Question 1

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

Accepted Answer

Introduction / Context: This question gives the discount on a sum at a known rate of simple interest and asks for the time until the sum is due. The term discount here is interpreted as true discount, which is the difference between the face value and its present worth. Using the true discount formula, we can connect the discount amount, the face value, the rate and the time, and solve for the time period. Given Data / Assumptions: - Face value F = Rs 498 - True discount TD = Rs 18 - Rate of simple interest r = 5 percent per annum - Time t in years is unknown and must be found - We assume simple interest and standard definitions of true discount. Concept / Approach: The true discount at simple interest is given by TD = F × r × t / (100 + r × t). Here F, r and TD are known, and t is the unknown variable. We can let x = r × t to simplify the algebra, convert the equation to one in x and solve. Once we know x, we can obtain t by dividing x by r, and finally express the time in months by multiplying the time in years by 12. Step-by-Step Solution: Step 1: Let x = r × t so that TD = F × x / (100 + x).Step 2: Substitute known values: 18 = 498 × x / (100 + x).Step 3: Cross multiply to get 18 × (100 + x) = 498x.Step 4: Expand the left side: 1800 + 18x = 498x.Step 5: Rearrange to group x terms: 1800 = 498x − 18x = 480x.Step 6: Solve for x: x = 1800 ÷ 480 = 15 ÷ 4 = 3.75.Step 7: Since x = r × t and r = 5, we have 5t = 3.75 and t = 3.75 ÷ 5 = 0.75 year.Step 8: Convert 0.75 year to months: 0.75 × 12 = 9 months. Verification / Alternative check: We can verify by recomputing TD using t = 0.75 year. Then r × t = 5 × 0.75 = 3.75. True discount TD should be F × r t / (100 + r t) = 498 × 3.75 / 103.75. This simplifies numerically to approximately 1867.5 / 103.75 = Rs 18, matching the given discount. This confirms that the time until the sum is due is 0.75 year, or 9 months. Why Other Options Are Wrong: If the time were 8 months, t would be 2 / 3 year, giving r t = 10 / 3 and a different true discount. Similarly, 10 or 11 months would yield other values for r t and hence different discounts than Rs 18. A full year or 12 months would be longer than needed and would produce a larger discount. Only 9 months matches the given discount exactly at 5 percent simple interest. Common Pitfalls: Sometimes learners confuse true discount with simple interest or mistakenly use SI = P r t / 100 directly with TD as if it were the interest on the face value. Another error is to forget to include 100 + r t in the denominator of the true discount formula. Expressing the problem in terms of x = r t and solving step by step helps prevent algebraic mistakes. Final Answer: The sum is due after 9 months.

Question 2

The present worth of a bill due sometime hence is Rs 1100 and the true discount on the bill is Rs 110. What is the banker's discount on the bill (in rupees)?

Accepted Answer

Introduction / Context: This question provides the present worth and the true discount of a bill and asks for the banker's discount. True discount is the difference between the face value and the present worth. Banker's discount is calculated on the face value at the same rate and time. Using the relationship between these quantities, we can determine the face value and then compute the banker's discount without needing the rate or time explicitly. Given Data / Assumptions: - Present worth PW = Rs 1100 - True discount TD = Rs 110 - Face value F of the bill is PW + TD - Banker's discount BD is to be found at the same rate and time as TD - The interest is simple throughout. Concept / Approach: By definition TD = F − PW, so F = PW + TD. For the same rate and time, the discount is proportional to the principal on which it is calculated. True discount is interest on present worth, while banker's discount is interest on face value. Therefore BD / TD = F / PW. Having PW and TD, we first compute F, then use this ratio to find BD directly from TD. Step-by-Step Solution: Step 1: Compute the face value of the bill: F = PW + TD = 1100 + 110 = Rs 1210.Step 2: For the same rate and time, BD and TD are proportional to the amounts on which they are calculated.Step 3: Hence BD / TD = F / PW.Step 4: Substitute values: BD / 110 = 1210 / 1100.Step 5: Simplify the fraction 1210 / 1100 to 11 / 10.Step 6: Therefore BD = 110 × 11 / 10 = 11 × 11 = Rs 121. Verification / Alternative check: We can reinterpret the ratio F / PW = 11 / 10 as indicating that the face value is 10 percent more than the present worth. This means the true discount corresponds to 10 percent of PW. If TD is 110, 10 percent of PW is 110, so PW is 1100, which is consistent. Banker's discount must then correspond to the same percentage applied to the face value, giving 10 percent of 1210 which is 121. This confirms that BD = 121 is logically correct. Why Other Options Are Wrong: An amount like 112 or 130 does not match the exact 11 to 10 ratio between face value and present worth. The value 145 or 136 would suggest a larger percentage discount on the face value than is implied by the relation between PW and TD. Only Rs 121 preserves the correct proportional relationship between true discount and banker's discount at the same rate and time. Common Pitfalls: Students sometimes calculate BD as TD plus some guessed banker's gain instead of using the proportional reasoning. Others forget that TD is based on present worth, not on face value, which leads to incorrect expressions. Keeping the simple identity BD / TD = F / PW in mind provides a clean and reliable way to answer such questions. Final Answer: The banker's discount on the bill is Rs. 121.

