If the true discount on a certain sum due 12 months hence at 25% per annum is Rs 1980, what is the banker's discount on the same sum for the same time and at the same rate?

Rs 2475. Banker's gain (B.G.) equals simple interest on the true discount, for the unexpired time. Here, T.D. = Rs 1980, the rate is R = 25% per annum, and the time is 12 months hence. B.G. = (T.D. x R x 12)/1200. Multiply first: 1980 x 25 x 12 = 594000. Then divide by 1200: 594000/1200 = Rs 495. Banker's discount is always true discount plus banker's gain. B.D. = T.D. + B.G. = 1980 + 495 = Rs 2475. Check: B.D. must exceed T.D., and Rs 2475 is indeed more than Rs 1980.

The banker's discount and the true discount on a sum of money due 4 months hence are Rs 560 and Rs 525 respectively. Find the sum.

Rs 8400. When both the banker's discount and true discount are known, a standard formula gives the sum. Sum = (B.D. x T.D.)/(B.D. - T.D.). Here, B.D. = Rs 560 and T.D. = Rs 525. First find the difference: B.D. - T.D. = 560 - 525 = 35. Multiply B.D. by T.D.: 560 x 525 = 294000. Divide: 294000/35 = Rs 8400. So the sum is Rs 8400. Check: this difference, Rs 35, is exactly the banker's gain on this sum.

The banker's discount on a sum of Rs 1400 due 3 months hence is Rs 42. Find the rate percent per annum.

12%. Banker's discount is simple interest on the sum due, for the unexpired time. So the rate can be found from Rate = (100 x B.D.)/(Sum x T). Here, Sum = Rs 1400, B.D. = Rs 42, and T = 3/12 year. Using T = 3/12, this simplifies to Rate = (1200 x B.D.)/(Sum x 3). Multiply first: 1200 x 42 = 50400. Multiply the sum by the months: 1400 x 3 = 4200. Divide: 50400/4200 = 12%. So the rate is 12% per annum.

The present worth of a bill due sometime hence is Rs 4800, and the true discount on the bill is Rs 360. Find the banker's gain.

Rs 27. Present worth (P.W.) is the sum which, invested now at the given rate, grows to the face value by the due date. Banker's gain equals the true discount squared, divided by the present worth. Here, T.D. = Rs 360 and P.W. = Rs 4800. First square the true discount: 360 x 360 = 129600. Then divide by the present worth: 129600/4800 = Rs 27. Banker's discount always equals true discount plus banker's gain. B.D. = T.D. + B.G. = 360 + 27 = Rs 387.

A bill of Rs 2500 is discounted 12 months before it is due, at 6% per annum. How much money does the holder of the bill receive?

Rs 2350. When a bill is discounted before its due date, the banker deducts the banker's discount upfront. The holder receives only the remaining amount, not the full face value. The face value (Amount) is Rs 2500. The rate is R = 6% per annum. The bill is discounted 12 months before it is due. First find B.D. = (Amount x R x 12)/1200 = 180000/1200 = Rs 150. Money received = Amount - B.D. = 2500 - 150 = Rs 2350. Check: Rs 2350 plus the discount of Rs 150 equals the face value, Rs 2500.

Banker's discount is defined as simple interest on which value, for the unexpired time?

The face value (amount) of the bill. A bill has a face value, the amount due on its due date. Banker's discount (B.D.) is simple interest calculated on this face value. It is charged for the unexpired time, from the date of discounting to the due date. This differs from true discount, which is interest on the present worth instead. So B.D. is always worked out on the face value, never on the present worth.

The banker's discount and the true discount on a sum of money due 8 months hence are Rs 2450 and Rs 2100 respectively. Find the sum.

Rs 14700. When both the banker's discount and true discount are known, a standard formula gives the sum. Sum = (B.D. x T.D.)/(B.D. - T.D.). Here, B.D. = Rs 2450 and T.D. = Rs 2100. First find the difference: B.D. - T.D. = 2450 - 2100 = 350. Multiply B.D. by T.D.: 2450 x 2100 = 5145000. Divide: 5145000/350 = Rs 14700. So the sum is Rs 14700. Check: this difference, Rs 350, is exactly the banker's gain on this sum.

Banker’s Discount: Formulas, Shortcuts and Solved Examples

A banker’s discount is the cut a bank takes when it buys a bill before the due date. It is simple interest on the full face value.

This lesson gives each formula and the reason it works. It then solves three examples and lists the traps that cost marks.

