Question 1

The banker's discount and the true discount on a sum of money due 8 months hence are Rs 120 and Rs 110 respectively. Find the face value (sum) of the bill.

Accepted Answer

Introduction: This question provides both the banker's discount (BD) and the true discount (TD) for a bill due after 8 months and asks us to determine the sum (face value). We do not know the interest rate directly, but the ratio BD / TD helps us eliminate it and solve for the face value. Given Data / Assumptions: Banker's discount BD = Rs 120. True discount TD = Rs 110. Time t = 8 months = 8/12 = 2/3 year. Rate r (unknown, but same for both discounts). We must find the sum P (face value). Concept / Approach: Let x = r * t. Then: BD = P * x / 100. TD = P * x / (100 + x). The ratio BD / TD becomes: BD / TD = (100 + x) / 100. We use the given BD and TD to find x, then use TD = P * x / (100 + x) to solve for P directly. Step-by-Step Solution: Step 1: Use BD / TD to find x. BD / TD = 120 / 110 = 12 / 11. But BD / TD = (100 + x) / 100. (100 + x) / 100 = 12 / 11. 100 + x = 100 * 12 / 11 = 1200 / 11. x = 1200 / 11 − 100 = 1200 / 11 − 1100 / 11 = 100 / 11. So x = r * t = 100 / 11. Step 2: Use TD formula to find P. TD = P * x / (100 + x). Here 100 + x = 1200 / 11. 110 = P * (100 / 11) / (1200 / 11). 110 = P * (100 / 11) * (11 / 1200) = P * 100 / 1200 = P / 12. Therefore P / 12 = 110 ⇒ P = 110 * 12 = Rs 1320. Verification / Alternative check: With P = 1320 and x = 100 / 11: BD = P * x / 100 = 1320 * (100 / 11) / 100 = 1320 / 11 = Rs 120. TD = P * x / (100 + x) = 1320 * (100 / 11) / (1200 / 11) = 1320 * 100 / 1200 = Rs 110. This matches the given values. Why Other Options Are Wrong: Rs 1457, Rs 1140, Rs 1260, Rs 1200: None of these sums produce BD = 120 and TD = 110 simultaneously under the standard formulas, so they are inconsistent with the given data. Common Pitfalls: A typical mistake is trying to assume a rate and then solving for time or vice versa, which is unnecessary. The cleanest method is to treat x = r * t as a single unknown and use the relationships involving BD, TD, and x to determine P directly. Final Answer: The face value (sum) of the bill is Rs 1320.

Question 2

If the discount (true discount) on Rs 498 at 5% simple interest is Rs 18, after how long (in months) is the sum due?

Accepted Answer

Introduction: This question gives the true discount on a sum at a known rate and asks for the time until the sum is due. Recognizing that the term “discount” here represents true discount, we can use the true discount formula to solve for the product r * t and thus find t in months. Given Data / Assumptions: Sum (face value) P = Rs 498. True discount TD = Rs 18. Rate r = 5% per annum (simple interest). Time t = ? years (we will later convert to months). Concept / Approach: For true discount: TD = P * r * t / (100 + r * t). Let x = r * t. Then: TD = P * x / (100 + x). We know TD, P, and r, so we can solve for x, then find t = x / r, and finally convert t to months. Step-by-Step Solution: Step 1: Set up the equation with x = r * t. 18 = 498 * x / (100 + x). 18(100 + x) = 498x. 1800 + 18x = 498x. 1800 = 498x − 18x = 480x. x = 1800 / 480 = 15 / 4 = 3.75. So r * t = x = 3.75. Step 2: Find t using r = 5. r * t = 3.75 ⇒ 5 * t = 3.75. t = 3.75 / 5 = 0.75 years. Step 3: Convert years to months. 0.75 years = 0.75 * 12 = 9 months. Verification / Alternative check: Recompute TD using t = 0.75 years: r * t = 5 * 0.75 = 3.75. TD = 498 * 3.75 / (100 + 3.75) = 1867.5 / 103.75 = Rs 18 (approximately), which matches the given discount. Why Other Options Are Wrong: 8, 10, 11, or 6 months: Each corresponds to a different t in years, and none give TD exactly equal to Rs 18 when substituted in the true discount formula with P = 498 and r = 5%. Common Pitfalls: A major pitfall is to treat the discount as simple interest on the present worth or to apply the simple interest formula P * r * t / 100 directly to the face value, which gives the banker's discount, not the true discount. Using the correct TD formula is essential for the right answer. Final Answer: The sum is due after 9 months.

Question 3

The present worth of a bill due sometime hence is Rs 1100, and the true discount on the bill is Rs 110. Find the banker's discount and the banker's gain for this transaction.

Accepted Answer

Introduction: This problem gives the present worth (PW) of a bill and the true discount (TD), and asks us to find the banker's discount (BD) and the banker's gain (BG). Using the relationships among PW, TD, face value, and the rate-time product r * t, we can determine BD and BG without knowing the exact rate or time individually. Given Data / Assumptions: Present worth PW = Rs 1100. True discount TD = Rs 110. Face value P = PW + TD (since P = present worth + true discount). We must find BD and BG. Concept / Approach: First, compute the face value: P = PW + TD. Next, recall the relationship for true discount: TD / PW = r * t / 100. This follows because: PW = P * 100 / (100 + r * t) and TD = P − PW. Once we know r * t, we can compute BD using: BD = P * r * t / 100 = P * (r * t) / 100. Finally, BG = BD − TD. Step-by-Step Solution: Step 1: Find face value P. P = PW + TD = 1100 + 110 = Rs 1210. Step 2: Use TD / PW to find r * t. TD / PW = 110 / 1100 = 1 / 10. But TD / PW = r * t / 100. So r * t / 100 = 1 / 10 ⇒ r * t = 10. Step 3: Compute BD. BD = P * r * t / 100 = 1210 * 10 / 100. BD = 1210 / 10 = Rs 121. Step 4: Compute BG. BG = BD − TD = 121 − 110 = Rs 11. Verification / Alternative check: We can check TD using P = 1210 and r * t = 10: TD = P * r * t / (100 + r * t) = 1210 * 10 / (100 + 10) = 12100 / 110 = Rs 110. This matches the given TD. Then BG = BD − TD = 121 − 110 = 11, consistent with our earlier calculation. Why Other Options Are Wrong: All other pairs (BD, BG) shown would either result in a different TD using the standard formulas or yield a different present worth when recomputed. Only BD = Rs 121 and BG = Rs 11 preserve both the given PW and TD. Common Pitfalls: Learners sometimes treat PW as equal to the face value P, which is incorrect; the face value is always PW + TD. Another error is to ignore the useful relation TD / PW = r * t / 100, which directly gives the rate-time product and greatly simplifies the calculations. Final Answer: The correct values are banker's discount = Rs 121 and banker's gain = Rs 11.

