. Divide: 12810 / 105 = 122. This matches the given true discount, confirming the rate is indeed 15 percent per annum. Why Other Options Are Wrong: Rates such as 12 percent, 13 and one third percent, 14 percent, or 10 percent produce different true discounts when substituted into the formula, not Rs. 122. Only 15 percent yields exactly the stated true discount for the given sum and time. Common Pitfalls: Students often forget to convert 4 months into 1 / 3 year and instead use t = 4, which leads to unrealistic values. Another pitfall is mishandling the algebraic step when clearing denominators or expanding 366 * (100 + r / 3). Careful manipulation of fractions and verifying the result by recomputing the true discount helps avoid such errors. Final Answer: The annual rate of simple interest is 15%.

Question 8

The simple interest and the true discount on a certain sum of money for a given time at a given rate are Rs. 85 and Rs. 80 respectively. What is the sum of money?

Accepted Answer

Introduction / Context: Here both simple interest and true discount are given for the same sum, same time, and same rate, but the time and rate themselves are not specified. The task is to find the principal sum. Using the known relationship between simple interest and true discount allows us to compute the effective rate time product and then the principal without explicitly knowing the rate or time separately. Given Data / Assumptions: Simple interest SI = Rs. 85. True discount TD = Rs. 80. Sum S is due after some time t at rate r percent per annum. The same S, t, and r are used for both SI and TD. Concept / Approach: We use the standard formulas: SI = S * r * t / 100 TD = S * r * t / (100 + r * t) Let R = r * t. Then: SI = S * R / 100, and TD = S * R / (100 + R). We also know that: SI / TD = (100 + R) / 100. From the numerical SI and TD, we can compute SI / TD, then find R, then use one of the expressions to solve for S. Step-by-Step Solution: Step 1: Compute the ratio SI / TD. SI / TD = 85 / 80 = 17 / 16 = 1.0625. Step 2: Set this equal to (100 + R) / 100. 17 / 16 = (100 + R) / 100. Step 3: Cross multiply: 17 * 100 = 16 * (100 + R). 1700 = 1600 + 16R. Step 4: 16R = 1700 − 1600 = 100. Step 5: R = 100 / 16 = 6.25. So R = r * t = 6.25 percent for the entire period. Step 6: Use SI formula to find S. SI = S * R / 100 = S * 6.25 / 100. 85 = S * 6.25 / 100. Step 7: S = 85 * 100 / 6.25. Note that 6.25 = 25 / 4, so 100 / 6.25 = 100 * 4 / 25 = 16. Thus S = 85 * 16 = 1360. Verification / Alternative check: Compute TD using S = 1360 and R = 6.25. Then: TD = S * R / (100 + R) = 1360 * 6.25 / 106.25. First, 1360 * 6.25 = 8500. 106.25 is 425 / 4, so 8500 divided by 106.25 is 80. Thus TD = 80 and SI = 85, exactly as given, confirming that S = 1360 is correct. Why Other Options Are Wrong: Other choices such as Rs. 1200, Rs. 1316, Rs. 1350, or Rs. 1500 do not satisfy the simultaneous conditions SI = 85 and TD = 80 when used with the same rate and time. They either produce different interest or different discount, not matching the given values. Common Pitfalls: Common errors include treating SI and TD as equal or attempting to guess rate and time separately without using the ratio method. Miscalculating the ratio 85 / 80 or mishandling the fractional rate 6.25 can also lead to wrong answers. Using the simplified parameter R = r * t and carefully solving the resulting algebraic equations avoids these issues. Final Answer: The sum of money is Rs. 1360.

Question 9

A trader owes a merchant Rs. 10028 due 1 year hence. The trader wants to settle the account after 3 months. If the rate of interest is 12% per annum simple interest, how much cash should he pay now?

