Question 1

The true discount on a sum due 2 years hence at 14% p.a. is Rs. 168. What is the sum due?

Accepted Answer

Concept / Approach With simple interest, for an amount A due after t years at r% p.a.: Present worth P = A / (1 + rt/100)True discount TD = A − P = A × (rt)/(100 + rt) Step-by-step calculationHere t = 2, r = 14 ⇒ rt = 28TD = A × 28/(100 + 28) = A × 28/128 = A × 7/32168 = A × 7/32 ⇒ A = 168 × 32/7 = 768 VerificationPresent worth P = 768 × 100/128 = 600; SI on Rs. 600 for 2 years at 14% = 600 × 14 × 2 / 100 = 168 = TD ✔ Final AnswerRs. 768

Question 2

The simple interest on Rs. 750 for 2 years equals the true discount on Rs. 960 due 2 years hence. Find the common rate per annum.

Accepted Answer

Interest side Simple interest = 750 × r × 2 / 100 = 15r True discount side For amount A = 960, time t = 2, rate r%:TD = A × (rt)/(100 + rt) = 960 × (2r)/(100 + 2r) Equate and solve15r = 960 × (2r)/(100 + 2r) ⇒ 15 = 1920/(100 + 2r)100 + 2r = 1920/15 = 128 ⇒ 2r = 28 ⇒ r = 14% VerificationSI = 15r = 210 when r = 14; TD = 960 × (28)/(128) = 210 — equal. Final Answer14%

Question 3

A trader owes a merchant Rs. 10,028 due 1 year hence and wants to settle after 3 months. At 12% simple interest per annum, how much should he pay now?

Accepted Answer

Given data Amount due after 1 year (A) = Rs. 10,028 Settlement occurs 3 months from now, i.e., 9 months before the due date Simple interest rate r = 12% p.a. Concept / Approach Present worth under simple interest for a future amount A due after t years: PW = A / (1 + r × t) Here, t = 9/12 = 0.75 year. Step-by-step calculation1 + r × t = 1 + 0.12 × 0.75 = 1 + 0.09 = 1.09PW = 10,028 / 1.09 = 9,200 Verification / AlternativeAccruing PW to the due date: 9,200 × 1.09 = 10,028 (matches A). Common pitfalls Discounting for the wrong time period (use 9 months, not 3 months). Using compound-interest formulas; the problem states simple interest. Final AnswerCash to be paid now = Rs. 9,200.

Question 4

Rs. 10 is the true discount on a bill of Rs. 110 due after time t. What is the true discount on the same sum due after double the time (rate unchanged)?

Accepted Answer

Given data A = Rs. 110 True discount TD1 = Rs. 10 at time t Find TD2 at time 2t with same rate r Concept / Approach For simple interest, true discount is: TD = A × (r t) / (1 + r t) Let x = r t. Step-by-step calculation10 = 110 × x / (1 + x)10 + 10x = 110x ⇒ 10 = 100x ⇒ x = 0.1At double time: x2 = 2x = 0.2TD2 = 110 × 0.2 / 1.2 = 110 / 6 = 18.33 VerificationPW at t: 110/1.1 = 100 ⇒ TD = 10. At 2t: 110/1.2 = 91.67 ⇒ TD ≈ 18.33. Common pitfalls Assuming TD doubles with time (it increases sublinearly because of the denominator 1 + r t). Final AnswerTrue discount after double time = Rs. 18.33.

Question 5

A watch is bought for Rs. 1,950 in cash and sold for Rs. 2,200 on 1-year credit at 10% simple interest. What is the gain or loss?

Accepted Answer

Given data Cost price (cash) = Rs. 1,950 Credit selling price after 1 year = Rs. 2,200 Rate r = 10% p.a., time t = 1 year Concept / Approach Compare cost with the present worth (PW) of the credit price: PW = A / (1 + r × t) CalculationPW = 2,200 / 1.10 = 2,000Gain = PW − Cost = 2,000 − 1,950 = 50 Common pitfalls Comparing cost with the future amount directly instead of discounting to present worth. Final AnswerGain of Rs. 50.

Question 6

A cow is bought for Rs. 3,000 and sold the same day for Rs. 3,600 with 2 years credit at 10% simple interest. What is the gain?

Accepted Answer

Given data Cost price (cash) = Rs. 3,000 Credit selling price after 2 years = Rs. 3,600 Rate r = 10% p.a., time t = 2 years Approach Compute the present worth (PW) of the credit price and compare with cost. CalculationPW = 3,600 / (1 + 0.10 × 2) = 3,600 / 1.20 = 3,000Gain = PW − Cost = 3,000 − 3,000 = 0 Final AnswerThere is no gain (Rs. 0).

Question 7

The true discount on Rs. 1,760 at 12% simple interest is Rs. 160. After how much time is it due?

