Question 1

The banker's discount on a sum due 2 years hence is 11/10 of the true discount. What is the annual rate of interest (simple)?

Accepted Answer

Given data Time t = 2 years; BD = (11÷10) × TD. Concept/Approach For simple interest, BD ÷ TD = 1 + r t where r is the annual rate (in decimals). Step-by-Step calculation 1 + r t = 11 ÷ 10 r × 2 = 1÷10 ⇒ r = 1÷20 = 0.05 Rate = 5% per annum Verification/Alternative True discount formula TD = S r t ÷ (1 + r t); Banker's discount BD = S r t. Their ratio is indeed 1 + r t. Common pitfalls Using compound interest relations; the problem uses simple discount/interest. Final Answer 5% per annum

Question 2

The banker's discount on ₹1600 at 15% p.a. equals the true discount on ₹1680 for the same time and rate. What is the time?

Accepted Answer

Given data BD on ₹1600 at r = 15% for time t equals TD on ₹1680 at same r and t. Concept/Approach Use BD = S r t and TD = S r t ÷ (1 + r t) (simple interest/discount). Step-by-Step calculation BD = 1600 × 0.15 × t = 240 t TD = 1680 × 0.15 × t ÷ (1 + 0.15 t) = 252 t ÷ (1 + 0.15 t) Set 240 t = 252 t ÷ (1 + 0.15 t) ⇒ 240 = 252 ÷ (1 + 0.15 t) 1 + 0.15 t = 252 ÷ 240 = 21 ÷ 20 = 1.05 0.15 t = 0.05 ⇒ t = 1÷3 year = 4 months Verification/Alternative At t = 1÷3, BD = 1600 × 0.15 × 1÷3 = ₹80; TD on 1680 = 1680 × 0.15 × 1÷3 ÷ (1 + 0.05) = 84 ÷ 1.05 = ₹80. Final Answer 4 months

Question 3

On a bill due 4 months hence at 15% p.a., the banker's discount is ₹420. What is the true discount?

Accepted Answer

Given data Time t = 4 months = 1÷3 year; rate r = 15% p.a. BD = ₹420. Concept/Approach BD = S r t ⇒ find face value S, then TD = S r t ÷ (1 + r t). Step-by-Step calculation BD = S × 0.15 × (1÷3) = S × 0.05 = 420 S = 420 ÷ 0.05 = ₹8400 TD = S × 0.05 ÷ (1 + 0.05) = 420 ÷ 1.05 = ₹400 Verification/Alternative Relation BD ÷ TD = 1 + r t = 1.05 ⇒ TD = BD ÷ 1.05 = 420 ÷ 1.05 = ₹400. Final Answer ₹400

Question 4

Present worth is Rs. 576 and banker's gain is Rs. 16. Find the true discount.

Accepted Answer

Given data Present worth (PW) = Rs. 576 Banker's gain (BG) = Rs. 16 True discount (TD) = ? Concept/Approach Key relation: BG = TD2 / PW ⇒ TD = √(BG × PW) Step-by-step calculation TD = √(16 × 576)= √(9216) = Rs. 96 Verification/Alternative BG should equal TD2 / PW = 962 / 576 = 9216 / 576 = 16 (matches). Common pitfalls Confusing banker's discount (BD) with banker's gain (BG); BG = BD − TD, not BD itself. Final Answer Rs. 96

Question 5

Banker's discount for 1½ years is Rs. 558 and true discount for 2 years is Rs. 600. Find the annual rate of interest.

Accepted Answer

Given data BD (1.5 years) = Rs. 558 TD (2 years) = Rs. 600 Rate r = ? per annum; Sum due (A) = unknown Concept/Approach Use definitions under simple interest:BD = A × r × t / 100TD = A × r × t / (100 + r t)Form two equations and eliminate A. Step-by-step calculation BD: A × r × 1.5 / 100 = 558 ⇒ A r = 558 × 100 / 1.5 = 37,200TD: A × (2r) / (100 + 2r) = 600 ⇒ 600(100 + 2r) = 2 A r600(100 + 2r) = 2 × 37,200 = 74,40060,000 + 1,200 r = 74,400 ⇒ r = 12% Verification/Alternative With r = 12%, TD (2y): A × 24 / (100 + 24) = A × 24/124 = 600 ⇒ A = 3,100; BD (1.5y): 3100 × 18/100 = 558 (checks). Common pitfalls Plugging BD formula for TD or mixing the times 1.5 and 2 incorrectly. Final Answer 12%

Question 6

The banker's gain on a sum due 1½ years hence is 3/25 of the banker's discount. Find the rate of interest per annum.