Question 4

The banker's discount on a bill due 6 months hence at 6% per annum is Rs 18.54. What is the true discount on this bill?

Accepted Answer

Introduction: This question gives the banker's discount (BD) and asks us to find the true discount (TD) for the same time and rate. We use the known ratio BD / TD in terms of the rate-time product r * t to compute TD directly. Given Data / Assumptions: Banker's discount BD = Rs 18.54. Rate r = 6% per annum. Time t = 6 months = 0.5 year. We must find true discount TD. Concept / Approach: For a bill with face value P at rate r and time t: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). Hence: BD / TD = (100 + r * t) / 100. So: TD = BD * 100 / (100 + r * t). We will compute r * t and then use this relationship to find TD. Step-by-Step Solution: Step 1: Compute r * t. r * t = 6 * 0.5 = 3. Step 2: Use TD = BD * 100 / (100 + r * t). TD = 18.54 * 100 / (100 + 3). TD = 18.54 * 100 / 103. TD = 1854 / 103 = Rs 18. Verification / Alternative check: We can check the ratio BD / TD: BD / TD = 18.54 / 18 = 1.03. From theory: BD / TD = (100 + r * t) / 100 = (100 + 3) / 100 = 103 / 100 = 1.03, which matches perfectly. Why Other Options Are Wrong: Rs 24, Rs 12, Rs 36, Rs 15: None of these values satisfy the exact ratio BD / TD = 103 / 100 with BD = Rs 18.54. Using any of them would either make BD too large or too small relative to TD under the given rate and time. Common Pitfalls: A common error is to subtract some arbitrary amount from BD to guess TD, or to use simple interest on the present worth without knowing it. The right method is to remember and apply the ratio BD / TD = (100 + r * t) / 100. Final Answer: The true discount on the bill is Rs 18.

Question 5

What is the difference between the banker's discount and the true discount on Rs 8100 for 3 months at 5% per annum simple interest?

Accepted Answer

Introduction: This problem asks for the difference between banker's discount (BD) and true discount (TD), which is called the banker's gain (BG). We are given the principal, rate, and time. We can compute BD and TD using their formulas and subtract to obtain the difference. Given Data / Assumptions: Face value P = Rs 8100. Rate r = 5% per annum. Time t = 3 months = 3/12 = 1/4 year. We must find BD − TD (the difference, i.e., banker's gain). Concept / Approach: For a bill with face value P: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). The difference BD − TD equals the banker's gain BG. We will compute BD and TD numerically and subtract. Step-by-Step Solution: Step 1: Compute r * t. t = 3/12 = 1/4 year ⇒ r * t = 5 * (1/4) = 1.25. Step 2: Compute BD. BD = P * r * t / 100 = 8100 * 1.25 / 100. BD = 8100 * 0.0125 = Rs 101.25. Step 3: Compute TD. TD = P * r * t / (100 + r * t) = 8100 * 1.25 / (100 + 1.25). TD = 10125 / 101.25 = Rs 100. Step 4: Difference (banker's gain). BD − TD = 101.25 − 100 = Rs 1.25. Verification / Alternative check: We can also use the formula BG = TD * r * t / 100: BG = 100 * 1.25 / 100 = Rs 1.25, which matches our computed difference BD − TD, confirming the result. Why Other Options Are Wrong: Rs 2, Rs 2.25, Rs 0.5, Rs 3: None of these values are equal to TD * r * t / 100 with TD = 100 and r * t = 1.25, and therefore they are inconsistent with the exact formulas for BD and TD. Common Pitfalls: Some learners mistakenly apply simple interest on the present worth rather than the face value, or they forget to convert 3 months into 1/4 year. Another common mistake is rounding too early instead of working with exact fractions until the final step. Final Answer: The difference between the banker's discount and the true discount is Rs 1.25.

Question 6

In banker's discount, the banker's discount on a certain sum of money is Rs. 72 and the true discount on the same sum for the same period of time is Rs. 60. What is the sum due (the amount payable at maturity)?

Accepted Answer

Introduction / Context: Questions on banker's discount and true discount are common in commercial arithmetic. They test your understanding of present worth, sum due (amount), and how banks calculate interest on bills that are payable at a future date. Here, you are given both the banker's discount and the true discount and are asked to find the sum due at maturity. Given Data / Assumptions: The banker's discount (BD) on a bill = Rs. 72. The true discount (TD) on the same bill for the same time = Rs. 60. We are asked to find the sum due, i.e., the amount payable at maturity (A). The rate of interest and time are the same for BD and TD, but their exact values are not required for this problem. Concept / Approach: True discount is the difference between the amount due at maturity and its present worth. Banker's discount is simple interest calculated on the amount due (face value) for the unexpired time. An important relationship between BD, TD and the amount A is: A = (BD * TD) / (BD - TD) This formula is derived from the basic simple interest formulas and the definition of banker's gain, but we can apply it directly here since BD and TD are known. Step-by-Step Solution: Step 1: Note BD and TD. BD = Rs. 72, TD = Rs. 60. Step 2: Use the formula for the amount (sum due). A = (BD * TD) / (BD - TD). Step 3: Substitute the values. A = (72 * 60) / (72 - 60). A = 4320 / 12. A = Rs. 360. Verification / Alternative check: We know banker's gain (BG) = BD - TD = 72 - 60 = Rs. 12. Another identity is BG * A = BD * TD. So 12 * A = 72 * 60 = 4320, hence A = 4320 / 12 = 360, which matches the result above. This cross-check confirms the correctness of the calculation. Why Other Options Are Wrong: Rs. 290 and Rs. 420 do not satisfy the relationship A = (BD * TD) / (BD - TD); if substituted, they would not preserve the necessary ratio between BD and TD. Rs. 480 is larger than the product-based value and contradicts the equations derived from simple interest relationships between BD and TD. Common Pitfalls: Students often confuse true discount with banker's discount and attempt to treat both as the same quantity. Others try to use simple interest directly without recalling the specific derived formula linking BD, TD, and amount. Remember that when both BD and TD are given, using A = (BD * TD) / (BD - TD) is the fastest and safest approach. Final Answer: The sum due at maturity is Rs. 360.

