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 bankers gain of a certain sum due 2 years hence at 10% per annum is Rs 24 .The percent worth is

Accepted Answer

T.D = (B.G * 100) / (Rate * Time) (24*100) / (10 * 2) = 120. P.W = (100 *T.D) / (Rate * Time) = (100 * 120) /(10 * 2) = 600

Question 10

The banker's discount on a certain sum due 2 years hence is11/10of the true discount. The rate percent is:

Accepted Answer

Let T.D. be Re. 1. Then, B.D. = Rs.11/10= Rs. 1.10. Sum = Rs. (1.10*1)/(1.10-1) = Rs. (110/10)= Rs. 11. S.I. on Rs. 11 for 2 years is Rs. 1.10 Rate = [(100 x 1.10)/(11 x 2 )]% = 5%.

Question 11

The banker's discount on Rs. 1650 due a certain time hence is Rs. 165. Find the true discount and the banker's gain.

Accepted Answer

Sum = [(B.D.xT.D.)/ (B.D.-T.D.)] = [(B.D.xT.D.)/B.G.] T.D./B.G. = Sum/ B.D. =1650/165 =10 Thus, if B.G. is Re 1, T.D. = Rs. 10. If B.D.is Rs. ll, T.D.=Rs. 10. If B.D. is Rs. 165, T.D. = Rs. [(10/11)xl65] =Rs.150 And, B.G. = Rs. (165 - 150) = Rs, 15.

Question 12

The true discount on a bill of Rs. 540 is Rs. 90. The banker's discount is:

Accepted Answer

P.W. = Rs. (540 - 90) = Rs. 450. S.I. on Rs. 450 = Rs. 90. S.I. on Rs. 540 = Rs.( 90 /450)x 540 = Rs. 108. B.D. = Rs. 108.

Question 13

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

Accepted Answer

B.G. = S.I. on T.D. = Rs.(120 x 15 x 1/2 x 1/100) = Rs. 9. (B.D.) - (T.D.) = Rs. 9. B.D. = Rs. (120 + 9) = Rs. 129.

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 at the same rate. Find the time.

Accepted Answer

S.I. on Rs. 1800 = T.D. on Rs. 1872. P.W. of Rs. 1872 is Rs. 1800. Rs. 72 is S.I. on Rs. 1800 at 12%. Time =[(100 x 72)/ (12x1800)]year =1/3year = 4 months.

Question 15

The banker's gain on a bill due 1 year hence at 12% per annum is Rs. 6. The true discount is:

Accepted Answer

T.D. =(B.G. x 100)/ (R x T)= Rs. (6 x 100)/( 12 x 1) = Rs. 50.

Question 16

The banker's gain on a sum due 3 years hence at 12% per annum is Rs. 270. The banker's discount is:

Accepted Answer

T.D. =(B.G*100)/(R*T)=Rs(270*100)/(12*3) = Rs. 750. B.D. = Rs.(750 + 270) = Rs. 1020.

Question 17

The B.G. on a certain sum 4 years hence at 5% is Rs. 200. What is the present worth?

Accepted Answer

T = 4 years R = 5% Banker's Gain, BG = Rs.200 T D = B G × 100 T R = 200 × 100 4 × 5 = R s . 1000 T D = P W × B G ⇒ 1000 = P W × 200 =>PW = Rs.5000

Question 18

The bankers 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 is the sum and the time?

Accepted Answer

BD = Rs.100 TD = Rs.80 R = 10% F = B D × T D B D - T D = 100 × 80 100 - 80 = R s . 400 BD = Simple interest on the face value of the bill for unexpired time= FTR/100 ⇒ 100 = 400 × T × 10 100 => T = 2.5 years

Question 19

A bill for Rs. 6000 is drawn on July 14 at 5 months. It is discounted on 5th October at 10%. Find the banker's discount, true discount, banker's gain and the money that the holder of the bill receives.

Accepted Answer

Face value of the bill = Rs. 6000. Date on which the bill was drawn = July 14 at 5 months. Nominally due date = December 14. Legally due date = December 17. Date on which the bill was discounted = October 5. Unexpired time : Oct. Nov. Dec. 26 + 30 + 17 = 73 days =1/ 5Years B.D. = S.I. on Rs. 6000 for 1/5 year = Rs. (6000 x 10 x1/5 x1/100)= Rs. 120. T.D. = Rs.[(6000 x 10 x1/5)/(100+(10*1/5))] =Rs.(12000/102)=Rs. 117.64. B.G. = (B.D.) - (T.D.) = Rs. (120 - 117.64) = Rs. 2.36. Money received by the holder of the bill = Rs. (6000 - 120) = Rs. 5880.

Question 20

The bankers discount and true discount on a sum of money due 8 months hence are Rs.120 & Rs.110 resp. Find the sum.

Accepted Answer

Sum = (B.D * T.D) / (B.D) -(T.D) = (120 * 110) / (120 -110) = 1320

Banker's Discount Questions