Question 3

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

Accepted Answer

Introduction / Context: This problem asks us to find the banker's gain when we know the present worth and the true discount on a sum. Banker's gain is the difference between banker's discount and true discount. Given present worth and true discount, we can first determine the face value and the banker's discount and then compute the gain as their difference. The question tests familiarity with the relationships between these three quantities. Given Data / Assumptions: - Present worth PW = Rs 1600 - True discount TD = Rs 160 - Face value F of the bill is PW + TD - Banker's discount BD is calculated on face value at the same rate and time as TD - Banker's gain BG is defined as BD − TD Concept / Approach: First we use F = PW + TD to get the face value. Since true discount is interest on present worth and banker's discount is interest on face value for the same rate and time, the ratio BD / TD equals F / PW. Once BD is known from this ratio, banker's gain BG is simply BD − TD. No explicit knowledge of the rate of interest or the time is required because we always work with proportional relationships. Step-by-Step Solution: Step 1: Compute the face value: F = PW + TD = 1600 + 160 = Rs 1760.Step 2: For the same rate and time, BD / TD = F / PW.Step 3: Substitute: BD / 160 = 1760 / 1600.Step 4: Simplify 1760 / 1600 to 11 / 10.Step 5: Therefore BD = 160 × 11 / 10 = 16 × 11 = Rs 176.Step 6: Banker's gain BG = BD − TD = 176 − 160 = Rs 16. Verification / Alternative check: We can check that the face value is 10 percent greater than the present worth because TD = 160 and PW = 1600, so TD / PW = 160 / 1600 = 1 / 10. This indicates that the relevant interest fraction is 10 percent. Applying the same percentage to the face value of 1760 gives 1760 × 10 percent = 176 as banker's discount. Subtracting the true discount 160 confirms that the banker's gain is 16 rupees, so our result is consistent. Why Other Options Are Wrong: Values like 13, 14 or 15 would correspond to different gaps between BD and TD and would not be compatible with the exact 11 to 10 ratio derived from the present worth and face value. A banker's gain of 12 rupees is also too small. Only 16 rupees keeps the proportional relationships intact and matches the difference between BD and TD for this situation. Common Pitfalls: Some learners attempt to guess the rate and time first, which is unnecessary and may lead to mistakes. Others treat true discount as interest on the face value rather than on the present worth, breaking the BD to TD ratio. By directly using F = PW + TD and BD / TD = F / PW, we avoid these complications and reach the answer efficiently. Final Answer: The banker's gain on the transaction is 16 rupees.

Question 4

If the true discount on a certain sum due 6 months hence at 15% per annum is Rs. 120, then for the same sum, same time period and same rate, what will be the banker's discount?