Question 7

The present worth of a certain sum of money due sometime hence is Rs. 1600 and the true discount on it is Rs. 160. What is the banker's gain on this transaction (in rupees)?

Accepted Answer

Introduction / Context: This question involves the concepts of present worth, true discount and banker's gain in commercial arithmetic. When a bill is discounted at a bank, the bank uses banker's discount, while true discount is the mathematically correct deduction based on present worth. The difference between these two is called banker's gain. Given Data / Assumptions: Present worth (P) of the sum = Rs. 1600. True discount (TD) = Rs. 160. Amount due at maturity (A) = P + TD = 1600 + 160 = Rs. 1760. We are required to find the banker's gain (BG). Rate of interest and time are the same for both types of discount, but their exact values are not needed. Concept / Approach: True discount is the difference between the future amount and its present worth. Banker's discount is simple interest on the amount due (A). Banker's gain is defined as: BG = Banker's discount - True discount. When the present worth P and true discount TD are known, there is a direct formula: BG = TD^2 / P This comes from the relationship between simple interest on present worth and on the amount due. Step-by-Step Solution: Step 1: Note the given values. P = Rs. 1600, TD = Rs. 160. Step 2: Apply the banker's gain formula. BG = TD^2 / P. Step 3: Substitute the numbers. BG = 160 * 160 / 1600. BG = 25600 / 1600. BG = 16. Verification / Alternative check: Let the simple interest rate * time factor be k. Then TD = P * k = 1600k = 160, so k = 0.1. Amount A = 1760, so banker's discount BD = A * k = 1760 * 0.1 = 176. Banker's gain BG = BD - TD = 176 - 160 = 16, confirming the result from the shortcut formula. Why Other Options Are Wrong: 13, 14 and 15 are all less than the computed banker's gain. None of them satisfies BD - TD when calculated consistently using the same rate and time derived from the given true discount and present worth. Common Pitfalls: A common mistake is to treat the given true discount as if it were banker's discount or to attempt guesswork with the rate of interest. Another frequent error is forgetting that BG depends on TD and present worth, not directly on the amount alone. Using BG = TD^2 / P when P and TD are known ensures a quick and reliable solution. Final Answer: The banker's gain is Rs. 16.

Question 8

Two successive trade discounts of x% and y% are allowed on the marked price of an article. Together, these two consecutive discounts are equivalent to a single discount of:

Accepted Answer

Introduction / Context: In percentage and commercial arithmetic, discounts are often given successively, such as a first discount followed by another one on the reduced price. This question asks for a formula that combines two successive discounts of x% and y% into one equivalent discount on the original marked price. Given Data / Assumptions: First discount rate = x% on the marked price. Second discount rate = y% on the already reduced price. We seek the single equivalent discount rate D% that gives the same final selling price as applying x% and then y% consecutively. The base (marked price) can be assumed to be any convenient value, commonly 100 units, since only proportions matter. Concept / Approach: Successive discounts do not simply add up, because the second discount is applied on a reduced price, not on the original price. If M is the marked price, after the first discount of x%, the price becomes M(1 - x/100). Applying a further y% discount reduces it to M(1 - x/100)(1 - y/100). An equivalent single discount D% must satisfy: M(1 - D/100) = M(1 - x/100)(1 - y/100). Cancelling M and expanding gives a direct expression for D in terms of x and y. Step-by-Step Solution: Step 1: Write the final price after successive discounts. Final price = M(1 - x/100)(1 - y/100). Step 2: Expand the product. (1 - x/100)(1 - y/100) = 1 - x/100 - y/100 + xy/10000. = 1 - (x + y)/100 + xy/10000. Step 3: Represent the final price with a single discount D%. Final price = M(1 - D/100). So 1 - D/100 = 1 - (x + y)/100 + xy/10000. Step 4: Compare coefficients to find D. D/100 = (x + y)/100 - xy/10000. D = x + y - xy/100. Verification / Alternative check: Take a simple numerical example: x = 20%, y = 10%, M = 100. Successive discounts: after 20%, price = 80; after 10% on 80, price = 72. Single discount from 100 to 72 is 28%. Using the formula, D = 20 + 10 - (20 * 10)/100 = 30 - 2 = 28%, confirming the result. Why Other Options Are Wrong: (x - y + xy/100)% and (x - y - xy/100)% incorrectly involve subtraction of y, which does not match the expansion of the product (1 - x/100)(1 - y/100). (x + y + xy/100)% overestimates the discount because it adds the interaction term xy/100 instead of subtracting it, giving a discount larger than both individual discounts combined in a realistic scenario. Common Pitfalls: A typical error is to simply add x% and y% and ignore the interaction term xy/100, which arises because the second discount is applied on a reduced base. Another mistake is to mis-handle the algebra when expanding the product of two binomials. Remember: for successive discounts, the effective discount is x + y - (xy/100). Final Answer: The equivalent single discount is (x + y - xy/100)%.

Question 9

The banker's gain on a certain sum due 2 years hence at 10% per annum is Rs 24. What is the present worth of this sum (in rupees)?

Accepted Answer

Introduction / Context: This question belongs to the topic of banker's discount, true discount, banker's gain and present worth. The banker's gain is the difference between the banker's discount and the true discount. By using standard formulas for a given rate and time, we can relate banker's gain to the face value and present worth of the bill. Here we are told the banker's gain and asked to find the present worth at a simple interest rate of 10 percent per annum for 2 years. Given Data / Assumptions: - Time until the sum is due = 2 years - Rate of simple interest = 10 percent per annum - Banker's gain (BG) = Rs 24 - Let face value of the bill be F - Present worth is the true value of the sum today at the given rate and time Concept / Approach: For a bill due after time t years at rate r percent per annum, banker's discount BD is calculated on the face value F as BD = F × r × t / 100. True discount TD is the difference between the face value and present worth. A useful relation is that banker's gain BG equals BD minus TD and for given r and t can be expressed as a fixed fraction of the face value. Once we find F from BG, we can compute present worth using the true discount relations. Step-by-Step Solution: Step 1: For r = 10 percent and t = 2 years, we have r × t = 20.Step 2: True discount TD = F × 20 / (100 + 20) = F × 20 / 120 = F / 6.Step 3: Banker's discount BD = F × 20 / 100 = F / 5.Step 4: Banker's gain BG = BD − TD = F / 5 − F / 6 = F / 30.Step 5: Given BG = 24, so F / 30 = 24 which implies F = 24 × 30 = Rs 720.Step 6: True discount TD = F / 6 = 720 ÷ 6 = Rs 120.Step 7: Present worth = face value − true discount = 720 − 120 = Rs 600. Verification / Alternative check: We can verify the result by recomputing BD from F. With F = 720, rate 10 percent and time 2 years, BD = 720 × 10 × 2 / 100 = 720 × 20 / 100 = Rs 144. Since TD = Rs 120, the banker's gain is BD − TD = 144 − 120 = Rs 24, which matches the given gain. Therefore the computed present worth of Rs 600 is consistent with all definitions and data. Why Other Options Are Wrong: A present worth of Rs 400 or Rs 500 would lead to a face value and discounts that do not produce a banker's gain of Rs 24 at 10 percent for 2 years. A value like Rs 800 is larger than the face value we found and is impossible for present worth. The face value itself is Rs 720, so present worth must be smaller than this. Only Rs 600 satisfies both the algebraic relations and the given banker's gain. Common Pitfalls: Students sometimes confuse banker's gain with banker's discount or take BG as a percentage instead of an amount. Others use BD directly to find present worth, ignoring the role of true discount. It is important to remember that BG = BD − TD and that true discount is linked to present worth, not directly to the face value in the same way as banker's discount. Final Answer: The present worth of the sum is 600 rupees.

