Q: True discount on Rs 440 becomes Rs 40 for a certain time. If the same sum becomes due at double that time, what is the new true discount (approx., to 2 decimals)?

Introduction / Context:In commercial arithmetic, true discount (TD) is the rebate for paying before the due date based on the present worth concept. For a fixed rate and sum, TD depends nonlinearly on time through the factor 1 + r t / 100. Given Data / Assumptions: Face value (amount due at maturity) S = Rs 440. At time t, TD₁ = Rs 40. We seek TD₂ at time 2t. Simple interest framework; rate r is constant (unknown but cancels). Concept / Approach:For simple interest, present worth PW = S / (1 + k), where k = r t / 100. Thus TD = S − PW = S · k/(1 + k). For doubled time we use 2k in place of k. Step-by-Step Solution: Let k = r t / 100.TD₁ = S · k/(1 + k) = 440 · k/(1 + k) = 40 ⇒ 11k = 1 + k ⇒ 10k = 1 ⇒ k = 0.1.For double time, use 2k = 0.2:TD₂ = 440 · 0.2/(1 + 0.2) = 440 · 0.2/1.2 = 440/6 = 73.333… ≈ 73.32. Verification / Alternative check:Using percentage arithmetic: at t the effective fraction = k/(1 + k) = 0.1/1.1 ≈ 0.0909; at 2t it is 0.2/1.2 ≈ 0.1667. Scaling Rs 440 gives ≈ Rs 40 and ≈ Rs 73.33, respectively. Why Other Options Are Wrong:72.00 and 72.80 understate the computed Rs 73.33; 73.20 is a rounding undershoot. Common Pitfalls:Doubling time does not double true discount because the denominator 1 + k also changes. Final Answer:Rs 73.32

Q: At 10% p.a. simple interest, the banker’s discount (BD) and true discount (TD) on the same bill for the same time are Rs 100 and Rs 80, respectively. Find the face value (sum due at maturity) and the time.

Introduction / Context:Banker’s discount (BD) is simple interest on the face value for the full term; true discount (TD) is the difference between face value and its present worth. The pair (BD, TD) at the same rate/time pins down both the time and the face value. Given Data / Assumptions: Rate r = 10% p.a. (simple interest). BD = Rs 100, TD = Rs 80, same time t. Let face value (amount due at maturity) be A. Concept / Approach:BD = A · r · t / 100. True discount TD = A · k/(1 + k) with k = r t / 100. Equivalently TD = BD/(1 + k). Two equations in k and A solve the problem. Step-by-Step Solution: BD = A · (0.1 t) = 100 ⇒ A t = 1000.TD = BD/(1 + 0.1 t) = 80 ⇒ 100/(1 + 0.1 t) = 80.Solve: 1 + 0.1 t = 100/80 = 1.25 ⇒ t = 2.5 years.Then A = 1000 / 2.5 = Rs 400. Verification / Alternative check:k = r t/100 = 0.25; BD = A k = 400 × 0.25 = 100; TD = A k/(1 + k) = 400 × 0.25/1.25 = 80 (consistent). Why Other Options Are Wrong:Other pairs mismatch BD/TD relationships with r = 10%. Common Pitfalls:Confusing present worth discounting (TD) with interest on face value (BD). Final Answer:Sum = Rs 400; Time = 2.5 years

Q: The banker’s gain (BG) on a bill due 6 years hence at 12% p.a. simple interest is Rs 540. Find the banker’s discount (BD).

Introduction / Context:Banker’s gain BG = BD − TD quantifies the excess of banker’s discount over true discount for a given bill. With rate and time known, BG and BD are linked by a fixed factor. Given Data / Assumptions: Rate r = 12% p.a., time t = 6 years. k = r t / 100 = 0.72. BG = Rs 540; we seek BD. Concept / Approach:For a face value A, BD = A k and TD = A k/(1 + k). Hence BG = BD − TD = A k − A k/(1 + k) = A k²/(1 + k). Also, TD = BD/(1 + k), so BG = BD · k/(1 + k). This directly relates BG and BD. Step-by-Step Solution: With k = 0.72, factor f = k/(1 + k) = 0.72/1.72.BG = BD · f ⇒ BD = BG · (1 + k)/k = 540 × 1.72/0.72.Compute: 540 × (172/72) = 540 × (43/18) = 30 × 43 = 1290. Verification / Alternative check:Once BD is known, TD = BD/(1 + k) = 1290/1.72 = 750; BG = 1290 − 750 = 540 (checks). Why Other Options Are Wrong:They do not satisfy BG = BD · 0.72/1.72. Common Pitfalls:Using compound interest or mixing up the definitions of BD and TD. Final Answer:Rs 1290

Q: BD on a certain sum for 3 years is Rs 1116; the true discount on the same sum for 4 years is Rs 1200. Find the simple interest rate (percent per annum).

Introduction / Context:Two time horizons with BD and TD on the same face value pin down the rate uniquely under simple interest. We eliminate the (unknown) face value to solve for r. Given Data / Assumptions: BD (3 years) = Rs 1116. TD (4 years) = Rs 1200. Let face value be A and rate r% p.a. Concept / Approach:BD = A·r·t/100. For 3 years: BD₃ = A·r·3/100 = 1116 ⇒ A r = 37200. For 4 years, with k′ = 4r/100 = r/25, TD₄ = A·k′/(1 + k′) = 1200. Substitute A from the first relation and solve for r. Step-by-Step Solution: Let AR = A r = 37200 (from BD₃).TD₄ = A·(r)/(25 + r) = 1200 ⇒ (AR)/(25 + r) = 1200.37200/(25 + r) = 1200 ⇒ 25 + r = 31 ⇒ r = 6%. Verification / Alternative check:Check: k′ = 0.24; TD₄ = A · 0.24/1.24. Since A r = 37200 → A = 37200/6 = 6200; TD₄ = 6200 · 0.24/1.24 = 1200. Why Other Options Are Wrong:Any other r fails the pair (1116, 1200) simultaneously. Common Pitfalls:Mistaking TD for A·r·t/100; that is BD, not TD. Final Answer:6%

Q: Selling the same article at Rs 900 yields a profit equal to twice the loss incurred when selling it at Rs 490. At what price should it be sold to earn a 25% profit?