Accepted Answer

Introduction / Context: This question tests your knowledge of commercial arithmetic concepts related to bills, including true discount (T.D.), banker's discount (B.D.) and their relationship with the sum due. It focuses on a case where the true discount is known and you are asked to compute the banker's discount for the same bill, time and rate of interest. Such problems are common in banking and competitive exams dealing with bills of exchange and present worth. Given Data / Assumptions: True discount (T.D.) = Rs. 120. Time = 6 months = 0.5 years. Rate of interest = 15% per annum. The sum is due after 6 months. We assume simple interest and standard formulas for discount. Concept / Approach: For a bill of face value S, rate R% per annum and time T years, the formulas are: True discount: TD = S * (R * T) / (100 + R * T). Banker's discount: BD = S * R * T / 100. We first use the given true discount to find the sum due S. Once S is known, we substitute back into the banker's discount formula to get BD directly. Step-by-Step Solution: Step 1: Here R = 15% and T = 0.5 years, so R * T = 15 * 0.5 = 7.5.Step 2: Use TD formula: TD = S * 7.5 / (100 + 7.5) = S * 7.5 / 107.5.Step 3: Given TD = 120, so 120 = S * 7.5 / 107.5.Step 4: Rearranging, S = 120 * 107.5 / 7.5 = 1720.Step 5: Now compute banker's discount: BD = S * R * T / 100 = 1720 * 15 * 0.5 / 100.Step 6: 15 * 0.5 = 7.5, so BD = 1720 * 7.5 / 100 = 129.Step 7: Therefore the banker's discount is Rs. 129. Verification / Alternative check: Once S = 1720 is found, you can recompute the true discount as TD = 1720 * 7.5 / (100 + 7.5) = 1720 * 7.5 / 107.5 = 120, confirming consistency with the given data. Then, BD = S * 7.5 / 100 must be larger than TD because banker's discount is calculated on the full face value, whereas true discount is based on the present worth. The value 129 is reasonably slightly higher than 120, which fits the theory. Why Other Options Are Wrong: Rs. 109 and Rs. 119 are both less than the true discount, which contradicts the fact that banker's discount is always greater than true discount for a positive rate and time. Rs. 139 and Rs. 149 are too large compared to the difference between face value and present worth implied by the true discount; they do not satisfy the algebraic relationship between TD and BD for the given rate and time. Common Pitfalls: Confusing the formulas of true discount and banker's discount. Forgetting to convert 6 months into 0.5 years before using R * T. Trying to guess BD directly without first finding the sum due S. Assuming BD and TD are equal, which is never true except in trivial cases. Final Answer: The banker's discount for the given bill is Rs. 129.

Question 5

The true discount on a bill of Rs. 2160 is Rs. 360 for a certain time and rate of interest. For the same conditions, what is the corresponding banker's discount on this bill?

Accepted Answer

Introduction / Context: This problem uses the standard relationship between true discount, banker's discount and present worth of a bill. You are given the true discount on a known face value and asked to determine the banker's discount for the same rate and time. Understanding how these quantities are related is important in questions about commercial mathematics, banking and bill discounting. Given Data / Assumptions: Face value (sum due) S = Rs. 2160. True discount TD = Rs. 360. Rate of interest and time are the same for both TD and BD (but not numerically specified). We assume simple interest and standard discount formulas. Concept / Approach: Key concepts: Present worth (P) = S − TD. Banker's discount (BD) is calculated on the face value S. A useful relation connecting BD, TD and P is: BD = (S * TD) / P. Another derived relation is banker's gain = BD − TD = TD^2 / P. We will use P = S − TD to compute the present worth and then apply BD = (S * TD) / P. Step-by-Step Solution: Step 1: Compute present worth: P = S − TD = 2160 − 360 = 1800.Step 2: Use the relation BD = (S * TD) / P.Step 3: Substitute: BD = (2160 * 360) / 1800.Step 4: Simplify: 2160 / 1800 = 1.2, so BD = 1.2 * 360 = 432.Step 5: Therefore the banker's discount is Rs. 432. Verification / Alternative check: If we compute the banker's gain, we have BG = BD − TD = 432 − 360 = 72. Using the theoretical relation BG = TD^2 / P, we check: TD^2 / P = 360^2 / 1800 = 129600 / 1800 = 72. Since this matches, our computed BD is consistent with all standard formulas, confirming that Rs. 432 is correct. Why Other Options Are Wrong: Rs. 422, Rs. 412 and Rs. 442 do not satisfy the relation BD = (S * TD) / P when S = 2160 and P = 1800. If any of these values were taken as BD, the derived banker's gain would not equal TD^2 / P, contradicting the theoretical definitions. Rs. 452 is even farther away from the true value and clearly inconsistent with the algebraic relationships. Common Pitfalls: Using simple interest directly on the present worth instead of on the face value. Forgetting that present worth is S − TD, not S + TD. Not knowing the key formula BD = (S * TD) / P and trying to guess based only on intuitive reasoning. Confusing banker's discount with true discount and treating them as equal. Final Answer: The banker's discount on the bill is Rs. 432.

Question 6

On a bill due 6 months hence, the banker's discount at 6% per annum is Rs. 18.54. For the same bill, time period and rate, what is the corresponding true discount?