Accepted Answer

Given data A = Rs. 1,760 TD = Rs. 160 Rate r = 12% p.a. Concept True discount: TD = A × (r t) / (1 + r t) Let x = r t. Steps160 = 1760 × x / (1 + x)160 + 160x = 1760x ⇒ 160 = 1600x ⇒ x = 0.1r t = 0.1 ⇒ t = 0.1 / 0.12 = 0.833… year = 10 months Final AnswerTime due = 10 months.

Question 8

Rs. 20 is the true discount on Rs. 260 due after time t. What is the true discount on the same sum due after t/2 (rate unchanged)?

Accepted Answer

Given data A = Rs. 260 TD at time t is Rs. 20 Find TD at time t/2 (same r) Concept TD = A × (r t) / (1 + r t). Let x = r t. Steps20 = 260 × x / (1 + x) ⇒ 20 + 20x = 260x ⇒ 20 = 240x ⇒ x = 1/12At t/2: x/2 = 1/24TD = 260 × (1/24) / (1 + 1/24) = 260 × (1/24) × (24/25) = 260/25 = 10.40 Final AnswerTrue discount = Rs. 10.40.

Question 9

Find the present worth of Rs. 1,404 due in two equal half-yearly instalments at 8% p.a. simple interest.

Accepted Answer

Given data Total due = Rs. 1,404 split equally into two instalments Each instalment = Rs. 702 Due times: 0.5 year and 1 year from now Rate r = 8% p.a. Concept / Approach Discount each instalment to present under simple interest and add: PW = Σ [Instalment / (1 + r × ti)] ComputationPW = 702/1.04 + 702/1.08= 675 + 650 = 1,325 Verification Accruing each PW part forward: 675 × 1.04 = 702 and 650 × 1.08 = 702. Final AnswerPresent worth = Rs. 1,325.

Question 10

The true discount on Rs. 2,562 due 4 months hence is Rs. 122. Find the annual simple interest rate.

Accepted Answer

Given data A = Rs. 2,562 TD = Rs. 122 t = 4 months = 1/3 year Concept TD = A × (r t) / (1 + r t). Let y = r/3. Steps122 = 2562 × y / (1 + y)122 + 122y = 2562y ⇒ 122 = (2562 − 122)y = 2440yy = 122/2440 = 0.05 ⇒ r = 3y = 0.15 = 15% p.a. Final AnswerRate = 15% per annum.

Question 11

The true discount on a bill due 9 months hence at 16% p.a. is Rs. 189. Find the amount (face value) of the bill.

Accepted Answer

Given data TD = Rs. 189 Rate r = 16% p.a. Time t = 9 months = 0.75 year Let amount = A Concept TD = A × (r t) / (1 + r t). Stepsr t = 0.16 × 0.75 = 0.12189 = A × 0.12 / 1.12 ⇒ A = 189 × (1.12 / 0.12) = 1,764 Final AnswerBill amount = Rs. 1,764.

Question 12

Goods cost Rs. 600 and are sold for Rs. 688.50 on 9 months credit with a 2% true gain. Find the annual simple interest rate.

Accepted Answer

Given data Cost price = Rs. 600 Credit selling price after 9 months = Rs. 688.50 True gain = 2% of cost = Rs. 12 Therefore, required present worth (PW) of proceeds = Rs. 612 Let annual simple interest rate = r Concept / Approach PW = A / (1 + r × t) with t = 9/12 = 0.75 year. StepsSet 688.50 / (1 + 0.75r) = 6121 + 0.75r = 688.50 / 612 = 1.1250.75r = 0.125 ⇒ r = 0.125 / 0.75 = 0.1666… = 16⅔% Verification At r = 16⅔%, PW = 688.50 / 1.125 = 612; gain = 612 − 600 = 12 = 2%. Final AnswerAnnual rate = 16⅔% p.a.

Question 13

The simple interest and the true discount on a certain sum for the same time and rate are ₹85 and ₹80. What is the sum due (face value)?

Accepted Answer

Concept / Approach The given 'simple interest' here is effectively the Banker's Discount (BD) because it is computed on the sum due (face value). The True Discount (TD) is the difference between the face value and the present worth. For the same time and rate with simple interest: BD = A × R, TD = A × R / (1 + R) which implies a handy result: A = (BD × TD) / (BD − TD) where A is the face value (sum due) and R is the time–rate product. Step-by-step calculation BD = ₹85, TD = ₹80 A = (BD × TD) / (BD − TD) = (85 × 80) / (85 − 80) = 6800 / 5 = ₹1,360 Verification Compute present worth (PW) and the effective R to check consistency: PW = A − TD = 1360 − 80 = ₹1,280 R = TD / PW = 80 / 1280 = 0.0625 = 6.25% Then BD should be A × R: BD = 1360 × 0.0625 = ₹85 which matches the given value. Common pitfalls Confusing BD with TD—BD is on face value; TD equals the interest on present worth. Forgetting to use the identity A = (BD × TD)/(BD − TD) for quick computation. Mixing up present worth with face value while plugging numbers. Final Answer ₹1,360

Question 14

Find the present worth of ₹2310 due 2½ years hence at 15% per annum (simple interest).