Introduction / Context:A linear relation links profit at one selling price to loss at another when the cost price (CP) is fixed. Once CP is found, any target-gain selling price is straightforward. Given Data / Assumptions: Selling price (SP₁) = Rs 900 with profit P. Selling price (SP₂) = Rs 490 with loss L. P = 2L (profit is double the loss). We want SP for 25% profit. Concept / Approach:Let CP = x. Then P = 900 − x and L = x − 490. Given P = 2L, solve for x. Then required price for 25% profit is 1.25x. Step-by-Step Solution: 900 − x = 2(x − 490)900 − x = 2x − 980 ⇒ 1880 = 3x ⇒ x = 1880/3 = 626.666…Price for 25% profit = 1.25x = (5/4) × 1880/3 = 2350/3 = Rs 783.33 (approx). Verification / Alternative check:Check P at Rs 900 is ≈ 273.33; L at Rs 490 is ≈ 136.67; indeed P ≈ 2L. Why Other Options Are Wrong:Rs 750, 775, 800, 825 do not equal 2350/3 and would not give exactly 25% on the computed CP. Common Pitfalls:Setting P = L instead of P = 2L or computing 25% profit on the selling price instead of CP. Final Answer:Rs 783.33

Q: Jaspal buys a cow for Rs 3000 and sells to Dharampal the same day for Rs 3600, allowing 2 years’ credit at 10% p.a. simple interest. What is Jaspal’s gain percentage?

Introduction / Context:When selling on credit, compare the present worth of the credit price at the going rate with the cash cost price to determine profit. Given Data / Assumptions: CP = Rs 3000 (cash today). SP (due in 2 years) = Rs 3600. Rate r = 10% p.a. simple; discounting to present worth applies. Concept / Approach:Present worth PW of an amount A due after t years at simple interest r% is A/(1 + r t/100). Profit is PW − CP if positive; else loss. Step-by-Step Solution: PW of Rs 3600 due in 2 years = 3600 / (1 + 0.10 × 2) = 3600 / 1.2 = Rs 3000.Compare with CP = Rs 3000 ⇒ Gain = 0. Verification / Alternative check:Viewed forward: If Jaspal invested Rs 3000 at 10% for 2 years, it would grow to 3600, equal to the credit selling price; hence zero advantage. Why Other Options Are Wrong:Any nonzero percentage contradicts the exact equality of PW and CP. Common Pitfalls:Treating Rs 3600 as if cash today without discounting. Final Answer:Nil

Q: The banker's discount on a certain sum due 2 years hence is 11/10 of the true discount. If the transaction is at simple interest, what is the annual rate of interest (rate percent)?

Introduction: This question compares the banker's discount with the true discount for the same sum of money and the same time period. By using the standard relationship between banker's discount and true discount, we can find the rate of interest per annum. Given Data / Assumptions: Time t = 2 years. Banker's discount (BD) = (11/10) * true discount (TD). Simple interest is used. We must find the rate of interest r (in percent per annum). Concept / Approach: For a sum with face value P: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). Therefore, BD / TD = (100 + r * t) / 100. We are given BD = (11/10) * TD, so BD / TD = 11/10. We equate these expressions and solve for r * t, then obtain r since t is known. Step-by-Step Solution: Step 1: Set up the ratio. BD / TD = (100 + r * t) / 100 = 11 / 10. Step 2: Cross-multiply to find r * t. 100 + r * t = 100 * (11 / 10) = 110. r * t = 110 − 100 = 10. Step 3: Time t = 2 years, so r * 2 = 10 ⇒ r = 10 / 2 = 5% per annum. Verification / Alternative check: If r = 5%, t = 2 years, then r * t = 10. The ratio BD / TD should be: (100 + 10) / 100 = 110 / 100 = 11 / 10, which exactly matches the condition BD = (11/10) * TD. Why Other Options Are Wrong: 10%: Would give r * t = 20, leading to BD / TD = 120 / 100 = 6/5, not 11/10. 7%: Gives r * t = 14, so BD / TD = 114 / 100 = 57/50, not 11/10. 8%: Leads to r * t = 16 and BD / TD = 116 / 100. 6%: Leads to r * t = 12 and BD / TD = 112 / 100. Common Pitfalls: A common mistake is to treat BD and TD as if they were both simple interest on the present worth, or to forget the exact BD / TD ratio formula. Always derive the relationship using the standard formulas for BD and TD before solving for the rate. Final Answer: The annual rate of interest is 5% per annum.

Question 1

Find time when BD(1600 at 15%) equals TD(1680 at 15%) for same term: The banker’s discount on Rs 1600 at 15% p.a. equals the true discount on Rs 1680 for the same time and rate. Find the time.

Accepted Answer

Introduction / Context:Use BD = S1 * r * t and TD = S2 * r * t / (1 + r * t). Given BD(1600) = TD(1680), solve for t. Given Data / Assumptions:S1 = 1600; S2 = 1680; r = 15% p.a.; t = ? Concept / Approach:1600 * r * t = 1680 * r * t / (1 + r t) → 1600(1 + r t) = 1680 → 1 + r t = 1680/1600 = 1.05 → r t = 0.05. Step-by-Step Solution: r = 0.15 → t = 0.05 / 0.15 = 1/3 year = 4 months. Verification / Alternative check:Check equality numerically at t = 1/3 year; both sides match. Why Other Options Are Wrong:3, 5, 6 months do not satisfy 1 + r t = 1.05. Common Pitfalls:Canceling t improperly; forgetting both sides share r * t. Final Answer:4 months

Question 2

Difference between SI and TD over 6 months at 12.5% is Rs 25: find face value: For a certain sum, the difference between Simple Interest (on the face value) and True Discount for 6 months at 12.5% p.a. equals Rs 25. Find the face value (sum due).

