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 calculation Here t = 2, r = 14 ⇒ rt = 28TD = A × 28/(100 + 28) = A × 28/128 = A × 7/32168 = A × 7/32 ⇒ A = 168 × 32/7 = 768 Verification Present worth P = 768 × 100/128 = 600; SI on Rs. 600 for 2 years at 14% = 600 × 14 × 2 / 100 = 168 = TD ✔ Final Answer Rs. 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 solve 15r = 960 × (2r)/(100 + 2r) ⇒ 15 = 1920/(100 + 2r)100 + 2r = 1920/15 = 128 ⇒ 2r = 28 ⇒ r = 14% Verification SI = 15r = 210 when r = 14; TD = 960 × (28)/(128) = 210 — equal. Final Answer 14%

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 calculation 1 + r × t = 1 + 0.12 × 0.75 = 1 + 0.09 = 1.09PW = 10,028 / 1.09 = 9,200 Verification / Alternative Accruing 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 Answer Cash 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 calculation 10 = 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 Verification PW 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 Answer True 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) Calculation PW = 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 Answer Gain 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. Calculation PW = 3,600 / (1 + 0.10 × 2) = 3,600 / 1.20 = 3,000Gain = PW − Cost = 3,000 − 3,000 = 0 Final Answer There 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. Steps 160 = 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 Answer Time 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. Steps 20 = 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 Answer True 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)] Computation PW = 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 Answer Present 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. Steps 122 = 2562 × y / (1 + y)122 + 122y = 2562y ⇒ 122 = (2562 − 122)y = 2440yy = 122/2440 = 0.05 ⇒ r = 3y = 0.15 = 15% p.a. Final Answer Rate = 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). Steps r t = 0.16 × 0.75 = 0.12189 = A × 0.12 / 1.12 ⇒ A = 189 × (1.12 / 0.12) = 1,764 Final Answer Bill 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. Steps Set 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 Answer Annual 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

If the interest on Rs. 50 at 4 1/ 2 be equal to the discount on Rs. 59 for the same time and at the same rate when is the latter sum due?

Accepted Answer

∵ 9/2 = [(59 - 50) x 100] / 50 x T ∴ T = 4 years

Question 17

The simple interest and the true discount on a certain sum for a given time and at a given rate are Rs. 25 and Rs. 20 respectively. The sum is?

Accepted Answer

Sum = [(S.I.) x (T.D.)] / [(S.I.) - (T.D.)] = Rs. (25 x 20) / (25 - 20) = Rs. 100

Question 18

The difference between the simple interest and the true discount on a certain sum of money foe 6 months at 6% is Rs. 27. Find the sum?

Accepted Answer

Req. amount = [27 x (100 + 6 x 1/2) x 100] / [(6 x 1/2)2] = (2700 x 103) / 9 = Rs. 30900

Question 19

A man buys a watch for Rs. 195 in cash and sells it for Rs. 220 at a credit of 1 year. If the rate of interest is 10%, the man?

Accepted Answer

P.W. of Rs. 220 due 1 year hence = Rs. (100 x 220) / (100 + 10) = Rs. 200 hence, the man gains Rs. 5

Question 20

What will be the true discount for the present worth of ₹ 600 for a period of 4 yr at 4% per annum rate of interest?

Accepted Answer

Given that, T = 4 yr, R = 4% PW = ₹ 600 and TD = ? According to the formula, (1) TD = (PW x R x T)/100 TD = (600 x 4 x 4)/100 = 6 x 4 x 4 = ₹ 96

True Discount Questions