Accepted Answer

Introduction / Context: This question asks for the present cash payment required to settle a debt earlier than its due date under simple interest conditions. The trader owes a certain amount 1 year later but wants to close the account after 3 months. Effectively, we need to find the present worth at the earlier date using the concept of true discount with the given annual rate of interest. Given Data / Assumptions: Future debt amount, S = Rs. 10028, due after 1 year. Trader settles the debt after 3 months, that is 0.25 year from now. This is 9 months, or 0.75 year, earlier than the due date. Rate of simple interest r = 12 percent per annum. We need the cash amount to be paid at 3 months that is equivalent in value to paying Rs. 10028 at the end of 1 year. Concept / Approach: Since the debt is paid earlier than due, the trader is entitled to a true discount for the time saved. Here, the time between settlement and due date is 9 months, or 0.75 year. True discount TD on S for time t at rate r is: TD = S * r * t / (100 + r * t) We first compute r * t and then the true discount. The present worth PW, which is the cash amount to be paid now (at 3 months), is: PW = S − TD. Step-by-Step Solution: Step 1: Compute the time between settlement and due date. Original due in 1 year, settlement at 3 months = 0.25 year, so t = 1 − 0.25 = 0.75 year. Step 2: Compute r * t = 12 * 0.75 = 9. Step 3: Use true discount formula. TD = S * r * t / (100 + r * t) = 10028 * 9 / (100 + 9). Step 4: Denominator is 109, so TD = 10028 * 9 / 109. Step 5: Multiply numerator: 10028 * 9 = 90252. Step 6: Divide: 90252 / 109 = 828. Thus the true discount is Rs. 828. Step 7: Present worth PW = S − TD = 10028 − 828 = 9200. So the trader should pay Rs. 9200 at 3 months to settle the debt. Verification / Alternative check: We can verify by growing the present worth forward for 9 months at 12 percent and showing that it equals Rs. 10028. Compute r * t again as 9: PW * (1 + r * t / 100) = 9200 * (1 + 9 / 100) = 9200 * 1.09 = 10028. This exactly matches the future debt amount, so our present worth is correct. Why Other Options Are Wrong: A payment of Rs. 9000, Rs. 9002, or Rs. 9120 would grow to less than Rs. 10028 at 12 percent for 9 months, so they would underpay the merchant relative to the agreed rate. A payment of Rs. 8800 is even lower and is clearly insufficient. Only Rs. 9200 grows to exactly Rs. 10028 at the given rate and time. Common Pitfalls: Typical mistakes include using the full year as the time period instead of 9 months, or applying the simple interest formula directly to compute interest on the present worth without accounting for the true discount structure. Some learners also confuse the number 10028 with the present worth and subtract interest incorrectly. Using the clear steps of computing r * t, then true discount TD, and finally PW = S − TD is a reliable method. Final Answer: The trader should pay Rs. 9200 in cash after 3 months to settle the debt.

Question 10

A man buys a watch for Rs 1950 in cash and sells it for Rs 2200 on a credit of 1 year. If the rate of interest in the market is 10% per annum, then, based on the present worth of Rs 2200, by how much amount does he gain or lose on this transaction?