Accepted Answer

Introduction / Context:A useful identity is that SI − TD for the same sum, time, and rate equals the banker’s gain (BG) = S * r^2 * t^2 / (1 + r t). Using the given difference, we can solve for S directly. Given Data / Assumptions:r = 12.5% = 0.125; t = 6 months = 0.5 year; SI − TD = Rs 25. Concept / Approach:BG = S * r^2 * t^2 / (1 + r t) = 25. Compute r t and r^2 t^2, then solve for S. Step-by-Step Solution: r t = 0.125 * 0.5 = 0.0625; r^2 t^2 = (0.0625)^2 = 0.00390625.25 = S * 0.00390625 / (1 + 0.0625) = S * 0.00390625 / 1.0625.S = 25 * 1.0625 / 0.00390625 = 6800. Verification / Alternative check:SI = S r t = 6800 * 0.0625 = 425. TD = SI / (1 + r t) = 425 / 1.0625 = 400. Difference = 25 ✔️. Why Other Options Are Wrong:6500, 6000, 6200 do not yield a difference of 25 between SI and TD at given r and t. Common Pitfalls:Confusing TD with BD; forgetting the 1 + r t denominator in BG. Final Answer:Rs 6,800

Question 3

Choose the better offer (cash vs 8-month credit at 18% p.a.): Vivek has two offers for selling his scooter: (i) Rs 12,000 cash; (ii) Rs 12,880 to be paid after 8 months. If money is worth 18% per annum (simple interest), which offer is better?

Accepted Answer

Introduction / Context:Compare offers by converting the credit offer to its present worth at the given simple interest rate. The better offer is the one with the higher present value today. Given Data / Assumptions: Cash offer = Rs 12,000. Credit offer = Rs 12,880 after 8 months. r = 18% p.a.; t = 8/12 year = 2/3 year. Concept / Approach:Present worth PW = F / (1 + r * t). Compute the PW of Rs 12,880 due after 8 months and compare with Rs 12,000. Step-by-Step Solution: 1 + r t = 1 + 0.18 * (2/3) = 1 + 0.12 = 1.12.PW (credit) = 12880 / 1.12 = Rs 11,500.Rs 11,500 < Rs 12,000 → better to take cash. Verification / Alternative check:Future value of Rs 12,000 in 8 months at 18% p.a. is 12000 * 1.12 = 13440, which exceeds 12880, confirming cash is superior in present terms. Why Other Options Are Wrong:12,800 is not an offer; “Both equal”/“Can’t say” are contradicted by the PW comparison. Common Pitfalls:Comparing face amounts without discounting; using compound interest accidentally. Final Answer:Choose the cash offer of Rs 12,000

Question 4

True discount on Rs 440 becomes Rs 40 for a certain time. If the same sum becomes due at double that time, what is the new true discount (approx., to 2 decimals)?

Accepted Answer

Introduction / Context:In commercial arithmetic, true discount (TD) is the rebate for paying before the due date based on the present worth concept. For a fixed rate and sum, TD depends nonlinearly on time through the factor 1 + r t / 100. Given Data / Assumptions: Face value (amount due at maturity) S = Rs 440. At time t, TD₁ = Rs 40. We seek TD₂ at time 2t. Simple interest framework; rate r is constant (unknown but cancels). Concept / Approach:For simple interest, present worth PW = S / (1 + k), where k = r t / 100. Thus TD = S − PW = S · k/(1 + k). For doubled time we use 2k in place of k. Step-by-Step Solution: Let k = r t / 100.TD₁ = S · k/(1 + k) = 440 · k/(1 + k) = 40 ⇒ 11k = 1 + k ⇒ 10k = 1 ⇒ k = 0.1.For double time, use 2k = 0.2:TD₂ = 440 · 0.2/(1 + 0.2) = 440 · 0.2/1.2 = 440/6 = 73.333… ≈ 73.32. Verification / Alternative check:Using percentage arithmetic: at t the effective fraction = k/(1 + k) = 0.1/1.1 ≈ 0.0909; at 2t it is 0.2/1.2 ≈ 0.1667. Scaling Rs 440 gives ≈ Rs 40 and ≈ Rs 73.33, respectively. Why Other Options Are Wrong:72.00 and 72.80 understate the computed Rs 73.33; 73.20 is a rounding undershoot. Common Pitfalls:Doubling time does not double true discount because the denominator 1 + k also changes. Final Answer:Rs 73.32

Question 5

At 10% p.a. simple interest, the banker’s discount (BD) and true discount (TD) on the same bill for the same time are Rs 100 and Rs 80, respectively. Find the face value (sum due at maturity) and the time.

Accepted Answer

Introduction / Context:Banker’s discount (BD) is simple interest on the face value for the full term; true discount (TD) is the difference between face value and its present worth. The pair (BD, TD) at the same rate/time pins down both the time and the face value. Given Data / Assumptions: Rate r = 10% p.a. (simple interest). BD = Rs 100, TD = Rs 80, same time t. Let face value (amount due at maturity) be A. Concept / Approach:BD = A · r · t / 100. True discount TD = A · k/(1 + k) with k = r t / 100. Equivalently TD = BD/(1 + k). Two equations in k and A solve the problem. Step-by-Step Solution: BD = A · (0.1 t) = 100 ⇒ A t = 1000.TD = BD/(1 + 0.1 t) = 80 ⇒ 100/(1 + 0.1 t) = 80.Solve: 1 + 0.1 t = 100/80 = 1.25 ⇒ t = 2.5 years.Then A = 1000 / 2.5 = Rs 400. Verification / Alternative check:k = r t/100 = 0.25; BD = A k = 400 × 0.25 = 100; TD = A k/(1 + k) = 400 × 0.25/1.25 = 80 (consistent). Why Other Options Are Wrong:Other pairs mismatch BD/TD relationships with r = 10%. Common Pitfalls:Confusing present worth discounting (TD) with interest on face value (BD). Final Answer:Sum = Rs 400; Time = 2.5 years

Question 6

The banker’s gain (BG) on a bill due 6 years hence at 12% p.a. simple interest is Rs 540. Find the banker’s discount (BD).