Accepted Answer

Concept / Approach Under simple interest, present worth (PW) is the discounted value of a future amount A using the time–rate product R = r × t: PW = A / (1 + R) Here, r = 15% per annum and t = 2½ years. Step-by-step calculation A = ₹2310 r = 15% = 0.15, t = 2.5 → R = 0.15 × 2.5 = 0.375 PW = 2310 / (1 + 0.375) = 2310 / 1.375 Note: 1.375 = 11/8, so PW = 2310 × (8/11) = ₹1,680 Verification True Discount (TD) = A − PW = 2310 − 1680 = ₹630. Also, interest on PW for R = 0.375 should equal TD: I = PW × R = 1680 × 0.375 = ₹630 Matches perfectly. Common pitfalls Using compound discount instead of simple interest discount. Incorrectly converting 2½ years (always use 2.5 in decimals). Rounding too early; here it divides cleanly via 8/11. Final Answer ₹1,680

Question 15

A scooter has two sale offers: ₹12,000 cash or ₹12,880 payable after 8 months. If money costs 18% per annum (simple interest), which offer is better?

Accepted Answer

Concept / Approach Compare the present worth (PW) of the credit price to the cash price using simple interest. For amount A due after time t at annual rate r: PW = A / (1 + r × t) Here, t = 8/12 years and r = 18% p.a. Step-by-step calculation A = ₹12,880 r = 0.18, t = 8/12 = 2/3 → r × t = 0.18 × (2/3) = 0.12 PW = 12,880 / (1 + 0.12) = 12,880 / 1.12 = ₹11,500 Decision The credit option is worth ₹11,500 today, which is less than the ₹12,000 cash price. Therefore, taking cash ₹12,000 is financially better. Common pitfalls Comparing future amounts directly to cash without discounting. Using 18% straight on 8 months (must scale by 8/12). Rounding 12,880/1.12 incorrectly (it is exactly 11,500). Final Answer Cash offer of ₹12,000 is better.

Question 16

Relating simple interest to true discount: If the simple interest on Rs. 50 at 4 1/2% per annum equals the true discount on Rs. 59 for the same time at the same rate, for how many years is the latter sum due?

Accepted Answer

Introduction / Context: True discount (TD) on a sum due at a future time is different from simple interest (SI) on a present sum, though both use the same rate and time. The relationship TD = S * r * t / (1 + r * t) is crucial, where S is the future due amount. This problem equates SI on a smaller principal to TD on a larger amount to find the time. Given Data / Assumptions: SI on Rs. 50 at r = 4.5% for time t equals TD on Rs. 59 for same r and t. r = 4.5% per annum = 0.045; t in years. Concept / Approach: SI on Rs. 50 = 50 * r * t. TD on Rs. 59 = 59 * r * t / (1 + r * t). Equate them and solve for r * t, then for t since r is known. Step-by-Step Solution: 50 * r * t = 59 * r * t / (1 + r * t).Cancel r * t (nonzero) ⇒ 50 = 59 / (1 + r * t).1 + r * t = 59/50 = 1.18 ⇒ r * t = 0.18.t = 0.18 / 0.045 = 4 years. Verification / Alternative check: Substitute back: r * t = 0.18 ⇒ TD factor = 0.18/1.18 = 0.152542…; 59 * 0.152542… ≈ 9; SI on 50 = 50 * 0.045 * 4 = 9; they match. Why Other Options Are Wrong: 2, 3, 5, and 6 years produce different r * t values that break the equality. Common Pitfalls: Using TD = S * r * t (ignoring the denominator), or forgetting that TD is computed on the future amount. Final Answer: 4 years

Question 17

Find the sum from SI and TD: The simple interest (SI) and the true discount (TD) on a certain future sum for a given time and rate are Rs. 25 and Rs. 20 respectively. What is the face value (sum due)?

Accepted Answer

Introduction / Context: For a future sum S due after a time t at rate r, the true discount (TD) relates to simple interest (SI) through standard formulas. A useful identity is S = (SI * TD) / (SI − TD) when both SI and TD refer to the same future sum S, same rate, and same time. Given Data / Assumptions: SI = Rs. 25 (on the due sum for the given t and r). TD = Rs. 20 (on the same due sum for the same t and r). Concept / Approach: Using the known relationship S = (SI * TD) / (SI − TD), substitute SI = 25 and TD = 20 to compute S directly without solving for r or t explicitly. Step-by-Step Solution: S = (25 * 20) / (25 − 20) = 500 / 5 = Rs. 100. Verification / Alternative check: From TD = S * r * t / (1 + r * t) and SI = S * r * t, the above identity is derived by algebraic elimination of r * t, ensuring consistency. Why Other Options Are Wrong: Rs. 250, Rs. 200, and Rs. 500 correspond to misapplications of the formula; Rs. 400 is also inconsistent. Common Pitfalls: Using SI − TD or SI + TD as the sum, or confusing present worth with the face value. Final Answer: Rs. 100

Question 18

Difference between SI and TD: The difference between simple interest and true discount on a certain sum due after 6 months at 6% per annum is Rs. 27. Find the face value (sum due).