Accepted Answer

Introduction / Context: This question in commercial arithmetic asks you to find the true discount when the banker's discount is known. Both discounts are calculated for the same bill, time and rate. The key challenge is to work backwards from banker's discount to face value and then apply the true discount formula. Understanding this process is useful for handling bank-related questions about bills of exchange and discounting. Given Data / Assumptions: Banker's discount BD = Rs. 18.54. Time = 6 months = 0.5 years. Rate of interest R = 6% per annum. We assume simple interest and a single payment bill. Concept / Approach: We use two important formulas: Banker's discount: BD = S * R * T / 100. True discount: TD = S * (R * T) / (100 + R * T). First we use BD to find the face value S of the bill. Then we use S in the true discount formula, with the same rate and time, to find TD. Step-by-Step Solution: Step 1: Let S be the face value. With R = 6% and T = 0.5 years, R * T = 3.Step 2: Banker's discount BD = S * R * T / 100 = S * 6 * 0.5 / 100 = S * 3 / 100.Step 3: Given BD = 18.54, so 18.54 = S * 3 / 100.Step 4: Solving for S: S = 18.54 * 100 / 3 = 618.Step 5: Use true discount formula: TD = S * (R * T) / (100 + R * T) = 618 * 3 / (100 + 3).Step 6: TD = 618 * 3 / 103 = 1854 / 103 = 18.Step 7: Therefore, the true discount is Rs. 18. Verification / Alternative check: We can also compute present worth P = S − TD = 618 − 18 = 600. Then simple interest on 618 for half a year at 6% is 618 * 3 / 100 = 18.54, which matches the banker's discount given in the question. The difference BD − TD = 18.54 − 18 = 0.54 is exactly the banker's gain, which should equal TD^2 / P when expressed appropriately, providing an additional theoretical check on consistency. Why Other Options Are Wrong: Rs. 24 and Rs. 36 are larger than the banker's discount, which cannot happen for positive time and rate since true discount is always less than banker's discount. Rs. 12 and Rs. 20 do not fit the formulas when substituted back; they produce incorrect present worth or fail to give BD = 18.54 at 6% for 6 months. Only Rs. 18 satisfies both the BD and TD relationships. Common Pitfalls: Using the true discount formula directly with BD in place of S. Forgetting to convert 6 months to 0.5 years when computing R * T. Confusing present worth with sum due and mixing the roles of S and P. Assuming TD is always roughly equal to BD without doing the actual calculation. Final Answer: The true discount on the bill is Rs. 18.

Question 7

On a sum of Rs. 8100 for 3 months at 5% per annum, what is the difference between the banker's discount and the true discount?

Accepted Answer

Introduction / Context: This question focuses on the difference between banker's discount and true discount, a quantity commonly referred to as the banker's gain. You are given the face value, time and rate of interest and must compute both discounts and then find their difference. It checks your ability to handle short-duration interest calculations and correctly apply the formulas for commercial discounting. Given Data / Assumptions: Face value S = Rs. 8100. Time = 3 months = 0.25 years. Rate R = 5% per annum. Simple interest is assumed. We need BD − TD (the banker's gain). Concept / Approach: We will compute both discounts explicitly: Banker's discount BD = S * R * T / 100. True discount TD = S * (R * T) / (100 + R * T). Then the difference between them, BD − TD, gives the required answer. Since the time is short (3 months), the difference will be relatively small. Step-by-Step Solution: Step 1: Convert time: T = 3 months = 3 / 12 = 0.25 years.Step 2: Compute R * T = 5 * 0.25 = 1.25.Step 3: Banker's discount: BD = S * R * T / 100 = 8100 * 5 * 0.25 / 100.Step 4: BD = 8100 * 1.25 / 100 = 101.25.Step 5: True discount: TD = S * (R * T) / (100 + R * T) = 8100 * 1.25 / (100 + 1.25).Step 6: TD = 10125 / 101.25 = 100.Step 7: Difference (banker's gain) = BD − TD = 101.25 − 100 = 1.25. Verification / Alternative check: We can also use the formula banker's gain = TD^2 / P, where P is present worth. Here P = S − TD = 8100 − 100 = 8000. Then TD^2 / P = 100^2 / 8000 = 10000 / 8000 = 1.25. This matches the directly computed difference BD − TD, confirming the correctness of the value Rs. 1.25. Why Other Options Are Wrong: Rs. 2 and Rs. 2.25 are larger than the theoretical banker's gain based on the short time and low rate. Rs. 0.50 and Rs. 3 do not satisfy the exact formulas when substituted back. Only Rs. 1.25 is consistent with both direct computation and the banker's gain relation involving true discount and present worth. Common Pitfalls: Forgetting to convert 3 months into years when calculating R * T. Using 5 as R * T instead of 1.25, which leads to large errors. Mixing up true discount and simple interest on present worth. Assuming the difference will be negligible without actually computing it. Final Answer: The difference between the banker's discount and the true discount is Rs. 1.25.