Accepted Answer

Introduction / Context:Banker’s gain BG = BD − TD quantifies the excess of banker’s discount over true discount for a given bill. With rate and time known, BG and BD are linked by a fixed factor. Given Data / Assumptions: Rate r = 12% p.a., time t = 6 years. k = r t / 100 = 0.72. BG = Rs 540; we seek BD. Concept / Approach:For a face value A, BD = A k and TD = A k/(1 + k). Hence BG = BD − TD = A k − A k/(1 + k) = A k²/(1 + k). Also, TD = BD/(1 + k), so BG = BD · k/(1 + k). This directly relates BG and BD. Step-by-Step Solution: With k = 0.72, factor f = k/(1 + k) = 0.72/1.72.BG = BD · f ⇒ BD = BG · (1 + k)/k = 540 × 1.72/0.72.Compute: 540 × (172/72) = 540 × (43/18) = 30 × 43 = 1290. Verification / Alternative check:Once BD is known, TD = BD/(1 + k) = 1290/1.72 = 750; BG = 1290 − 750 = 540 (checks). Why Other Options Are Wrong:They do not satisfy BG = BD · 0.72/1.72. Common Pitfalls:Using compound interest or mixing up the definitions of BD and TD. Final Answer:Rs 1290

Question 7

BD on a certain sum for 3 years is Rs 1116; the true discount on the same sum for 4 years is Rs 1200. Find the simple interest rate (percent per annum).

Accepted Answer

Introduction / Context:Two time horizons with BD and TD on the same face value pin down the rate uniquely under simple interest. We eliminate the (unknown) face value to solve for r. Given Data / Assumptions: BD (3 years) = Rs 1116. TD (4 years) = Rs 1200. Let face value be A and rate r% p.a. Concept / Approach:BD = A·r·t/100. For 3 years: BD₃ = A·r·3/100 = 1116 ⇒ A r = 37200. For 4 years, with k′ = 4r/100 = r/25, TD₄ = A·k′/(1 + k′) = 1200. Substitute A from the first relation and solve for r. Step-by-Step Solution: Let AR = A r = 37200 (from BD₃).TD₄ = A·(r)/(25 + r) = 1200 ⇒ (AR)/(25 + r) = 1200.37200/(25 + r) = 1200 ⇒ 25 + r = 31 ⇒ r = 6%. Verification / Alternative check:Check: k′ = 0.24; TD₄ = A · 0.24/1.24. Since A r = 37200 → A = 37200/6 = 6200; TD₄ = 6200 · 0.24/1.24 = 1200. Why Other Options Are Wrong:Any other r fails the pair (1116, 1200) simultaneously. Common Pitfalls:Mistaking TD for A·r·t/100; that is BD, not TD. Final Answer:6%

Question 8

Selling the same article at Rs 900 yields a profit equal to twice the loss incurred when selling it at Rs 490. At what price should it be sold to earn a 25% profit?

Accepted Answer

Introduction / Context:A linear relation links profit at one selling price to loss at another when the cost price (CP) is fixed. Once CP is found, any target-gain selling price is straightforward. Given Data / Assumptions: Selling price (SP₁) = Rs 900 with profit P. Selling price (SP₂) = Rs 490 with loss L. P = 2L (profit is double the loss). We want SP for 25% profit. Concept / Approach:Let CP = x. Then P = 900 − x and L = x − 490. Given P = 2L, solve for x. Then required price for 25% profit is 1.25x. Step-by-Step Solution: 900 − x = 2(x − 490)900 − x = 2x − 980 ⇒ 1880 = 3x ⇒ x = 1880/3 = 626.666…Price for 25% profit = 1.25x = (5/4) × 1880/3 = 2350/3 = Rs 783.33 (approx). Verification / Alternative check:Check P at Rs 900 is ≈ 273.33; L at Rs 490 is ≈ 136.67; indeed P ≈ 2L. Why Other Options Are Wrong:Rs 750, 775, 800, 825 do not equal 2350/3 and would not give exactly 25% on the computed CP. Common Pitfalls:Setting P = L instead of P = 2L or computing 25% profit on the selling price instead of CP. Final Answer:Rs 783.33

Question 9

Jaspal buys a cow for Rs 3000 and sells to Dharampal the same day for Rs 3600, allowing 2 years’ credit at 10% p.a. simple interest. What is Jaspal’s gain percentage?

Accepted Answer

Introduction / Context:When selling on credit, compare the present worth of the credit price at the going rate with the cash cost price to determine profit. Given Data / Assumptions: CP = Rs 3000 (cash today). SP (due in 2 years) = Rs 3600. Rate r = 10% p.a. simple; discounting to present worth applies. Concept / Approach:Present worth PW of an amount A due after t years at simple interest r% is A/(1 + r t/100). Profit is PW − CP if positive; else loss. Step-by-Step Solution: PW of Rs 3600 due in 2 years = 3600 / (1 + 0.10 × 2) = 3600 / 1.2 = Rs 3000.Compare with CP = Rs 3000 ⇒ Gain = 0. Verification / Alternative check:Viewed forward: If Jaspal invested Rs 3000 at 10% for 2 years, it would grow to 3600, equal to the credit selling price; hence zero advantage. Why Other Options Are Wrong:Any nonzero percentage contradicts the exact equality of PW and CP. Common Pitfalls:Treating Rs 3600 as if cash today without discounting. Final Answer:Nil

Question 10

The banker's gain on a certain sum of money, due 2 years hence at 10% per annum simple interest, is Rs 24. What is the present worth (that is, the present value) of this sum?