Question 10

The banker's discount on a certain sum due 2 years hence is eleven-tenths of the true discount. If the rate of simple interest is the same for both discounts, what is the rate percent per annum?

Accepted Answer

Introduction / Context: This problem deals with the relationship between banker's discount and true discount at the same rate and time. We are told that the banker's discount is eleven-tenths of the true discount for a bill due 2 years hence, and we are asked to determine the rate of interest. This is a formula based question but requires understanding of how the ratio BD over TD depends on rate and time. Given Data / Assumptions: - Time until the bill is due t = 2 years - Banker's discount BD is equal to 11 divided by 10 times the true discount TD, that is BD = (11 / 10) × TD - Rate of simple interest per annum is r percent, same for both BD and TD - We assume simple interest throughout. Concept / Approach: For a bill of face value F due after t years at rate r percent, true discount TD is given by TD = F × r × t / (100 + r × t), while banker's discount BD is BD = F × r × t / 100. The ratio BD / TD simplifies nicely to (100 + r × t) / 100. Here we know BD / TD as 11 / 10 and time t as 2 years, so we can set up an equation in terms of r × t and solve for r. Step-by-Step Solution: Step 1: Use the formula BD / TD = (100 + r × t) / 100.Step 2: Given BD = (11 / 10) TD, so BD / TD = 11 / 10.Step 3: Therefore (100 + r × t) / 100 = 11 / 10.Step 4: Cross multiply: 10 × (100 + r × t) = 11 × 100.Step 5: This gives 1000 + 10 r t = 1100, so 10 r t = 100 and r t = 10.Step 6: Time t = 2 years, so r × 2 = 10 and r = 10 ÷ 2 = 5 percent per annum. Verification / Alternative check: To confirm, take an assumed face value, for example F = 1000. With r = 5 percent and t = 2 years, we have r t = 10. Then BD = 1000 × 10 / 100 = 100. True discount TD = 1000 × 10 / (100 + 10) = 1000 × 10 / 110 = 10000 / 110 ≈ 90.91. The ratio BD / TD equals 100 / 90.91 which is approximately 1.1, that is 11 / 10. This matches the condition in the question, confirming that 5 percent is correct. Why Other Options Are Wrong: If r were 10 percent, then r t would be 20 and BD / TD would equal (100 + 20) / 100 = 1.2 which is 6 / 5, not 11 / 10. If r were 7 or 8 percent, the product r t would differ and the ratio would no longer be 11 / 10. Only r = 5 percent gives r t = 10 and exactly satisfies the required ratio. Common Pitfalls: Many learners try to apply direct percentage difference between BD and TD without recalling the correct formula for their ratio. Another mistake is to confuse the time as 10 years instead of interpreting r t = 10 correctly. Always express BD / TD in terms of r and t first, then substitute the given ratio and time to solve cleanly for the rate. Final Answer: The required rate of simple interest is 5% per annum.

Question 11

The banker's discount on a bill of Rs 1650 due after some time is Rs 165. What is the bank's gain (in rupees) on this transaction?

Accepted Answer

Introduction / Context: This problem combines the ideas of banker's discount, true discount and banker's gain. We are given the banker's discount on a bill and the face value of that bill. Using the relationship between banker's discount and true discount at the same rate and time, we can compute true discount and then obtain the banker's gain as the difference between them. The focus here is on using ratios cleanly without needing the explicit rate or time. Given Data / Assumptions: - Face value of the bill F = Rs 1650 - Banker's discount BD = Rs 165 - Rate of interest and time are the same for both BD and true discount TD, but not given explicitly - Banker's gain BG is defined as BD minus TD Concept / Approach: For a bill with face value F, the banker's discount at rate r percent for time t years is BD = F × r × t / 100. The true discount is TD = F × r × t / (100 + r × t). The ratio BD / TD simplifies to (100 + r t) / 100, which also equals F / present worth. However, in this question it is simpler to first compute the product r t using BD and F, then use the TD formula. Once TD is known, banker's gain BG is simply BD minus TD. Step-by-Step Solution: Step 1: Use BD = F × r × t / 100 to find r t: 165 = 1650 × r t / 100.Step 2: Simplify: r t = (165 × 100) ÷ 1650 = 1000 ÷ 10 = 10.Step 3: True discount TD = F × r t / (100 + r t) = 1650 × 10 / (100 + 10) = 1650 × 10 / 110.Step 4: Compute TD = 16500 ÷ 110 = Rs 150.Step 5: Banker's gain BG = BD − TD = 165 − 150 = Rs 15. Verification / Alternative check: We can check using the relation BD / TD = (100 + r t) / 100. Here BD / TD = 165 / 150 = 11 / 10. Therefore (100 + r t) / 100 = 11 / 10, giving 100 + r t = 110 and r t = 10, which matches the earlier computation. Substituting r t = 10 into the true discount formula again yields TD = 150 and BG = 15, so the value is consistent from both viewpoints. Why Other Options Are Wrong: Values like 20 or 18 would correspond to different relationships between BD and TD and would not maintain the correct equality with the given face value and rate time product. Thirteen is too small relative to the difference between a discount of 165 and a true discount close to 150. Only 15 is compatible with the formulas for banker's discount and true discount at some positive rate and time. Common Pitfalls: One common error is to treat 165 as the true discount directly and try to find BG without using the proper formulas. Another mistake is mixing up face value and present worth, leading to incorrect calculations of the product r t. Always start by finding r t from BD and F and then compute TD and BG systematically. Final Answer: The banker's gain on the bill is 15 rupees.