Accepted Answer

Introduction / Context: This question tests the concept of true profit or loss when an item is sold on credit. In aptitude exams, it is common to compare the present worth of a future payment with the actual cash cost price in order to judge whether the seller gains or loses. Here, a man buys a watch for a cash price and sells it on a one year credit. Because money has a time value, the future payment of Rs 2200 is not equal to receiving Rs 2200 in cash today. We must discount the future amount at the given interest rate to find its present worth and then compare it with the cash cost price. Given Data / Assumptions: Cash cost price of the watch = Rs 1950. Credit selling price after 1 year = Rs 2200. Rate of interest in the market = 10 percent per annum (simple interest). We compare present worth of Rs 2200 with Rs 1950 to decide gain or loss. Concept / Approach: The key idea is present worth. A future payment of S after t years at simple interest rate r percent per year has present worth given by: Present worth = S / (1 + r * t / 100) For a one year credit, t = 1. After finding the present worth of Rs 2200, we compare it with the cash cost price. If present worth is greater than cost price, there is a gain equal to that difference. If it is lower, there is a loss. Step-by-Step Solution: Step 1: Identify S, r and t. S = 2200, r = 10, t = 1 year. Step 2: Compute present worth of Rs 2200 payable after 1 year. Present worth = 2200 / (1 + 10 * 1 / 100) Present worth = 2200 / (1 + 0.10) Present worth = 2200 / 1.10 Present worth = 2000. Step 3: Compare present worth with cash cost price. Cost price (cash) = 1950. Effective selling price in present terms = 2000. Gain = 2000 - 1950 = 50. Thus he gains Rs 50 on the transaction when we correctly account for the time value of money. Verification / Alternative check: We can also think in terms of what amount invested today at 10 percent per year would become Rs 2200 after 1 year. Let P be that amount. Then: P * (1 + 10 / 100) = 2200 P * 1.10 = 2200 P = 2200 / 1.10 = 2000. So, effectively, he is receiving Rs 2000 today for a watch that cost Rs 1950 in cash. The difference again is Rs 50, confirming the gain. Why Other Options Are Wrong: Option B (Gain of Rs 55) assumes a different interest or arithmetic error in discounting the future amount. Option C (Loss of Rs 15) wrongly assumes that cash cost is greater than present worth, which is not true here. Option D (Loss of Rs 45) is also incorrect since we clearly found a gain, not a loss. Option E (No profit no loss) would only be correct if the present worth equaled the cash cost of Rs 1950, which is not the case. Common Pitfalls: Many learners mistakenly compare the nominal future selling price of Rs 2200 with the cash cost of Rs 1950 and conclude a gain of Rs 250. This ignores the fact that the Rs 2200 will be received after 1 year and must therefore be discounted. Another common mistake is to wrongly apply simple interest to the cost price instead of discounting the future value. Careful attention to which amount is at present and which is in the future is essential. Final Answer: The man makes a Gain of Rs 50 on the transaction when the time value of money is considered at 10 percent per annum.

Question 11

The true discount on a bill of Rs 1600 due after a certain time at 5 percent per annum simple interest is Rs 160. After how much time, expressed in months, will the amount fall due?

Accepted Answer

Introduction / Context: This problem belongs to the topic of true discount and present worth under simple interest. In commercial mathematics, a bill that is due in the future can be discounted today. The difference between the sum due at maturity and its present worth is called the true discount. Here we are given the true discount, the sum due and the rate of interest, and we are asked to find the time after which the bill is due, expressed in months. This is a classic reverse application of the true discount formula, which is very important for bank discount and bill of exchange questions in aptitude exams. Given Data / Assumptions: Sum due at maturity S = Rs 1600. True discount TD = Rs 160. Rate of interest r = 5 percent per annum (simple interest). Time period t is to be found in years and then converted to months. Interest is calculated on simple interest basis. Concept / Approach: For simple interest, if a sum S is due after t years at rate r percent per annum, then the present worth PW is: PW = S / (1 + r * t / 100) The true discount TD is the difference between S and PW: TD = S - PW Using the above formulas we can express TD directly in terms of S, r and t as: TD = S * r * t / (100 + r * t) Here S, r and TD are known, so we can solve this equation for t. Step-by-Step Solution: Step 1: Write the equation for true discount. 160 = 1600 * 5 * t / (100 + 5 * t) Step 2: Simplify the numerator. 160 = 8000 * t / (100 + 5 * t) Step 3: Multiply both sides by (100 + 5 * t). 160 * (100 + 5 * t) = 8000 * t 16000 + 800 * t = 8000 * t Step 4: Bring like terms together. 16000 = 8000 * t - 800 * t 16000 = 7200 * t t = 16000 / 7200 = 20 / 9 years Step 5: Convert years to months. t in months = (20 / 9) * 12 = 240 / 9 = 80 / 3 months 80 / 3 months is approximately 26.67 months. Rounded to the nearest whole month, the due date is after 27 months. Verification / Alternative check: We can quickly verify by converting 27 months into years. That is 27 / 12 = 2.25 years. Substitute t = 2.25 into the formula: TD approximate = 1600 * 5 * 2.25 / (100 + 5 * 2.25) TD approximate = 1600 * 11.25 / (100 + 11.25) = 18000 / 111.25 This is close to 161.8, slightly higher than 160 because we rounded the exact time. When we use the exact value 20 / 9 years, we get exactly Rs 160 as shown earlier, so the reasoning is sound. Why Other Options Are Wrong: Option B (23 months) would give a smaller t and hence a smaller denominator, resulting in a true discount larger than Rs 160. Option C (20 months) gives even smaller time and an even larger discount. Option D (12 months) corresponds to only one year and would make the true discount far too small. Option E (30 months) gives time larger than 27 months and would produce a discount larger than the required value when calculated exactly. Common Pitfalls: A common mistake is to confuse true discount with simple interest. True discount is calculated on the sum due, but the formula includes the present worth indirectly through the denominator 100 plus r times t. Another frequent error is to mix months and years without proper conversion. Always convert time to years when applying the formula with annual rate and then convert back to months only at the end. Careful algebraic manipulation avoids errors when solving for t. Final Answer: The bill is due after approximately 27 months to produce a true discount of Rs 160 at 5 percent per annum simple interest.