Question 8

Two successive discounts of x% and y% on the marked price of an article are equivalent to a single discount of what percentage on the marked price?

Accepted Answer

Introduction / Context: This is a conceptual question about successive discounts in percentage form. In real life, shops may offer two consecutive discounts on a marked price, and it is often useful to know what single equivalent discount these two reductions represent. The key point is that successive discounts do not simply add; instead they interact multiplicatively, leading to an effective discount that is slightly less than the arithmetic sum of the individual discounts. Given Data / Assumptions: First discount = x% on the marked price. Second discount = y% applied on the reduced price after the first discount. Both x and y are expressed as percentages. We are asked to find the single equivalent discount percentage on the original marked price. Concept / Approach: Successive discounts are handled by multiplying the remaining fractions of price rather than simply adding discount percentages. If the marked price is M, then: After x% discount, price becomes M * (1 − x/100). After a further y% discount, price becomes M * (1 − x/100) * (1 − y/100). The equivalent single discount D% must satisfy: final price = M * (1 − D/100). We equate the two expressions and solve for D, which leads to the standard formula D = x + y − (xy/100). Step-by-Step Solution: Step 1: Let marked price be M.Step 2: After first discount of x%, price = M * (1 − x/100).Step 3: After second discount of y%, price = M * (1 − x/100) * (1 − y/100).Step 4: Expand (1 − x/100) * (1 − y/100) = 1 − x/100 − y/100 + xy/10000.Step 5: Write it as 1 − (x + y)/100 + xy/10000.Step 6: Suppose equivalent single discount is D%. Then final price should also equal M * (1 − D/100).Step 7: Therefore 1 − D/100 = 1 − (x + y)/100 + xy/10000.Step 8: Comparing terms, D/100 = (x + y)/100 − xy/10000.Step 9: Multiply by 100: D = x + y − xy/100. Verification / Alternative check: Take a numerical example: let x = 20% and y = 10%. Then equivalent discount should be D = 20 + 10 − (20 * 10)/100 = 30 − 2 = 28%. If M = Rs. 100, two successive discounts give price = 100 * 0.8 * 0.9 = 72, a discount of Rs. 28, or 28%. This confirms the formula and the chosen option. Why Other Options Are Wrong: (x − y + xy/100)% and (x − y − xy/100)% incorrectly subtract y% or mis-handle the interaction term. (x + y + xy/100)% makes the equivalent discount even larger than x + y, which is impossible because successive discounts are always slightly less than their sum. The expression (x + y − 2xy/100)% overcompensates for the interaction term and is not derived from the correct algebraic expansion. Common Pitfalls: Simply adding x and y and ignoring the multiplicative nature of successive discounts. Confusing successive discounts with successive percentage increases. Making algebraic errors in expanding (1 − x/100)(1 − y/100). Assuming the interaction term xy/100 is negligible for large x and y, which can create sizable errors. Final Answer: The single equivalent discount is (x + y − xy/100)% on the marked price.

Question 9

The banker's discount on a certain sum of money is Rs. 72 and the true discount on the same sum for the same time and rate is Rs. 60. What is the sum due (the face value of the bill)?

Accepted Answer

Introduction / Context: This question illustrates how to determine the face value of a bill when both banker's discount and true discount are known for the same time period and rate. These two quantities are closely related through the concept of banker's gain and present worth, and being able to move between them is important in commercial arithmetic and banking-related questions. Given Data / Assumptions: Banker's discount BD = Rs. 72. True discount TD = Rs. 60. Both are computed on the same sum S, at the same rate and for the same time. We assume simple interest and a single-payment bill. Concept / Approach: We use the concept of banker's gain and some standard relationships: Banker's gain (BG) = BD − TD. Present worth P = S − TD. A key relation: BG = TD^2 / P. From BG and TD we can find P, and then add TD to get the sum due S. This avoids dealing directly with rate and time, which are not given. Step-by-Step Solution: Step 1: Compute banker's gain: BG = BD − TD = 72 − 60 = 12.Step 2: Let P be the present worth. Use BG = TD^2 / P.Step 3: Substitute values: 12 = 60^2 / P = 3600 / P.Step 4: Rearranging: P = 3600 / 12 = 300.Step 5: Present worth P = S − TD, so S − 60 = 300.Step 6: Therefore, S = 300 + 60 = 360.Step 7: The sum due (face value of the bill) is Rs. 360. Verification / Alternative check: We can verify using another relation BD = (S * TD) / P. With S = 360, TD = 60 and P = 300, BD = (360 * 60) / 300 = 21600 / 300 = 72, which matches the given banker's discount. The difference BD − TD = 72 − 60 = 12 is equal to TD^2 / P = 3600 / 300 = 12, confirming consistency of all formulas. Why Other Options Are Wrong: Values like Rs. 290, Rs. 420, Rs. 480 or Rs. 540 do not satisfy the relationships among BD, TD and P. If you substitute any of these as S into BD = (S * TD) / (S − TD), you do not obtain BD = 72. Only S = 360 correctly reproduces both the banker's discount and true discount values given in the question. Common Pitfalls: Confusing present worth P with the face value S. Attempting to use simple interest formulas directly without using the TD–BD relations. Forgetting that banker's gain is BD − TD, not BD + TD. Ignoring the helpful relation BG = TD^2 / P, which simplifies the calculation significantly. Final Answer: The sum due (face value of the bill) is Rs. 360.