Question 12

The true discount on a bill of Rs 540 is Rs 90. What is the banker's discount (in rupees) on this bill at the same rate and for the same time?

Accepted Answer

Introduction / Context: This question asks us to find the banker's discount given the true discount and the face value of a bill. The rate of simple interest and the time until maturity are the same for both discounts, but they are not provided explicitly. The key insight is that the ratio of banker's discount to true discount depends only on the relationship between face value and present worth, and not directly on the numerical values of rate and time. Given Data / Assumptions: - Face value of the bill F = Rs 540 - True discount TD = Rs 90 - Let present worth be PW - Banker's discount BD is calculated at the same rate and time as TD - Banker's gain is BD minus TD but is not directly required here Concept / Approach: By definition, true discount TD is the difference between the face value and the present worth, so TD = F − PW. Therefore PW = F − TD. For simple interest, the ratio BD / TD equals F / PW because BD is interest on the face value while TD is interest on the present worth for the same rate and time. Once we compute PW from F and TD, we can obtain this ratio and thus find BD directly from TD without needing rate or time. Step-by-Step Solution: Step 1: Compute present worth PW: PW = F − TD = 540 − 90 = Rs 450.Step 2: For the same rate and time, BD and TD are proportional to the principals on which they are calculated.Step 3: Therefore BD / TD = F / PW.Step 4: Substitute values: BD / 90 = 540 / 450 = 6 / 5.Step 5: So BD = 90 × (6 / 5) = 90 × 1.2 = Rs 108. Verification / Alternative check: We can interpret the factor 6 / 5 as showing that banker's discount is 20 percent greater than true discount in this case. Since TD is 90, twenty percent of 90 is 18, so BD should be 90 + 18 = 108. This matches the computed result precisely. We could also set up equations using an unknown rate and time, but the proportional method is faster and less error prone. Why Other Options Are Wrong: The value 100 is slightly higher than the true discount but does not correspond to the exact ratio F / PW. A value like 120 would make BD one third larger than TD, which contradicts F / PW = 6 / 5. The option 115 is arbitrary and not linked to any simple ratio. The value 90 is simply the true discount itself, not the banker's discount. Only 108 fits the exact relationship between the face value, present worth and the given true discount. Common Pitfalls: Some learners mistakenly assume BD equals TD or try to compute both using a guessed rate of interest, which is unnecessary and may lead to inconsistency. Others forget that true discount is based on present worth while banker's discount is based on face value. Remembering the ratio BD / TD = F / PW gives a clean and reliable route to the answer. Final Answer: The banker's discount on the bill is 108 rupees.

Question 13

If the true discount on a certain sum due 6 months hence at 15% simple interest is Rs 120, what is the banker's discount (in rupees) on the same sum for the same time and at the same rate?

Accepted Answer

Introduction / Context: Here we are given the true discount on a sum due after a known time at a known rate of simple interest. Our task is to compute the banker's discount for the same sum, time and rate. The question combines knowledge of true discount formulas with the formula for banker's discount and requires careful algebraic manipulation to find the face value first. Given Data / Assumptions: - True discount TD = Rs 120 - Time t = 6 months = 0.5 year - Rate of simple interest r = 15 percent per annum - Let face value of the bill be F - Banker's discount BD is to be found for the same rate and time Concept / Approach: True discount at simple interest is given by TD = F × r × t / (100 + r × t). Since we know TD, r and t, we can solve this formula for the face value F. Once F is known, banker's discount is calculated as BD = F × r × t / 100, because it is simple interest on the face value for the same period. The difference BD − TD would give banker's gain, but the question only asks for BD. Step-by-Step Solution: Step 1: Compute r × t = 15 × 0.5 = 7.5.Step 2: Use the true discount formula TD = F × r × t / (100 + r × t).Step 3: Substitute known values: 120 = F × 7.5 / (100 + 7.5) = F × 7.5 / 107.5.Step 4: Rearrange to find F: F = 120 × 107.5 / 7.5 = 120 × (1075 / 75) = 1720.Step 5: Now compute banker's discount BD = F × r × t / 100 = 1720 × 15 × 0.5 / 100.Step 6: Simplify BD = 1720 × 7.5 / 100 = 12900 / 100 = Rs 129. Verification / Alternative check: We can cross check using the ratio BD / TD. Here BD = 129 and TD = 120, so BD / TD = 129 / 120 = 1.075. For r t = 7.5, the ratio BD / TD should be (100 + 7.5) / 100 = 107.5 / 100 = 1.075, which matches exactly. This confirms that the computed face value and banker's discount are consistent with the standard formulas. Why Other Options Are Wrong: An answer like 100 or 50 ignores the fact that banker's discount must be slightly larger than the true discount because it is calculated on the face value. A value such as 160 is too large and would violate the relation between TD and BD at the given rate and time. Only Rs 129 satisfies both the algebraic relationship and the inequality BD greater than TD. Common Pitfalls: Some learners mistakenly apply the simple interest formula directly to TD instead of to the face value, or they treat 6 months as 6 years when multiplying r by t. Another common error is to forget to add r t in the denominator when using the true discount formula, leading to an incorrect face value. Carefully writing down the formulas and substituting the values step by step helps avoid these mistakes. Final Answer: The banker's discount for the given sum is 129 rupees.

Question 14

The banker's discount on Rs 1800 at 12% per annum is equal to the true discount on Rs 1872 for the same time and at the same rate. What is the time period (in months)?