Accepted Answer

Introduction: In banker's discount problems, three related quantities are very important: the banker's discount (BD), the true discount (TD), and the banker's gain (BG). This question gives us the banker's gain on a bill and asks for the present worth, or present value, of the sum using the given rate and time period. Given Data / Assumptions: Banker's gain (BG) = Rs 24. Time period t = 2 years. Rate of simple interest r = 10% per annum. We must find: Present worth (PW) of the sum. Concept / Approach: For a sum with face value P due after time t at rate r: True discount: TD = P * r * t / (100 + r * t). Banker's discount: BD = P * r * t / 100. Banker's gain: BG = BD − TD. There is also a direct relationship: BG = TD * r * t / 100. We will first find TD from the banker's gain, then find the face value P, and finally subtract TD from P to get the present worth. Step-by-Step Solution: Step 1: Compute r * t = 10 * 2 = 20. Step 2: Use BG = TD * r * t / 100. 24 = TD * 20 / 100. TD = 24 * 100 / 20 = 24 * 5 = Rs 120. Step 3: Use TD formula to find the face value P. TD = P * r * t / (100 + r * t). 120 = P * 20 / (100 + 20) = P * 20 / 120. 120 = P / 6 ⇒ P = 120 * 6 = Rs 720. Step 4: Present worth PW = P − TD = 720 − 120 = Rs 600. Verification / Alternative check: We can verify quickly: BD = P * r * t / 100 = 720 * 20 / 100 = Rs 144. TD = Rs 120 (as found). BG = BD − TD = 144 − 120 = Rs 24, which matches the given banker's gain. Why Other Options Are Wrong: Rs 400: This would give inconsistent discount and gain values when formulas are applied. Rs 800: This would lead to a much larger banker's discount and a different banker's gain. Rs 500: Does not satisfy the exact banker's gain relationship BG = Rs 24. Rs 720: This is actually the face value, not the present worth. Common Pitfalls: Many learners confuse true discount with banker's discount or forget the relation BG = TD * r * t / 100. Another common mistake is to use the simple interest formula directly on the present worth instead of solving step by step for TD, P, and finally PW. Final Answer: The present worth of the sum is Rs 600.

Question 11

The banker's discount on a certain sum due 2 years hence is 11/10 of the true discount. If the transaction is at simple interest, what is the annual rate of interest (rate percent)?

Accepted Answer

Introduction: This question compares the banker's discount with the true discount for the same sum of money and the same time period. By using the standard relationship between banker's discount and true discount, we can find the rate of interest per annum. Given Data / Assumptions: Time t = 2 years. Banker's discount (BD) = (11/10) * true discount (TD). Simple interest is used. We must find the rate of interest r (in percent per annum). Concept / Approach: For a sum with face value P: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). Therefore, BD / TD = (100 + r * t) / 100. We are given BD = (11/10) * TD, so BD / TD = 11/10. We equate these expressions and solve for r * t, then obtain r since t is known. Step-by-Step Solution: Step 1: Set up the ratio. BD / TD = (100 + r * t) / 100 = 11 / 10. Step 2: Cross-multiply to find r * t. 100 + r * t = 100 * (11 / 10) = 110. r * t = 110 − 100 = 10. Step 3: Time t = 2 years, so r * 2 = 10 ⇒ r = 10 / 2 = 5% per annum. Verification / Alternative check: If r = 5%, t = 2 years, then r * t = 10. The ratio BD / TD should be: (100 + 10) / 100 = 110 / 100 = 11 / 10, which exactly matches the condition BD = (11/10) * TD. Why Other Options Are Wrong: 10%: Would give r * t = 20, leading to BD / TD = 120 / 100 = 6/5, not 11/10. 7%: Gives r * t = 14, so BD / TD = 114 / 100 = 57/50, not 11/10. 8%: Leads to r * t = 16 and BD / TD = 116 / 100. 6%: Leads to r * t = 12 and BD / TD = 112 / 100. Common Pitfalls: A common mistake is to treat BD and TD as if they were both simple interest on the present worth, or to forget the exact BD / TD ratio formula. Always derive the relationship using the standard formulas for BD and TD before solving for the rate. Final Answer: The annual rate of interest is 5% per annum.

Question 12

The banker's discount on a bill of Rs 1650, due a certain time hence, is Rs 165. Find both the true discount and the banker's gain on this transaction.

Accepted Answer

Introduction: Here we are given the banker's discount (BD) on a bill and asked to compute both the true discount (TD) and the banker's gain (BG). This is a standard banker's discount question that tests your understanding of the relationships between BD, TD, BG, and the underlying rate-time product r * t. Given Data / Assumptions: Face value of the bill P = Rs 1650. Banker's discount BD = Rs 165. Bill is due after some time t at rate r (simple interest). We must find: True discount (TD) and banker's gain (BG). Concept / Approach: Key formulas for banker's discount problems: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). BG = BD − TD. Also, there is a handy relation: BG = TD * r * t / 100. From BD we first find the product r * t, then use that to find TD, and finally compute BG = BD − TD. Step-by-Step Solution: Step 1: Use BD to find r * t. BD = P * r * t / 100 ⇒ 165 = 1650 * r * t / 100. 165 = 16.5 * r * t ⇒ r * t = 165 / 16.5 = 10. So r * t = 10. Step 2: Use TD formula. TD = P * r * t / (100 + r * t) = 1650 * 10 / (100 + 10). TD = 16500 / 110 = Rs 150. Step 3: Banker's gain. BG = BD − TD = 165 − 150 = Rs 15. Verification / Alternative check: We can also use BG = TD * r * t / 100: BG = 150 * 10 / 100 = 1500 / 100 = Rs 15, which agrees with our earlier result. This confirms that our TD and BG values are consistent with the rate-time product r * t = 10. Why Other Options Are Wrong: True discount = Rs 165, banker's gain = Rs 0: This would mean BD = TD, which is never true for positive r and t. True discount = Rs 135, banker's gain = Rs 30: Here BD − TD ≠ 165 − 135 = 30, but this pair does not match the required TD formula for the given P. True discount = Rs 120, banker's gain = Rs 45: The implied BD would be 165, but TD = 120 does not satisfy the correct ratio. True discount = Rs 155, banker's gain = Rs 10: Again, this does not match the exact formula-based computation for the given BD. Common Pitfalls: Students often confuse the roles of BD and TD and may try to treat TD as simple interest on the present worth without using the formula TD = P * r * t / (100 + r * t). Another mistake is to assume BG is a fixed percentage of BD, which is not generally true unless derived from the given data. Final Answer: The true discount is Rs 150 and the banker's gain is Rs 15.