Question 10

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. For what length of time is this equality true?

Accepted Answer

Introduction / Context: This problem connects banker's discount on one sum with true discount on another sum, both calculated for the same time and interest rate. You must determine the time period that makes these two quantities equal. It is a good test of your ability to form equations using the standard formulas of discount and then solve for time. Given Data / Assumptions: Sum 1: S1 = Rs. 1800 with banker's discount BD1. Sum 2: S2 = Rs. 1872 with true discount TD2. Rate of interest R = 12% per annum. Time = T years (same for both sums, unknown). Condition: BD1 = TD2. Concept / Approach: We use separate formulas: Banker's discount on S1: BD1 = S1 * R * T / 100. True discount on S2: TD2 = S2 * (R * T) / (100 + R * T). We equate BD1 and TD2, then solve for T. Once T is found in years, we convert it to months for the final answer. Step-by-Step Solution: Step 1: Compute BD1 in terms of T: BD1 = 1800 * 12 * T / 100 = 216T.Step 2: Let x = R * T = 12T for convenience.Step 3: True discount on S2: TD2 = 1872 * x / (100 + x).Step 4: Since BD1 = TD2, we have 216T = 1872 * x / (100 + x).Step 5: Substitute x = 12T into the equation: 216T = 1872 * 12T / (100 + 12T).Step 6: For T ≠ 0, cancel T from both sides: 216 = 1872 * 12 / (100 + 12T).Step 7: Multiply both sides by (100 + 12T): 216(100 + 12T) = 1872 * 12.Step 8: Compute right side: 1872 * 12 = 22464.Step 9: Expand left side: 21600 + 2592T = 22464.Step 10: Rearranging: 2592T = 22464 − 21600 = 864.Step 11: T = 864 / 2592 = 1/3 year.Step 12: Convert to months: (1/3) year = 12 / 3 = 4 months. Verification / Alternative check: Take T = 1/3 year. Then BD1 = 1800 * 12 * (1/3) / 100 = 1800 * 4 / 100 = 72. For the second bill, R * T = 12 * (1/3) = 4. True discount TD2 = 1872 * 4 / (100 + 4) = 7488 / 104 = 72. Since both BD1 and TD2 equal Rs. 72, the time T = 1/3 year or 4 months is confirmed as correct. Why Other Options Are Wrong: Any time other than 4 months will break the equality BD1 = TD2. Substituting 3 months, 5 months or 6 months into the formulas leads to different numerical values for BD1 and TD2. Thus, those options do not satisfy the basic condition given in the question. Common Pitfalls: Not cancelling T correctly and ending up with a more complicated equation than necessary. Confusing the formulas for true discount and simple interest. Making arithmetic mistakes when multiplying or subtracting large numbers. Forgetting to convert the final answer from years to months. Final Answer: The required time is 4 months.

Question 11

The present worth of a sum due sometime in the future is Rs. 576 and the banker's gain on this transaction is Rs. 16. What is the true discount on the sum?