Accepted Answer

Given data Time t = 1½ years (assumption clarified from the wording) BG = (3/25) × BD Rate r = ? Concept/Approach Let x = r t (in percent-years). A standard relation is:BG / BD = x / (100 + x)because BD = A x/100 and BG = BD − TD with TD = A x/(100 + x). Step-by-step calculation x / (100 + x) = 3/2525x = 300 + 3x ⇒ 22x = 300 ⇒ x = 150/11r = x / t = (150/11) / 1.5 = (150/11) × (2/3) = 100/11% ≈ 9.09% Verification/Alternative Compute BG/BD with r = 100/11% and t = 1.5 ⇒ x = 150/11; BG/BD = x/(100+x) = (150/11) / (1250/11) = 3/25 (checks). Common pitfalls Using BG/TD instead of BG/BD. Note BG/TD = x/100 while BG/BD = x/(100 + x). Final Answer 9.09% per annum

Question 7

Present worth is Rs. 1600 and true discount is Rs. 160. Find the banker's gain.

Accepted Answer

Given data PW = Rs. 1600 TD = Rs. 160 BG = ? Concept/Approach Use the identity BG = TD2 / PW. Step-by-step calculation BG = 1602 / 1600 = 25,600 / 1600 = Rs. 16 Verification/Alternative Also, BD = TD + BG = 160 + 16 = 176; consistency checks can be made with the implied rate-time if needed. Common pitfalls Applying BG = BD − TD without first finding BD; the squared relation is faster here. Final Answer Rs. 16

Question 8

Banker's gain for a sum due 2 years hence at 10% p.a. is Rs. 24. Find the present worth.

Accepted Answer

Given data t = 2 years, r = 10% p.a. ⇒ r t = 20 BG = Rs. 24 PW = ? Concept/Approach Since TD / PW = r t / 100 = 20/100 = 0.2 we have TD = 0.2 × PW. Also BG = TD2 / PW. Step-by-step calculation BG = (0.2 PW)2 / PW = 0.04 PW0.04 PW = 24 ⇒ PW = Rs. 600 Verification/Alternative Then TD = 0.2 × 600 = 120; BG = TD2 / PW = 14,400 / 600 = 24 (checks). Common pitfalls Confusing PW with sum due (face value A). Here we only need PW. Final Answer Rs. 600

Question 9

Banker's gain on a bill due 1 year hence at 12% p.a. is Rs. 6. Find the true discount.

Accepted Answer

Given data t = 1 year, r = 12% ⇒ r t = 12 BG = Rs. 6 TD = ? Concept/Approach Use the relation BG / TD = r t / 100 ⇒ TD = BG × 100 / (r t). Step-by-step calculation TD = 6 × 100 / 12 = Rs. 50 Verification/Alternative With TD = 50 and r t = 12, BG should be 50 × 12/100 = 6 (matches). Common pitfalls Using BD instead of BG in the ratio with TD; only BG/TD equals r t / 100. Final Answer Rs. 50

Question 10

Present worth is Rs. 800 and true discount is Rs. 36. Find the banker's discount.

Accepted Answer

Given data PW = Rs. 800 TD = Rs. 36 BD = ? Concept/Approach First find banker's gain using BG = TD2 / PW, then BD = TD + BG. Step-by-step calculation BG = 362 / 800 = 1296 / 800 = 1.62BD = 36 + 1.62 = Rs. 37.62 Verification/Alternative Implied sum due A = PW + TD = 836; check that BD − TD = BG = 1.62 as computed. Common pitfalls Taking BD = TD; remember BD exceeds TD by the banker's gain. Final Answer Rs. 37.62

Question 11

Banker's gain on a sum due 3 years hence at 12% p.a. is Rs. 270. Find the banker's discount.