Question 12

A trader buys a chair for ₹ 600 and sells it for ₹ 765 at a credit of 4 months. Reckoning money worth 6% p.a., his gain percent is

Accepted Answer

Concept & True Discount (Present Value): When an item is sold on credit, the future selling price must be discounted to its "Present Worth" (PW) to determine the true profit made today. PW = \frac{Amount due}{1 + ((R × T/100))} Step-by-Step Solution: • Identify Given Values: • Cost Price (CP) = ₹ 600 • Future Value of Selling Price = ₹ 765 • Time (T) = 4 months = (4/12) years = (1/3) years. • Rate of Interest (R) = 6% p.a. • Calculate Present Worth of Selling Price: • PW = (765/1 + ( \frac{6 × 1/3){100} )} • PW = (765/1 + ( \frac{2){100} )} = (765/1.02) • PW = 750 rupees. • The effective immediate selling price is ₹ 750. • Calculate True Gain Percentage: • Gain = Effective SP - CP = 750 - 600 = 150 rupees. • Gain % = ( (150/600) ) × 100 = (1/4) × 100 = 25%. Exam Strategy & Shortcut: For simple numbers, note that 6% per year is equivalent to 2% for 4 months (since 4 months is a third of a year). So, the future amount represents 102% of the present value. 765 / 1.02 = 750. Comparing 750 (SP) to 600 (CP) yields a 150 profit on 600, instantly giving (1/4) or 25%. Common Pitfall: A standard trap is ignoring the credit period entirely and calculating profit as (765 - 600) / 600 = 27.5%. This fails to account for the time value of money, which the prompt explicitly states by saying "Reckoning money worth 6% p.a.". Final Answer: Therefore, the correct answer is 25%.

Question 13

The true discount on ₹ 2562 due 4 months hence is ₹ 122. The rate percent is

Accepted Answer

Concept & True Discount Logic: The true discount is the simple interest on the Present Worth. We can find the Present Worth by subtracting the true discount from the total amount due. PW = Amount - TD R = \frac{100 × TD}{PW × T} Step-by-Step Solution: - Given Amount = ₹ 2562 - True Discount (TD) = ₹ 122 - Time (T) = 4 months = (4/12) years = (1/3) year - Find the Present Worth (PW): PW = 2562 - 122 = 2440 - Now substitute PW, TD, and T into the rate formula: R = (100 × 122/2440 × (\frac{1){3})} R = (12200/\frac{2440){3}} = (12200 × 3/2440) R = 5 × 3 = 15% Exam Strategy & Shortcut: Instead of writing complex fractions, observe that 122 is exactly 1/20th (or 5%) of 2440. If the interest for 1/3 of a year is 5%, the interest for a full year is 5% × 3 = 15%. Common Pitfall: Forgetting to convert months into years. Using T = 4 instead of T = 1/3 will yield a drastically incorrect, tiny rate percentage. Final Answer: Therefore, the correct answer is 15%.