Question 13

The true discount on a certain sum due 6 months (half a year) hence at 15% per annum is Rs 120. What is the banker's discount on the same sum for the same time and at the same rate?

Accepted Answer

Introduction: This is a classic relationship question between true discount (TD) and banker's discount (BD). We are given the true discount and asked to compute the banker's discount for the same sum, the same time, and the same simple interest rate. Given Data / Assumptions: True discount TD = Rs 120. Time t = 6 months = 0.5 year. Rate r = 15% per annum. We must find banker's discount BD. Concept / Approach: For a bill with face value P, at rate r and time t: TD = P * r * t / (100 + r * t). BD = P * r * t / 100. The ratio BD / TD is: BD / TD = (100 + r * t) / 100. Therefore: BD = TD * (100 + r * t) / 100. We will compute r * t, plug it into this formula, and obtain BD directly without needing the actual face value P. Step-by-Step Solution: Step 1: Compute r * t. r * t = 15 * 0.5 = 7.5. Step 2: Use BD = TD * (100 + r * t) / 100. BD = 120 * (100 + 7.5) / 100. BD = 120 * 107.5 / 100. BD = 120 * 1.075 = Rs 129. Verification / Alternative check: We can check the ratio: BD / TD = 129 / 120 = 1.075. From theory, BD / TD = (100 + r * t) / 100 = (100 + 7.5) / 100 = 107.5 / 100 = 1.075. The two values match, confirming our calculation. Why Other Options Are Wrong: Rs 50 and Rs 100: These are too small and do not satisfy the theoretical ratio between BD and TD. Rs 160: Too large; it would imply an incorrect effective (100 + r * t) value. Rs 120: This would make BD equal to TD, which is impossible for positive rate and time. Common Pitfalls: A frequent error is to treat BD and TD as equal or to attempt to compute BD using only simple interest on present worth. The safe method is always to use the ratio BD / TD = (100 + r * t) / 100 once TD is given. Final Answer: The banker's discount on the bill is Rs 129.

Question 14

The banker's discount on Rs 1800 at 12% per annum simple interest is equal to the true discount on Rs 1872 for the same time and at the same rate. Find the time period for which the amounts are due.

Accepted Answer

Introduction: This problem compares a banker's discount on one sum with the true discount on another sum, at the same rate and for the same time. Using the formulas for BD and TD, we can form an equation and solve for the time period t. Given Data / Assumptions: Sum 1 (for BD) P1 = Rs 1800. Sum 2 (for TD) P2 = Rs 1872. Rate r = 12% per annum. Time t = ? years (same for both). Banker's discount on P1 equals true discount on P2. Concept / Approach: For the first sum P1: BD = P1 * r * t / 100. For the second sum P2: TD = P2 * r * t / (100 + r * t). Given that BD = TD, we set these expressions equal and solve for t. Once t (in years) is found, we convert it into months. Step-by-Step Solution: Step 1: Write the equality. P1 * r * t / 100 = P2 * r * t / (100 + r * t). For r = 12%, P1 = 1800, P2 = 1872. 1800 * 12 * t / 100 = 1872 * 12 * t / (100 + 12 * t). Simplify left side: 1800 * 12 / 100 = 216. So 216 * t = 1872 * 12 * t / (100 + 12 * t). Cancel t (t ≠ 0): 216 = 1872 * 12 / (100 + 12 * t). Compute numerator: 1872 * 12 = 22464. So 216 * (100 + 12 * t) = 22464. 100 + 12 * t = 22464 / 216 = 104. 12 * t = 104 − 100 = 4 ⇒ t = 4 / 12 = 1/3 year. Convert to months: (1/3) year = 4 months. Verification / Alternative check: Substitute t = 1/3 into both expressions: BD = 1800 * 12 * (1/3) / 100 = 1800 * 4 / 100 = Rs 72. TD = 1872 * 12 * (1/3) / (100 + 12 * (1/3)) = 1872 * 4 / (100 + 4) = 7488 / 104 = Rs 72. Since BD = TD = Rs 72, the time t = 4 months is validated. Why Other Options Are Wrong: 7 months, 6 months, 3 months, 5 months: None of these values, when converted to years and substituted into the formulas, make BD equal TD. Common Pitfalls: Students often forget to cancel t correctly or mis-handle the algebra when solving for t. Another mistake is to assume that the time must be a whole year, which is not necessary; here it is a fraction of a year, corresponding to 4 months. 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 simple interest is Rs 6. What is the true discount on this bill?

Accepted Answer

Introduction: In this question, the banker's gain (BG) is given and we are asked to find the true discount (TD). Using the relationship between BG and TD in banker's discount problems, we can solve this quickly without knowing the face value explicitly. Given Data / Assumptions: Banker's gain BG = Rs 6. Rate r = 12% per annum. Time t = 1 year. We must find the true discount TD. Concept / Approach: For a bill with face value P at rate r for time t: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). BG = BD − TD. There is a useful relationship: BG = TD * r * t / 100. Given BG, and knowing r and t, we can solve directly for TD. Step-by-Step Solution: Step 1: Compute r * t. r * t = 12 * 1 = 12. Step 2: Use BG = TD * r * t / 100. 6 = TD * 12 / 100. TD = 6 * 100 / 12. TD = 600 / 12 = Rs 50. Verification / Alternative check: If TD = Rs 50 and r * t = 12, then the banker's gain should be: BG = TD * r * t / 100 = 50 * 12 / 100 = 600 / 100 = Rs 6, which matches the given value, confirming our answer. Why Other Options Are Wrong: Rs 62, Rs 58, Rs 47, Rs 60: None of these values satisfy the exact relation BG = TD * 12 / 100 = 6. Any other TD would either give a larger or smaller banker's gain than Rs 6. Common Pitfalls: A common error is to assume BG is simply BD * (some fraction) without recalling the precise formula. Another mistake is trying to find the face value first, which is unnecessary and leads to more complicated algebra. Final Answer: The true discount on the bill is Rs 50.