The idea in brief

BD: simple interest on the face value.
TD: simple interest on the present worth.
BG: BD − TD = TD²/PW.
Sum: BD × TD / (BD − TD).
Cash: face value − BD.

What Banker’s Discount Means

What is a banker’s discount?

A bill is a promise to pay a fixed amount on a future date. A bank may buy it early and pay you less than its face value.

The cut the bank takes is the banker’s discount. It is simple interest on the face value, for the time left.

Which terms must you know?

Face value (A): the amount due on the due date. Present worth (PW): the sum that grows to A by then. True discount (TD): A − PW.Banker’s discount (BD): simple interest on A. Banker’s gain (BG): BD − TD.

Sources: NCERT Class 8 Mathematics, Comparing Quantities.

Check 1

Banker’s discount is simple interest on the present worth.

Core Formulas and Why They Work

How do you find the banker’s discount?

BD = A × R × T / 100. Here R is the yearly rate and T is the time in years.

Why it works: it is the plain simple-interest rule, applied to the face value.

The face value splits into present worth and true discount. True discount is the interest on the present worth.

How do you find the true discount?

TD = A × R × T / (100 + R × T).

Why it works: the present worth PW earns TD as interest. So A = PW + PW × R × T/100, which gives this result.

How do the three discounts connect?

BD = TD + BG.BG = TD × R × T / 100 = TD² / PW.

Why it works: BD is interest on A, and TD is interest on PW. The gap, A − PW, equals TD. So the extra interest is interest on TD.

How do you find the sum from BD and TD?

Sum = (BD × TD) / (BD − TD).

Why it works: BD − TD is BG, and BG = TD²/PW. So PW = TD²/BG. Add TD to get A. The two forms are the same.

How much does the holder receive?

Cash received = A − BD.

Why it works: the bank takes its discount first and hands over the rest.

Banker’s gain is the small extra the banker keeps. It is the interest on the true discount.

Check 2

Banker’s discount is always more than the true discount.

Three Solved Examples

Easy: a bill of Rs 4000 is discounted 9 months early at 8% a year. What does the holder receive?

T = 9/12 = 0.75 year.

BD = 4000 × 8 × 0.75 / 100 = Rs 240.

Cash received = 4000 − 240 = Rs 3760.

Medium: the true discount is Rs 200 and the present worth is Rs 2000. Find the banker’s gain and banker’s discount.

BG = TD² / PW = 200 × 200 / 2000 = Rs 20.

BD = TD + BG = 200 + 20 = Rs 220.

The banker’s gain is Rs 20.

Exam level: BD is Rs 330 and TD is Rs 300, six months before the due date. Find the sum and the rate.

Sum = 330 × 300 / (330 − 300) = 99000 / 30 = Rs 3300.

BD is simple interest on 3300 for half a year. So 330 = 3300 × R × 0.5 / 100.

That gives R = 20% a year.

Check: TD = 3300 × 20 × 0.5 / 110 = 300.

Step by step: a bill of Rs 2100 is due in 6 months. The rate is 10% a year. Find BD, TD and BG.

Step 1: BD is simple interest on the face value. BD = 2100 × 10 × 0.5/100 = Rs 105.

Step 2: present worth = 2100 × 100/(100 + 5) = Rs 2000.

Step 3: TD = 2100 − 2000 = Rs 100.

Step 4: BG = BD − TD = Rs 5.

Check: BG is also the interest on TD. 100 × 10 × 0.5/100 = 5.

Check 3

A sum of Rs 3300 with BD Rs 330 and TD Rs 300 has a banker’s gain of Rs 30.

Common Traps and Shortcuts

Which traps cost marks?

Using the present worth as the base for BD. BD always uses the face value.

Leaving the time in months. Convert it to years.

Reversing BD and TD. The larger one is the banker’s discount.

Forgetting to subtract BD from the face value when the question asks for cash received.

Which shortcuts save time?

If BD and TD are given, subtract first. That difference is the banker’s gain.

If TD and PW are given, BG = TD²/PW. Square, then divide.

Keep the rate and time together as R × T/100 and reuse it.

Check that your BD is larger than your TD.

How do you keep BD, TD and BG straight?

BD is always the larger of the two. It is interest on the face value. TD is interest on the present worth.

Banker’s gain is the difference, BG = BD − TD. It is also the interest on TD for the same time and rate.

If a question gives BD and TD, subtract to get BG. If it gives BG and the rate and time, find TD first.

Always convert the time to years before you use the rate.

Check 4

If TD is Rs 360 and PW is Rs 4800, then BG is Rs 27.

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