Accepted Answer

Given data t = 3 years, r = 12% ⇒ r t = 36 BG = Rs. 270 BD = ? Concept/Approach Use BG = A (r t)2 / (100(100 + r t)) to find the sum due A, then BD = A × r t / 100. Step-by-step calculation BG = A × 362 / (100 × 136) = A × 1296 / 13,600 = A × 81 / 850270 = A × 81 / 850 ⇒ A = 270 × 850 / 81 = 8500 / 3BD = A × 36 / 100 = (8500 / 3) × 0.36 = Rs. 1,020 Verification/Alternative Compute TD = BD − BG = 1020 − 270 = 750; then PW = A − TD = 8500/3 − 750 = 1600/3; check BG = TD2/PW = 270. Common pitfalls Using PW instead of A in the BD formula; BD is computed on the face value A. Final Answer Rs. 1,020

Question 12

True discount on a bill of Rs. 540 is Rs. 90. Find the banker's discount.

Accepted Answer

Given data Face value A = Rs. 540 TD = Rs. 90 BD = ? Concept/Approach Present worth PW = A − TD. Also TD / PW = r t / 100. Once r t is known, BD = A × r t / 100. Step-by-step calculation PW = 540 − 90 = 450r t / 100 = TD / PW = 90 / 450 = 0.2 ⇒ r t = 20BD = A × 20 / 100 = 540 × 0.20 = Rs. 108 Verification/Alternative BG = BD − TD = 108 − 90 = 18; also BG = TD2 / PW = 8100 / 450 = 18 (checks). Common pitfalls Computing BD on PW rather than on A; BD is always on the face value. Final Answer Rs. 108

Question 13

Banker's discount is Rs. 72 and true discount is Rs. 60 for the same sum and time. Find the sum due (face value).

Accepted Answer

Given data BD = Rs. 72 TD = Rs. 60 Sum due A = ? Concept/Approach Ratio BD / TD = (100 + r t) / 100 ⇒ find r t. Then use TD = A × r t / (100 + r t) to get A. Step-by-step calculation BD / TD = 72 / 60 = 6 / 5 = 1.2 ⇒ (100 + r t) / 100 = 1.2 ⇒ r t = 20TD = A × 20 / 120 = A / 660 = A / 6 ⇒ A = Rs. 360 Verification/Alternative BD = A × 20 / 100 = 360 × 0.2 = 72 (matches). Common pitfalls Subtracting BD − TD to get A; remember A = PW + TD, and formulas above are cleaner. Final Answer Rs. 360

Question 14

The banker's gain of a certain sum of money is Rs. 36 and the true discount on the same sum for the same time and at the same rate is Rs. 30. The sum is?

Accepted Answer

Sum = (B.D. x T.D.) / (B.D. - T.D.) = Rs. (36 x 30) / 6 = Rs. 180

Question 15

The banker's gain of a certain sum due 2 years hence at 5% per annum is Rs. 80. The present worth is?

Accepted Answer

T.D. = (B.G. X 100) / (Rate x Time) = Rs. (80 x 100) / (5 x 2) = Rs. 800

Question 16

The present worth of a certain due sometime hence is Rs. 1600 and the true discount is Rs. 160. The banker's gain is?

Accepted Answer

B.G. = (T.D.)2 / P.W. = Rs. (160 x 160) / 1600 = Rs. 16

Question 17

The banker's gain on a sum due 3 years hence at 5% is Rs. 90. The banker's discount is?

Accepted Answer

T.D. = (B.G. x 100) / (R x T) = Rs. (90 x 100) / (5 x 3) = Rs. 600 ∴ B.D. = Rs. (600 +90) = Rs. 690

Question 18

The banker's discount on a bill due 6 months hence at 6% is Rs. 37.08. The true discount is?

Accepted Answer

T.D. = (B.D. X 100) / {100 + (R X T)} = Rs. (37.08 x 100) / {100 + (6 x 1/2)} = Rs. 36

Question 19

The true discount on a bill of Rs. 1860 due after 8 months is Rs. 60. Find the banker's discount?

Accepted Answer

Amount = Rs. 1860. True discount = Rs. 60 ∴ Present worth = Rs. 1860 - Rs. 60 = Rs. 1800 S.I. on Rs. 1800 for 8 months = Rs. 60 ∴ Rate = [(100 x 60) / (1800 x 2/3)]% = 5% ∴ Banker's discount = (1860 x 5 x 2/3)/100 = Rs. 62

Question 20

The present worth of a sum due some times hence is Rs. 576 and the banker's gain is Re. 1. The true discount is?

Accepted Answer

T.D. √(P.W. x B.G. ) = √(576 x 1) = Rs. 24

Banker's Discount Questions