Accepted Answer

Introduction / Context: This question links the banker's discount on one sum to the true discount on another sum when both are calculated at the same rate of simple interest and for the same time period. By equating these two quantities and using the standard formulas, we can solve for the time. It is a good test of understanding how BD and TD depend differently on the face value, rate and time. Given Data / Assumptions: - Face value for banker's discount F1 = Rs 1800 - Face value for true discount F2 = Rs 1872 - Rate of simple interest r = 12 percent per annum - Time period t is the same for both and is to be found - Banker's discount on Rs 1800 equals the true discount on Rs 1872 Concept / Approach: Banker's discount is calculated on the face value as BD = F1 × r × t / 100. True discount on F2 is given by TD = F2 × r × t / (100 + r × t). The condition BD = TD gives an equation in terms of r, t and the two face values. We can cancel the common factor r × t and solve for r t, then convert the resulting time in years to months. Step-by-Step Solution: Step 1: Write BD on Rs 1800: BD = 1800 × r × t / 100.Step 2: Write TD on Rs 1872: TD = 1872 × r × t / (100 + r × t).Step 3: Since BD = TD, we have 1800 × r × t / 100 = 1872 × r × t / (100 + r × t).Step 4: Cancel r × t from both sides (it is non zero), giving 1800 / 100 = 1872 / (100 + r × t).Step 5: Simplify 1800 / 100 to 18, so 18 = 1872 / (100 + r × t).Step 6: Cross multiply: 18 × (100 + r × t) = 1872 which gives 1800 + 18 r t = 1872.Step 7: Therefore 18 r t = 72 and r t = 4.Step 8: With r = 12 percent, 12 × t = 4, so t = 4 ÷ 12 = 1 / 3 year.Step 9: Convert to months: time = (1 / 3) × 12 = 4 months. Verification / Alternative check: To verify, substitute t = 4 months = 1 / 3 year back into the formulas. For Rs 1800, BD = 1800 × 12 × (1 / 3) / 100 = 1800 × 4 / 100 = Rs 72. For Rs 1872, r t = 4, so TD = 1872 × 4 / (100 + 4) = 7488 / 104 = Rs 72. Both discounts are equal at Rs 72, which confirms that a time of 4 months is correct. Why Other Options Are Wrong: If t were 3 months, time in years would be 1 / 4 and r t would be 3, not 4, which would break the equality between BD and TD. Values such as 5, 6 or 7 months would also produce different r t products and no longer satisfy the derived equation. Only 4 months leads to BD and TD being numerically equal for the given sums. Common Pitfalls: Students sometimes forget to cancel r × t from both sides and instead try to solve for r and t separately, which is unnecessary. Another mistake is to misconvert years to months or to treat 4 as the time directly in years instead of as the product r t. Carefully distinguishing between the product r t and the actual time in years is crucial. Final Answer: The required time period is 4 months.

Question 15

The banker's gain on a bill due 1 year hence at 12% per annum is Rs 6. What is the true discount on the bill (in rupees)?

Accepted Answer

Introduction / Context: This question focuses on banker's gain and true discount for a bill under simple interest conditions. Banker's gain is the difference between banker's discount and true discount. Knowing the rate and time, we can express both BD and TD in terms of the face value of the bill and then relate banker's gain to that face value. With banker's gain given, we can work backwards to find the true discount. Given Data / Assumptions: - Time until the bill is due t = 1 year - Rate of simple interest r = 12 percent per annum - Banker's gain BG = Rs 6 - Let face value of the bill be F - We need to find true discount TD on this bill Concept / Approach: For a bill of face value F at rate r percent for time t years, banker's discount is BD = F × r × t / 100. True discount is TD = F × r × t / (100 + r × t). Banker's gain is BG = BD − TD. For known r t, BG becomes a constant fraction of F. We can find F from BG, then compute TD from its formula. Finally, we report TD as the answer. Step-by-Step Solution: Step 1: Compute r × t = 12 × 1 = 12.Step 2: Banker's discount BD = F × 12 / 100 = 3F / 25.Step 3: True discount TD = F × 12 / (100 + 12) = F × 12 / 112 = 3F / 28.Step 4: Banker's gain BG = BD − TD = 3F / 25 − 3F / 28.Step 5: Simplify BG = 3F × (1 / 25 − 1 / 28) = 3F × (28 − 25) / (25 × 28) = 3F × 3 / 700 = 9F / 700.Step 6: Given BG = 6, so 9F / 700 = 6 which implies F = 6 × 700 / 9 = 1400 / 3.Step 7: Now compute TD = 3F / 28 = 3 × (1400 / 3) / 28 = 1400 / 28 = Rs 50. Verification / Alternative check: We can check the result by computing BD as well. With F = 1400 / 3, BD = 3F / 25 = 3 × (1400 / 3) / 25 = 1400 / 25 = Rs 56. True discount TD is 50, so banker's gain BG = BD − TD = 56 − 50 = Rs 6, which matches the given banker's gain. This confirms that the computed true discount is correct even though the face value is a fraction. Why Other Options Are Wrong: Values such as 47, 58 or 62 would lead to a banker's gain different from Rs 6 when combined with the required BD at 12 percent for 1 year. For example, if TD were 62, the relationship between BD, TD and BG would not satisfy the equations derived from the rate and time. Only 50 fits consistently with the given banker's gain and interest conditions. Common Pitfalls: Students often assume that the face value must be a round integer and then force BD and TD into those values, which can cause errors. Others confuse banker's gain with true discount or neglect to compute both BD and TD before subtracting. It is essential to set up the algebra carefully, solve for the face value first and then compute the true discount from its standard formula. Final Answer: The true discount on the bill is 50 rupees.

Question 16

The banker's gain on a sum due 3 years hence at 12% per annum is Rs 270. What is the banker's discount (in rupees) on this sum?

Accepted Answer

Introduction / Context: This problem asks us to determine the banker's discount when the banker's gain is given for a loan or bill due after 3 years at a rate of 12 percent per annum. We use the relationships among banker's discount, true discount and banker's gain for a fixed rate and time. Once we express banker's gain as a fraction of the face value, we can find the face value and then compute the banker's discount. Given Data / Assumptions: - Time until the sum is due t = 3 years - Rate of simple interest r = 12 percent per annum - Banker's gain BG = Rs 270 - Let face value of the bill be F - Banker's discount BD and true discount TD are both based on simple interest at this rate and time Concept / Approach: For a bill with face value F at rate r and time t, banker's discount is BD = F × r × t / 100 and true discount is TD = F × r × t / (100 + r × t). Banker's gain BG equals BD − TD. For fixed r and t, BG simplifies to a fixed fraction of F. With r = 12 and t = 3, we can compute r t and express BD and TD accordingly, then solve for F from BG, and finally find BD from its formula. Step-by-Step Solution: Step 1: Compute r × t = 12 × 3 = 36.Step 2: Banker's discount BD = F × 36 / 100 = 9F / 25.Step 3: True discount TD = F × 36 / (100 + 36) = F × 36 / 136 = 9F / 34.Step 4: Banker's gain BG = BD − TD = 9F / 25 − 9F / 34.Step 5: Simplify BG = 9F × (1 / 25 − 1 / 34) = 9F × (34 − 25) / (25 × 34) = 9F × 9 / 850 = 81F / 850.Step 6: Given BG = 270, so 81F / 850 = 270 and therefore F = 270 × 850 / 81 = 8500 / 3.Step 7: Banker's discount BD = 9F / 25 = 9 × (8500 / 3) / 25 = 8500 / 25 = Rs 1020. Verification / Alternative check: We can verify by computing TD from F. True discount TD = 9F / 34 = 9 × (8500 / 3) / 34 = 8500 / 34 = Rs 250. Then BD − TD = 1020 − 250 = Rs 770 which is not equal to 270, so we must be careful. Let us recalculate TD carefully: 36 / 136 simplifies to 9 / 34, so TD = F × 9 / 34. With F = 8500 / 3, TD = (8500 / 3) × 9 / 34 = 8500 × 3 / 34 = 25500 / 34 ≈ 750. Therefore BD = 1020, TD ≈ 750, and BG = BD − TD = 1020 − 750 = Rs 270, which matches the given banker's gain. This confirms BD = 1020 as correct. Why Other Options Are Wrong: Values like 980 or 1150 correspond to different combinations of BD and TD that would not yield a banker's gain of 270 at 12 percent over 3 years. The option 1315 is much too large and would make BG far greater than the stated 270. Only 1020 is consistent with both the formulas and the given banker's gain. Common Pitfalls: One common error is simplifying fractions incorrectly when computing BD and TD, which leads to a wrong expression for BG. Another mistake is forgetting that BG is BD minus TD and not the other way round. Working systematically with r t, expressing all quantities in terms of F, and checking calculations avoids such slips. Final Answer: The banker's discount on the sum is 1020 rupees.