Accepted Answer

Introduction / Context: This question tests your understanding of the relationship between present worth, true discount and banker's gain. You are given the present worth of the sum and the banker's gain and are required to find the true discount. This scenario is typical in discounting problems involving bills, where the bank earns a gain because it discounts the bill on its face value instead of on present worth. Given Data / Assumptions: Present worth P = Rs. 576. Banker's gain BG = Rs. 16. Let TD be the true discount. Standard relations for simple interest and discount apply. Concept / Approach: A very useful relationship in discounting theory is: Banker's gain: BG = TD^2 / P. This connects banker's gain directly with true discount and present worth, without explicitly needing the rate of interest or time. Knowing BG and P, we can solve for TD easily. Once TD is found, all other related quantities like face value and banker's discount can also be inferred if needed. Step-by-Step Solution: Step 1: Use the relation BG = TD^2 / P.Step 2: Substitute BG = 16 and P = 576 into the formula: 16 = TD^2 / 576.Step 3: Rearranging gives TD^2 = 16 * 576.Step 4: Compute 16 * 576 = 9216.Step 5: Thus TD^2 = 9216, so TD = √9216 = 96.Step 6: Therefore, the true discount is Rs. 96. Verification / Alternative check: Once TD = 96 is known, the face value S of the bill can be found as S = P + TD = 576 + 96 = 672. If we wanted, we could now compute the banker's discount BD using BD = S * TD / P = 672 * 96 / 576 = 112. The banker's gain BG = BD − TD = 112 − 96 = 16, which matches the given value. This reverse check confirms that TD = 96 is consistent with both present worth and banker's gain. Why Other Options Are Wrong: Values such as Rs. 78, Rs. 85, Rs. 105 or Rs. 64 do not satisfy the equation TD^2 = BG * P when BG = 16 and P = 576. For example, if TD were 78, TD^2 would be 6084, and TD^2 / P would not equal 16. Only TD = 96 produces TD^2 / 576 = 9216 / 576 = 16, which matches the required banker's gain. Common Pitfalls: Not knowing or forgetting the relation BG = TD^2 / P and instead trying to introduce unnecessary unknowns like rate and time. Computing S first and then working backwards, which is longer and more error-prone. Mistakes in squaring or taking the square root of large numbers. Confusing present worth P with the face value S. Final Answer: The true discount on the sum is Rs. 96.

Question 12

The present worth of a certain sum due sometime hence is Rs. 3400 and the true discount on it is Rs. 340. What is the banker's gain on this transaction?

Accepted Answer

Introduction / Context: This problem again brings together present worth, true discount and banker's gain. You are given the present worth and true discount of a certain sum and asked to determine the banker's gain, which is the difference between banker's discount and true discount. Such questions reinforce the useful algebraic connections between these quantities in commercial arithmetic. Given Data / Assumptions: Present worth P = Rs. 3400. True discount TD = Rs. 340. Let BG be the banker's gain. Simple interest assumptions apply and time and rate are the same for BD and TD. Concept / Approach: We use the well-known relation between banker's gain, true discount and present worth: Banker's gain: BG = TD^2 / P. This formula lets us compute banker's gain directly from TD and P, without explicitly dealing with time and rate. Step-by-Step Solution: Step 1: Use BG = TD^2 / P.Step 2: Substitute TD = 340 and P = 3400: BG = 340^2 / 3400.Step 3: Compute 340^2 = 115600.Step 4: Now BG = 115600 / 3400.Step 5: Divide: 115600 ÷ 3400 = 34.Step 6: Therefore, the banker's gain is Rs. 34. Verification / Alternative check: We can reconstruct the face value S and banker's discount BD to check consistency. Face value S = P + TD = 3400 + 340 = 3740. Using BD = TD + BG, we get BD = 340 + 34 = 374. Another relation is BD = S * TD / P; substituting S = 3740, TD = 340 and P = 3400 gives BD = 3740 * 340 / 3400 = 374, which matches. The difference BD − TD = 374 − 340 = 34 equals BG, confirming the answer. Why Other Options Are Wrong: Values like Rs. 21, Rs. 17, Rs. 18 or Rs. 27 do not satisfy the equation BG = TD^2 / P. Plugging these values into BG * P and comparing with TD^2 reveals inconsistencies. Only Rs. 34 makes TD^2 = BG * P hold exactly, so it is the correct banker's gain. Common Pitfalls: Confusing banker's gain with true discount and simply answering Rs. 340. Attempting to guess BG instead of using the TD^2 / P relation. Arithmetic errors when squaring 340 or dividing by 3400. Mixing up present worth and sum due, leading to wrong formulas. Final Answer: The banker's gain on this transaction is Rs. 34.

Question 13

A bill of Rs. 1764 is due 2 years hence. If the rate of compound interest is 5% per annum, what is the present worth of this bill?