Question 14

The true discount on ₹ 1760 due after a certain time at 12% per annum is ₹ 160. The time after which it is due is

Accepted Answer

Concept & Time Calculation: To find the time period, isolate it from the simple interest formula applied to the present worth. PW = Amount - TD T = \frac{100 × TD}{PW × R} Step-by-Step Solution: - Given Amount = ₹ 1760 - True Discount (TD) = ₹ 160 - Rate (R) = 12% per annum - Calculate the Present Worth (PW): PW = 1760 - 160 = 1600 - Use the formula to find Time (T) in years: T = (100 × 160/1600 × 12) T = (16000/19200) = (5/6) years - Convert years into months by multiplying by 12: T = (5/6) × 12 = 10 months Exam Strategy & Shortcut: Observe that the TD is exactly 10% of the PW (160 / 1600). If the annual rate is 12%, then a 10% return requires a fraction of the year equal to 10/12. 10/12 of 12 months is trivially 10 months. Common Pitfall: Leaving the final answer as a fraction of a year (5/6) and guessing incorrectly among the options instead of converting it to months. Final Answer: Therefore, the correct answer is 10 months.

Question 15

A has to pay ₹ 220 to B after 1 year. B asks A to pay ₹ 110 in cash and defer the payment of ₹ 110 for 2 years. A agrees to it. If the rate of interest be 10% per annum, in this mode of payment (a) There is no gain or loss to any one (b) A gains ₹ 7…

Accepted Answer

Concept & Time Value of Money Equivalence: To determine a gain or loss, we must compare the equivalent future values of both payment streams at a common point in time. Let's use the end of year 2 (t=2). Amount = Principal + (\frac{Principal × R × T}{100}) Step-by-Step Solution: - Original Agreement (Value at t=2): A pays ₹ 220 at the end of year 1. B can reinvest this ₹ 220 for the remaining 1 year at 10% simple interest. Future Value at t=2: 220 + (220 × 10 × 1/100) = 220 + 22 = 242. - New Agreement (Value at t=2): A pays ₹ 110 immediately (t=0). B can reinvest this ₹ 110 for the full 2 years at 10% simple interest. Future Value of immediate payment at t=2: 110 + (110 × 10 × 2/100) = 110 + 22 = 132. A also pays the deferred ₹ 110 at the end of year 2. Total Future Value at t=2: 132 + 110 = 242. - Conclusion: Since B ends up with exactly ₹ 242 in both scenarios, and A shells out the equivalent value of ₹ 242 in both scenarios, neither party gains or loses. Exam Strategy & Shortcut: Recognize this classic setup. If a payment is split such that the early payment's interest exactly covers the deferred payment's lack of interest over symmetric periods, the net result is always zero sum. Common Pitfall: Trying to calculate just the Present Worth of the new payments without accounting for the opportunity cost (interest earned) of the ₹ 110 paid in cash upfront. Final Answer: Therefore, the correct answer is (a) There is no gain or loss to any one.

Question 16

₹ 20 is the true discount on ₹ 260 due after a certain time. What will be the true discount on the same sum due after half of the former time, the rate of interest being the same? (a) ₹ 15.20 (b) ₹ 10.40 (c) ₹ 10.83 (d) ₹ 13

Accepted Answer

Concept & Time Scaling of True Discount: True discount is calculated on the Present Worth. When the time frame is halved, the present worth changes, making the scaling non-linear. We extract the Rate × Time (RT) value to solve for the new state. TD_2 = \frac{Amount × (RT/2)}{100 + (RT/2)} Step-by-Step Solution: - Initial State: Amount = ₹ 260, True Discount = ₹ 20. Present Worth (PW_1) = 260 - 20 = 240. Calculate the RT product from the TD_1 interest equation: 240 × (R × T/100) = 20 \Rightarrow RT = (2000/240) = (25/3) - Halved Time State: New time factor is T/2. Thus the new multiplier is R × (T/2) = (1/2) × (25/3) = (25/6). Substitute into the TD_2 formula: TD_2 = (260 × (\frac{25/6))}{100 + (25/6)} = (260 × \frac{25/6)}{(625/6)} TD_2 = (260 × 25/625) = (260/25) = 10.40 Exam Strategy & Shortcut: Recognize that true discount behaves inversely to present worth. If time is halved, simple interest halves (from 20 to 10), but true discount remains slightly higher than half because the new present worth (base) is larger. 10.40 is the only logical slight increase over 10. Common Pitfall: A trap answer is ₹ 10, assuming TD scales linearly like simple interest. Final Answer: Therefore, the correct answer is (b) ₹ 10.40.

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