Accepted Answer

Introduction / Context: For a sum S due after time t at rate r, simple interest (SI) on S and true discount (TD) on S have a known difference: SI − TD = S * (r * t)^2 / (1 + r * t). This relation enables solving for S directly when SI − TD, r, and t are known. Given Data / Assumptions: Time t = 6 months = 0.5 years. Rate r = 6% per annum = 0.06. Difference SI − TD = Rs. 27. Concept / Approach: Use SI − TD = S * (r t)^2 / (1 + r t). Compute r t, then rearrange to S = (SI − TD) * (1 + r t) / (r t)^2. Step-by-Step Solution: r t = 0.06 * 0.5 = 0.03.SI − TD factor = (0.03)^2 / (1 + 0.03) = 0.0009 / 1.03 ≈ 0.000873786.S = 27 / 0.000873786 ≈ Rs. 30,900. Verification / Alternative check: Plug S ≈ 30,900 back: SI − TD ≈ 30,900 * 0.000873786 ≈ 27. It matches the given difference. Why Other Options Are Wrong: Rs. 39,000, Rs. 30,600, and Rs. 20,900 deviate from the precise computation; Rs. 25,000 is also incorrect. Common Pitfalls: Forgetting to convert months to years; squaring r t incorrectly; using SI + TD instead of the required difference formula. Final Answer: Rs. 30900

Question 19

Cash vs. credit sale with interest: A man buys a watch for Rs. 195 in cash and sells it for Rs. 220 on 1-year credit. If the simple interest rate is 10% per annum, does he gain or lose, and by how much (in rupees)?

Accepted Answer

Introduction / Context: To fairly compare a cash purchase with a credit sale involving a stated interest rate, convert the credit sale price to its present worth (discounted value) and compare with the cash cost. The difference shows the real profit or loss. Given Data / Assumptions: Cash cost price (CP) = Rs. 195. Credit sale price after 1 year = Rs. 220. Interest rate r = 10% per annum; time t = 1 year. Concept / Approach: Present worth PW of the credit price = Future value / (1 + r t). Profit = PW − CP. If positive, it is a gain; if negative, a loss. Step-by-Step Solution: PW of Rs. 220 due in 1 year = 220 / (1 + 0.10) = Rs. 200.Real gain = PW − CP = 200 − 195 = Rs. 5. Verification / Alternative check: If the buyer invested Rs. 195 at 10%, it would become Rs. 214.5 in a year. Selling effectively for PW Rs. 200 shows a modest gain over CP; the problem, however, uses PW directly as the benchmark. Why Other Options Are Wrong: Rs. 15 and Rs. 3 do not match the present-worth calculation; “Loses” options invert the sign incorrectly. Common Pitfalls: Comparing 220 directly to 195 without discounting; using the interest on CP rather than discounting the credit price to present. Final Answer: Gains Rs. 5

Question 20

Compute true discount from present worth: What is the true discount if the present worth (PW) is ₹ 600 for a period of 4 years at 4% per annum simple interest? (Find the difference between the amount due and PW.)

Accepted Answer

Introduction / Context: True discount (TD) is the reduction from a future amount due to obtain its present worth under simple interest. If the present worth is known, multiply by (1 + r t) to get the future amount, then subtract PW to get TD. Given Data / Assumptions: PW = ₹ 600. r = 4% per annum; t = 4 years. Concept / Approach: Amount due S = PW * (1 + r t). True discount TD = S − PW. Plug r and t into the formula to compute S and then TD. Step-by-Step Solution: 1 + r t = 1 + 0.04 * 4 = 1 + 0.16 = 1.16.S = 600 * 1.16 = ₹ 696.TD = S − PW = 696 − 600 = ₹ 96. Verification / Alternative check: Using TD formula TD = S * r t / (1 + r t): TD = 696 * 0.16 / 1.16 = 96 confirms the same result. Why Other Options Are Wrong: ₹ 100, ₹ 86, ₹ 84, and ₹ 32 do not match the exact computation for the given PW, rate, and time. Common Pitfalls: Applying the rate directly to PW as if TD = PW * r t (that gives SI on PW, not TD); forgetting to reconstruct S first. Final Answer: ₹ 96

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