Question 16

The true discount on a bill of face value Rs 540 is Rs 90. What is the banker's discount on this bill for the same rate and time?

Accepted Answer

Introduction: We are told the true discount (TD) on a bill and asked to compute the banker's discount (BD). This is a typical banker's discount question that uses the relationship between the present worth, the face value, and the true discount, and then uses the ratio between BD and TD. Given Data / Assumptions: Face value P = Rs 540. True discount TD = Rs 90. Rate and time are the same for both BD and TD (exact values need not be found explicitly). Concept / Approach: Let r be the rate percent and t be the time in years. Then: TD = P * r * t / (100 + r * t). BD = P * r * t / 100. Let x = r * t. Then: TD = P * x / (100 + x). We can find x from the TD equation and then compute BD = P * x / 100. Step-by-Step Solution: Step 1: Set TD equation. 90 = 540 * x / (100 + x). 90(100 + x) = 540x. 9000 + 90x = 540x. 9000 = 450x ⇒ x = 9000 / 450 = 20. So r * t = x = 20. Step 2: Compute BD. BD = P * x / 100 = 540 * 20 / 100. BD = 540 * 0.2 = Rs 108. Verification / Alternative check: We can also check the ratio BD / TD: BD / TD = 108 / 90 = 1.2. From theory: BD / TD = (100 + x) / 100 = (100 + 20) / 100 = 120 / 100 = 1.2. Both values match, confirming the correctness of BD = Rs 108. Why Other Options Are Wrong: Rs 115, Rs 100, Rs 120, Rs 96: Each of these amounts would change the ratio BD / TD away from 1.2 and thus violate the standard formula relationship between BD, TD, and r * t. Common Pitfalls: Learners sometimes confuse face value with present worth, or they try to guess the rate and time separately instead of working with x = r * t. It is more efficient to solve for x and use it directly to compute BD. Final Answer: The banker's discount on the bill is Rs 108.

Question 17

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

Accepted Answer

Introduction: Here we are given the banker's gain over a 3-year period at 12% simple interest and must determine the banker's discount. This involves using the link between banker's gain and true discount, and then connecting true discount to banker's discount. Given Data / Assumptions: Banker's gain BG = Rs 270. Rate r = 12% per annum. Time t = 3 years. We must find the banker's discount BD. 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). BG = BD − TD. The key relation: BG = TD * r * t / 100. Hence: TD = BG * 100 / (r * t). Once TD is known, we use: BD / TD = (100 + r * t) / 100, or simply: BD = TD * (100 + r * t) / 100. Step-by-Step Solution: Step 1: Compute r * t. r * t = 12 * 3 = 36. Step 2: Find TD from BG. BG = TD * r * t / 100 ⇒ 270 = TD * 36 / 100. TD = 270 * 100 / 36 = 27000 / 36 = Rs 750. Step 3: Compute BD from TD. BD = TD * (100 + r * t) / 100. BD = 750 * (100 + 36) / 100 = 750 * 136 / 100. BD = 750 * 1.36 = Rs 1020. Verification / Alternative check: We can confirm BG: BG = BD − TD = 1020 − 750 = Rs 270, which is exactly the given banker's gain. This check confirms that our BD value is correct. Why Other Options Are Wrong: Rs 1315, Rs 1150, Rs 980, Rs 900: None of these values, when used as BD with TD = 750, produce a banker's gain of exactly Rs 270, nor do they satisfy the exact rate-time relationships. Common Pitfalls: Many learners try to guess BD directly or incorrectly assume BG is a certain percentage of BD without calculation. It is safer to compute TD from BG first using BG = TD * r * t / 100 and then compute BD using BD = TD * (100 + r * t) / 100. Final Answer: The banker's discount on the sum is Rs 1020.

Question 18

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

Accepted Answer

Introduction: This question involves the banker's gain (BG) on a bill at a given rate and time, and asks us to find the present worth (PW) of the sum. We must use the relationships between banker's gain, true discount, face value, and present worth in simple-interest discounting. Given Data / Assumptions: Banker's gain BG = Rs 200. Rate r = 5% per annum. Time t = 4 years. We must find: Present worth PW. Concept / Approach: Let P be the face value of the bill. Then: TD = true discount. BD = banker's discount. BG = BD − TD. Key formulas: BG = TD * r * t / 100. TD = P * r * t / (100 + r * t). PW = P − TD. We will first find TD from BG, then find P using the TD formula, and finally subtract TD from P to obtain the present worth PW. Step-by-Step Solution: Step 1: Compute r * t. r * t = 5 * 4 = 20. Step 2: Find TD from BG. BG = TD * r * t / 100 ⇒ 200 = TD * 20 / 100. TD = 200 * 100 / 20 = 200 * 5 = Rs 1000. Step 3: Use TD to find P. TD = P * r * t / (100 + r * t). 1000 = P * 20 / (100 + 20) = P * 20 / 120. 1000 = P / 6 ⇒ P = 1000 * 6 = Rs 6000. Step 4: Present worth PW = P − TD = 6000 − 1000 = Rs 5000. Verification / Alternative check: We can compute BD from P: BD = P * r * t / 100 = 6000 * 20 / 100 = Rs 1200. Then BG should be: BG = BD − TD = 1200 − 1000 = Rs 200, which matches the given banker's gain, confirming that PW = Rs 5000 is correct. Why Other Options Are Wrong: Rs 4500, Rs 4000: Both are lower than the correct present worth and would imply different values of TD and BG. Rs 6000: This is the face value, not the present worth. Rs 5200: Does not satisfy the exact relationships among TD, BD, and BG at 5% for 4 years. Common Pitfalls: Often, learners confuse face value with present worth, or they attempt to find BD directly from BG without first finding TD. Remember, BG is tied most directly to TD through BG = TD * r * t / 100. Once TD is known, the rest of the values follow systematically. Final Answer: The present worth of the sum is Rs 5000.