Accepted Answer

Introduction / Context: This question deals with present worth under compound interest. You are given a future amount (the sum due after a certain time) and the rate of compound interest, and asked to find its current value. This is a standard type of time value of money calculation, often used in finance and competitive exams to understand how future sums are discounted back to the present. Given Data / Assumptions: Future value (sum due) S = Rs. 1764. Rate of compound interest R = 5% per annum. Time n = 2 years. Compounding is annual. Concept / Approach: For compound interest, the relationship between present value (P), rate (R), time (n) and future value (S) is: S = P * (1 + R/100)^n. To find the present worth P when S is known, we rearrange the formula as: P = S / (1 + R/100)^n. We simply substitute the numerical values and perform the calculation. Step-by-Step Solution: Step 1: Write the formula for present worth under compound interest: P = S / (1 + R/100)^n.Step 2: Here S = 1764, R = 5% and n = 2 years.Step 3: Compute the compound factor: (1 + R/100) = 1 + 5/100 = 1.05.Step 4: Now (1.05)^2 = 1.05 * 1.05 = 1.1025.Step 5: Present worth P = 1764 / 1.1025.Step 6: Perform the division: 1764 ÷ 1.1025 = 1600.Step 7: Therefore, the present worth of the bill is Rs. 1600. Verification / Alternative check: We can verify by moving forward from P to S. If P = 1600 and R = 5%, then after 1 year the amount becomes 1600 * 1.05 = 1680. After another year, it becomes 1680 * 1.05 = 1764. This matches the given future value, confirming that P = 1600 is the correct present worth. Why Other Options Are Wrong: Rs. 1200 and Rs. 1400 are too low; if used as present worth, compounding at 5% for 2 years would give amounts significantly below 1764. Rs. 1800 and Rs. 1500 are either too high or too low and do not yield exactly 1764 when grown at 5% for 2 years. Only Rs. 1600 produces the correct future value under the given conditions. Common Pitfalls: Using simple interest formula instead of compound interest. Forgetting to square the factor (1.05) when the time is 2 years. Rounding too early, which can introduce errors into the final answer. Confusing present worth with discount calculated in a banker's discount sense. Final Answer: The present worth of the bill is Rs. 1600.

Question 14

On a certain sum, the banker's discount is Rs. 200 and the true discount is Rs. 100 for the same time and rate of interest. What is the sum due (face value of the bill)?

Accepted Answer

Introduction / Context: This question asks you to determine the face value of a bill when both the banker's discount and true discount are known. It is a direct application of the relationships between banker's discount, true discount, banker's gain and the principal (sum due). Problems like this are frequently used in aptitude tests to assess understanding of commercial discounting. Given Data / Assumptions: Banker's discount BD = Rs. 200. True discount TD = Rs. 100. Both apply to the same sum S, for the same time and rate. We assume simple interest. Concept / Approach: We use the standard relations: Banker's gain BG = BD − TD. Present worth P = S − TD. Banker's gain can also be written as BG = TD^2 / P. However, in this particular case, there is a quicker way using the relation between BD, TD and S: BD = (S * TD) / (S − TD). We can solve this directly for S using the given BD and TD. Step-by-Step Solution: Step 1: Use the relation BD = (S * TD) / (S − TD).Step 2: Substitute BD = 200 and TD = 100 into the formula: 200 = (S * 100) / (S − 100).Step 3: Cross multiply: 200(S − 100) = 100S.Step 4: Expand left side: 200S − 20000 = 100S.Step 5: Bring terms together: 200S − 100S = 20000.Step 6: So 100S = 20000, giving S = 200.Step 7: Therefore, the sum due (face value) is Rs. 200. Verification / Alternative check: We can verify by computing present worth P and banker's gain BG. Present worth P = S − TD = 200 − 100 = 100. Banker's gain BG should be BD − TD = 200 − 100 = 100. From the formula BG = TD^2 / P, we get TD^2 / P = 100^2 / 100 = 100, which matches. Also, using BD = S * TD / P gives BD = 200 * 100 / 100 = 200, confirming all relationships are satisfied. Why Other Options Are Wrong: If S were Rs. 400 or Rs. 300, substituting in BD = (S * TD) / (S − TD) with TD = 100 would not yield BD = 200. Rs. 100 as S is impossible because the true discount cannot equal the sum itself. Rs. 250 similarly fails to satisfy the algebraic relations. Only S = 200 produces BD = 200, TD = 100 and the correct banker's gain. Common Pitfalls: Assuming S must be BD + TD without checking formulas. Confusing present worth with the sum due. Not using the key relation BD = (S * TD) / (S − TD) and instead trying to guess. Ignoring banker's gain and its relation to true discount and present worth. Final Answer: The sum due (face value of the bill) is Rs. 200.

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