Question 17

The banker's gain on a certain sum due 4 years hence at 5% per annum is Rs 200. What is the present worth of this sum (in rupees)?

Accepted Answer

Introduction / Context: This problem focuses on computing the present worth of a sum when banker's gain is known for a fixed rate and time. Banker's gain is the excess of banker's discount over true discount. For fixed rate and time, banker's gain is a fixed fraction of the face value of the bill. Using this relationship we can determine face value first and then true discount and present worth. Given Data / Assumptions: - Time until the sum is due t = 4 years - Rate of simple interest r = 5 percent per annum - Banker's gain BG = Rs 200 - Let face value of the bill be F - Present worth PW must be determined Concept / Approach: For a bill of face value F at rate r for time t, banker's discount is BD = F × r × t / 100 and true discount is TD = F × r × t / (100 + r × t). Banker's gain is BG = BD − TD. This expression can be rewritten as a constant fraction of F that depends only on r t. Once we find F using BG, the true discount and present worth follow from standard true discount relations: PW = F − TD, and TD = F × r t / (100 + r t). Step-by-Step Solution: Step 1: Compute r × t = 5 × 4 = 20.Step 2: Banker's discount BD = F × 20 / 100 = F / 5.Step 3: True discount TD = F × 20 / (100 + 20) = F × 20 / 120 = F / 6.Step 4: Banker's gain BG = BD − TD = F / 5 − F / 6 = F / 30.Step 5: Given BG = 200, so F / 30 = 200 and therefore F = 200 × 30 = Rs 6000.Step 6: True discount TD = F / 6 = 6000 ÷ 6 = Rs 1000.Step 7: Present worth PW = face value − true discount = 6000 − 1000 = Rs 5000. Verification / Alternative check: We can verify by recomputing BD and BG from the found values. Banker's discount BD = F / 5 = 6000 ÷ 5 = Rs 1200. True discount TD is Rs 1000. Therefore banker's gain BG = BD − TD = 1200 − 1000 = Rs 200, which matches the given value. Since both BD and TD satisfy the interest relationships at 5 percent for 4 years, the present worth of Rs 5000 is fully consistent. Why Other Options Are Wrong: A present worth of Rs 4500 or Rs 4000 would give a much smaller true discount and would no longer produce a banker's gain of Rs 200 at the given rate and time. A value of Rs 6000 is the face value itself and would wrongly imply zero true discount. Rs 4800 is also inconsistent when checked through the formulas. Only Rs 5000 leads to internal consistency between BD, TD and BG. Common Pitfalls: Students sometimes confuse present worth with face value or try to compute BD directly from BG without first deriving the correct relationship. Another mistake is to compute true discount on the wrong base. Always express BD, TD and BG in terms of F using r t first, then use the given banker's gain to find F and from that determine present worth accurately. Final Answer: The present worth of the sum is Rs. 5000.

Question 18

The banker's discount and the true discount of a sum at 10% per annum simple interest for the same time are Rs 100 and Rs 80 respectively. What are the sum and the time period?

Accepted Answer

Introduction / Context: This question combines all three main quantities in banker's discount problems: banker's discount, true discount and the underlying sum with its time period. We know BD and TD at a given rate and for the same time. From their ratio we can find the product of rate and time, and from BD we can then obtain the face value of the bill. Finally, we deduce the time in years from the product r t. Given Data / Assumptions: - Banker's discount BD = Rs 100 - True discount TD = Rs 80 - Rate of simple interest r = 10 percent per annum - Time t is the same for BD and TD and must be found - Face value of the bill is the sum to be found Concept / Approach: For a bill of face value F, banker's discount is BD = F × r × t / 100 and true discount is TD = F × r × t / (100 + r × t). Their ratio BD / TD simplifies to (100 + r t) / 100. Hence BD / TD allows us to find r t without knowing F. Once r t is known, we can use BD to find F. Then, r and t are related by r t, so we can solve for t using the given rate of 10 percent. Step-by-Step Solution: Step 1: Compute BD / TD = 100 ÷ 80 = 5 / 4.Step 2: Use formula BD / TD = (100 + r t) / 100.Step 3: Therefore (100 + r t) / 100 = 5 / 4, so 100 + r t = 125.Step 4: Thus r t = 125 − 100 = 25.Step 5: Since r = 10 percent, 10 × t = 25 and t = 25 ÷ 10 = 2.5 years.Step 6: Now use BD = F × r × t / 100 with BD = 100, r = 10, t = 2.5.Step 7: So 100 = F × 10 × 2.5 / 100 = F × 25 / 100, which gives F = 100 × 100 / 25 = Rs 400. Verification / Alternative check: We can confirm by recomputing TD from F and t. With F = 400, r = 10 and t = 2.5, we have r t = 25. True discount TD = F × r t / (100 + r t) = 400 × 25 / 125 = 400 × 1 / 5 = Rs 80, which matches the given TD. Banker's discount BD = F × r t / 100 = 400 × 25 / 100 = Rs 100, which matches the given BD. This consistency across both values confirms that F = 400 and t = 2.5 years is correct. Why Other Options Are Wrong: If the sum were Rs 200 or Rs 600 with any of the listed times, substituting into the formulas for BD and TD would not reproduce BD = 100 and TD = 80 at 10 percent. For example, with sum Rs 200 and t = 5 years, r t would be 50, not 25, and both discounts would be different. Only the pair sum Rs 400 and time 2.5 years fits all the given conditions exactly. Common Pitfalls: Some learners attempt to guess the time directly or mistakenly use BD − TD instead of BD / TD to find r t. Others forget to use the combined factor r t and try to solve for rate and time independently even though rate is already given. Treating r t as a single variable first simplifies the algebra and reduces errors. Final Answer: The sum is Rs. 400 and the time period is 2.5 years.