Question 19

The banker's discount and the true discount on a certain sum at 10% per annum simple interest for the same time are Rs 100 and Rs 80 respectively. Find both the sum (face value) and the time period.

Accepted Answer

Introduction: This question gives both the banker's discount (BD) and the true discount (TD) at a known rate and asks us to determine the face value of the sum and the time period. It nicely illustrates how the ratio BD / TD relates to the product r * t in banker's discount theory. Given Data / Assumptions: Banker's discount BD = Rs 100. True discount TD = Rs 80. Rate r = 10% per annum. Time t = ? years. Face value (sum) P = ?. Concept / Approach: For face value P, rate r and time t: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). Hence: BD / TD = (100 + r * t) / 100. We can use BD / TD to find r * t, then use BD to find P. Finally we solve for t using r. Step-by-Step Solution: Step 1: Use BD / TD. BD / TD = 100 / 80 = 5 / 4. But BD / TD = (100 + r * t) / 100. (100 + r * t) / 100 = 5 / 4. 100 + r * t = 100 * (5 / 4) = 125. r * t = 125 − 100 = 25. Step 2: Since r = 10, we get 10 * t = 25 ⇒ t = 25 / 10 = 2.5 years. Step 3: Use BD to find P. BD = P * r * t / 100 ⇒ 100 = P * 10 * 2.5 / 100. 100 = P * 25 / 100 ⇒ P = 100 * 100 / 25 = Rs 400. Verification / Alternative check: Check TD with P = 400, r = 10, t = 2.5: TD = P * r * t / (100 + r * t) = 400 * 25 / (100 + 25) = 10000 / 125 = Rs 80. BD with P = 400: BD = 400 * 25 / 100 = Rs 100. Both match the given values exactly. Why Other Options Are Wrong: Sum = Rs 400 and Time = 5 years: Time is too long; r * t would be 50, not 25. Sum = Rs 200 and Time = 2.5 years: BD and TD would be half of the required amounts. Sum = Rs 200 and Time = 5 years: Neither BD nor TD matches the given values. Sum = Rs 500 and Time = 2 years: Gives a different pair of BD and TD entirely. Common Pitfalls: Mistakes include adding or subtracting BD and TD in arbitrary ways or trying to guess P and t. The clean route is always via the ratio BD / TD and the formulas that connect them to r * t. Final Answer: The correct combination is Sum = Rs 400 and Time = 2.5 years.

Question 20

A bill for Rs 6000 is drawn on July 14 at 5 months. It is discounted on October 5 at 10% per annum simple interest. Find the banker's discount, the true discount, the banker's gain, and the net amount (money) that the holder of the bill receives.

Accepted Answer

Introduction: This is a comprehensive banker's discount problem involving a dated bill with a term in months, a discount date, and a given annual interest rate. We must carefully determine the time from the date of discount to the bill's due date (including days of grace), then calculate the banker's discount (BD), true discount (TD), banker's gain (BG), and the net amount received by the holder. Given Data / Assumptions: Face value of the bill P = Rs 6000. Bill drawn on = July 14. Term of the bill = 5 months. Rate of interest r = 10% per annum (simple interest). Date of discount = October 5. Bills allow 3 days of grace, so due date is term end + 3 days. Concept / Approach: Steps required: 1. Find the nominal due date: add 5 months to July 14. 2. Add 3 days of grace to get the final due date. 3. Find the number of days between the discount date and the final due date. 4. Convert this into years for simple interest calculations. 5. Compute banker's discount using BD = P * r * t / 100. 6. Compute true discount using TD = P * r * t / (100 + r * t). 7. Banker's gain BG = BD − TD. 8. Amount received = P − BD. Step-by-Step Solution: Step 1: Nominal due date. 5 months after July 14 is December 14. Step 2: Add 3 days of grace. Final due date = December 17. Step 3: Time from discount date (October 5) to final due date (December 17). From October 5 to October 31 = 26 days. November = 30 days. December 1 to 17 = 17 days. Total days = 26 + 30 + 17 = 73 days. Step 4: Convert to years. t = 73 / 365 = 1/5 year (since 73 * 5 = 365). Step 5: Banker's discount. BD = P * r * t / 100 = 6000 * 10 * (1/5) / 100. BD = 6000 * 2 / 100 = Rs 120. Step 6: True discount. Compute r * t: r * t = 10 * (1/5) = 2. TD = P * r * t / (100 + r * t) = 6000 * 2 / (100 + 2). TD = 12000 / 102 ≈ Rs 117.65. Step 7: Banker's gain. BG = BD − TD ≈ 120 − 117.65 ≈ Rs 2.35. Step 8: Amount received by the holder. Net amount received = P − BD = 6000 − 120 = Rs 5880. Verification / Alternative check: We can check the relation BG = TD * r * t / 100: BG ≈ 117.65 * 2 / 100 ≈ 2.353 ≈ Rs 2.35. This is consistent with BD − TD, supporting the calculations. Why Other Options Are Wrong: Option B, C, D, E: All use incorrect values for BD, TD, BG, or the final amount. They do not respect the precise time interval of 73 days, nor the exact formulas for true discount and banker's discount, and thus lead to incorrect net proceeds. Common Pitfalls: Common mistakes include forgetting the 3 days of grace, miscounting the days between the discount date and the due date, or using months as 1/12 year instead of the exact number of days. Another error is assuming TD ≈ BD for short periods, which ignores the subtle difference captured in the formula TD = P * r * t / (100 + r * t). Final Answer: The correct set of values is Banker's discount = Rs 120; true discount ≈ Rs 117.65; banker's gain ≈ Rs 2.35; amount received = Rs 5880.

Banker's Discount Questions

Questions

Practice smarter