Question 19

A bill for Rs 6000 is drawn on 14 July at 5 months' term. It is discounted on 5 October at 10% per annum simple interest. Find the net amount that the holder of the bill receives (in rupees).

Accepted Answer

Introduction / Context: This question tests understanding of banker's discount in practical date based situations. A bill is drawn with a certain term in months, and then discounted at the bank before its due date. We must carefully determine the remaining time between the date of discount and the legally due date of the bill, including days of grace, and then apply simple interest on the face value to find the banker's discount. Subtracting this discount from the face value gives the net amount received by the holder, also called the proceeds. Given Data / Assumptions: - Face value of the bill F = Rs 6000 - Date of drawing the bill = 14 July - Term of the bill = 5 months - Date of discount = 5 October - Rate of simple interest r = 10 percent per annum - We assume standard practice of adding 3 days of grace to the due date when computing time for discount. Concept / Approach: The nominal due date of the bill is obtained by adding the term of 5 months to the date of drawing. Legal due date is three days after the nominal due date. The time for which the bank charges interest is the interval from the date of discount to the legal due date. The banker's discount BD is then BD = F × r × t / 100, where t is measured in years. The proceeds or net amount received by the holder is face value minus banker's discount. Step-by-Step Solution: Step 1: Add 5 months to 14 July to get the nominal due date: 14 December.Step 2: Add 3 days of grace to find the legal due date: 17 December.Step 3: Count the number of days from 5 October to 17 December: from 5 October to 31 October is 26 days, plus 30 days in November, plus 17 days in December, for a total of 26 + 30 + 17 = 73 days.Step 4: Convert 73 days to years using 365 days in a year: t = 73 / 365 = 0.2 year.Step 5: Compute banker's discount BD = F × r × t / 100 = 6000 × 10 × 0.2 / 100 = 6000 × 2 / 100 = Rs 120.Step 6: Proceeds or net amount received by the holder = F − BD = 6000 − 120 = Rs 5880. Verification / Alternative check: We can confirm that the time fraction is exactly 0.2 year because 73 ÷ 365 equals 0.2. At 10 percent per annum, interest for one year on Rs 6000 would be Rs 600. For one fifth of a year, the interest is one fifth of Rs 600, which is Rs 120. This matches the computed banker's discount. Subtracting 120 from the face value 6000 yields Rs 5880 consistently as the proceeds. Why Other Options Are Wrong: An amount like 6580 would exceed the face value, which is impossible as the bank always deducts a discount, not adds to the amount. Values like 4390 or 5350 imply a discount much larger than Rs 120 and are not consistent with a modest 10 percent annual rate for only 73 days. The option 5700 also does not match the exact simple interest calculation. Only Rs 5880 fits the correct banker's discount computation. Common Pitfalls: Many students forget to include 3 days of grace in the time computation or mistakenly count the term from the date of discount instead of from the date of drawing. Another error is to treat 73 days as 73 months or to divide by 360 instead of 365, leading to slightly different results. Being systematic about dates and always converting the exact number of days to a year fraction avoids these mistakes. Final Answer: The holder of the bill receives a net amount of 5880 rupees.

Question 20

The banker's discount and the true discount on a sum of money due 8 months hence are Rs 120 and Rs 110 respectively. What is the sum (face value) of the bill (in rupees)?

Accepted Answer

Introduction / Context: This question provides both the banker's discount and the true discount on a sum due after 8 months and asks us to find the underlying face value. Using the standard formulas and the given pair of discounts, we can eliminate the interest rate and time by working with ratios and then solve for the face value without being told the rate explicitly. Given Data / Assumptions: - Banker's discount BD = Rs 120 - True discount TD = Rs 110 - Time until the sum is due = 8 months = 2 / 3 year - Rate of simple interest r is unknown but same for both BD and TD - Let face value of the bill be F Concept / Approach: Banker's discount is given by BD = F × r × t / 100, while true discount is TD = F × r × t / (100 + r × t). The ratio BD / TD equals (100 + r t) / 100. We can also use the true discount equation to link F, r t and TD. By expressing everything in terms of the single product r t, we can first find r t from one equation and then solve for F using either the BD or TD expression. Step-by-Step Solution: Step 1: Let x = r × t. Then BD = F × x / 100 = 120 and TD = F × x / (100 + x) = 110.Step 2: From BD we get F = 120 × 100 / x = 12000 / x.Step 3: Substitute this into the TD formula: 110 = (12000 / x) × x / (100 + x) = 12000 / (100 + x).Step 4: Rearranging, 12000 = 110 × (100 + x) = 11000 + 110x.Step 5: So 12000 − 11000 = 110x and 1000 = 110x, giving x = 1000 ÷ 110 = 100 ÷ 11.Step 6: Substitute x back into F = 12000 / x: F = 12000 × 11 / 100 = 1320. Verification / Alternative check: We can verify the answer by recomputing BD and TD using F = 1320. Since x = r t = 100 / 11, BD = F × x / 100 = 1320 × (100 / 11) / 100 = 1320 ÷ 11 = 120, which matches the given BD. True discount TD = F × x / (100 + x) = 1320 × (100 / 11) / (100 + 100 / 11), which simplifies to 1320 × (100 / 11) × 11 / 120 = 1320 × 100 / 120 = 132000 / 120 = 110, matching the given TD. This confirms that F = 1320 is correct. Why Other Options Are Wrong: Values like 1140, 1260 or 1457 would lead to BD and TD that do not equal 120 and 110 when calculated using any positive simple interest rate over 8 months. The option 1540 is also incompatible with the exact ratio between BD and TD implied by the problem. Only 1320 satisfies both discount equations simultaneously. Common Pitfalls: Students may try to assume a rate r and then work numerically, which is unnecessary and can introduce error. Others confuse BD − TD with the sum or use simple interest formulas incorrectly. Working with the product r t and keeping equations symbolic until the final substitution is a safer and more general method. Final Answer: The face value of the bill is 1320 rupees.

Banker's Discount Questions