You owe Rs 6,600 in two years. What lump sum clears the debt today? That fair sum is the present worth. The gap is the true discount.
This article gives the formulas, the shortcuts and five solved examples. Each example says why the method works.
- True discount (TD) is the simple interest on the present worth.
- Present worth (PW) is the sum that grows to the amount by the due date.
- Amount = PW + TD. This holds in every problem.
- PW = 100A / (100 + RT). A is the amount, R the rate, T the time.
- Sum = (SI × TD) / (SI − TD) when simple interest and true discount are both known.
What are present worth and true discount?
What is the present worth of a bill?
A bill of Rs 6,600 is due in two years. Present worth is the sum that, put at interest today, grows to exactly Rs 6,600 by then.
So it is smaller than the bill. Money paid early is worth less than money due later.
Sources: NCERT Class 8 Maths: Comparing Quantities, simple interest and amount; Practice question: formula for present worth.
What is true discount?
True discount (TD) is the amount cut off the bill for paying early. It equals amount minus present worth.
It is also the simple interest on the present worth for the given time and rate.
Sources: NCERT Class 8 Maths: Comparing Quantities, principal, rate and time; Practice question: relation of amount, PW and TD.
Which formulas should you remember?
Let A be the amount due, R the rate percent and T the time in years.
- PW = 100 × A / (100 + R × T)
- TD = A − PW = A × R × T / (100 + R × T)
- TD = PW × R × T / 100, because TD is interest on PW
- A = PW + TD
Sources: NCERT Class 8 Maths: Comparing Quantities; Practice question: PW and TD for a sum due later.
How is true discount different from banker’s discount?
Banker’s discount (BD) is the simple interest on the whole amount. It is a little bigger than TD.
The gap BD − TD is the interest on the true discount. That is why BD is always larger.
Sources: NCERT Class 8 Maths: Comparing Quantities, simple interest and amount.
Which shortcuts save time in exams?
How do you find PW without a long fraction?
Write RT as a percent first. Then PW = A × 100 / (100 + RT).
At 10% for 2 years, RT = 20. So PW = A × 100 / 120 = 5A/6. Learn the ratios below.
- RT = 20 gives PW = 5/6 of A
- RT = 25 gives PW = 4/5 of A
- RT = 50 gives PW = 2/3 of A
- RT = 100 gives PW = 1/2 of A
Sources: Practice question: true discount on Rs 500 due 4 years.
How do you get TD when PW is known?
TD is simple interest on PW. So TD = PW × RT / 100.
If PW = 3700 and RT = 20, TD = 3700 × 20 / 100 = 740. This avoids working back from the amount.
Sources: NCERT Class 8 Maths: Comparing Quantities, principal, rate and time.
How do you find the sum from SI and TD?
SI is interest on the sum. TD is interest on the present worth. So SI − TD is interest on TD.
That gives Sum = (SI × TD) / (SI − TD). Then rate = SI × 100 / (Sum × T).
Sources: Practice question: sum from SI and TD; Practice question: rate from TD and SI.
How do you find the time when TD and PW are given?
From TD = PW × RT / 100, we get T = TD × 100 / (PW × R).
First find PW as amount minus TD. Then put the numbers in.
Check 1
The true discount is always smaller than the simple interest on the amount.
Check 2
Present worth is larger than the amount due.
Check 3
If PW = 100 and RT = 20, then TD = 20.
Check 4
TD is the simple interest on the amount.
How do you solve five typical problems step by step?
Example 1: Find the present worth and true discount of Rs 6,600 due in 2 years at 10 percent.
Step 1. RT = 10 × 2 = 20.
Step 2. PW = 6600 × 100 / 120 = Rs 5,500.
Step 3. TD = 6600 − 5500 = Rs 1,100.
Why this works. The amount is 120 parts. Rs 100 grows to 120 in two years. So PW is 100 of those 120 parts.
Example 2: The true discount on a bill due in 4 years at 5 percent is Rs 740. Find the present worth and the amount.
Step 1. RT = 5 × 4 = 20.
Step 2. PW = TD × 100 / RT = 740 × 100 / 20 = Rs 3,700.
Step 3. Amount = PW + TD = 3700 + 740 = Rs 4,440.
Why this works. TD is 20 percent of PW, since interest is on PW. So PW is TD divided by 0.20.
Example 3: The true discount on a sum due in 5 years is Rs 60. The simple interest on it is Rs 96. Find the sum and the rate.
Step 1. Sum = (96 × 60) / (96 − 60) = 5760 / 36 = Rs 160.
Step 2. Rate = 96 × 100 / (160 × 5) = 12 percent.
Check. SI − TD = 36. Interest on TD = 60 × 12 × 5 / 100 = 36. It matches.
Why this works. The gap of Rs 36 is interest on TD. Dividing SI × TD by that gap gives the sum.
Example 4: The true discount on Rs 7,440 is Rs 1,440 at 6 percent. In how many years is it due?
Step 1. PW = 7440 − 1440 = Rs 6,000.
Step 2. TD = PW × R × T / 100, so 1440 = 6000 × 6 × T / 100 = 360 T.
Step 3. T = 1440 / 360 = 4 years.
Why this works. TD is just simple interest on PW. So the usual simple-interest time formula applies.
Example 5: Find the present worth of Rs 7,198 due in 3 years at 6 percent.
Step 1. RT = 6 × 3 = 18.
Step 2. PW = 7198 × 100 / 118 = 61 × 100 = Rs 6,100.
Step 3. TD = 7198 − 6100 = Rs 1,098.
Why this works. 7198 is 118 × 61. So the division is exact. Spot such factors before you divide.
How do you check your answer?
What quick check catches big mistakes?
Add PW and TD. The total must equal the amount. In Example 1, 5500 + 1100 = 6600.
Also check the size. PW must be less than the amount. TD must be less than the simple interest on the amount.
Check 5
Amount = present worth + true discount.
Check 6
Sum = SI + TD.
Exam traps
Base trap. TD is interest on PW, not on the amount. Interest on the amount is banker’s discount.
Divide trap. The divisor is 100 plus RT, never 100 minus RT.
Order trap. When TD is given, PW = TD × 100 / RT. Add TD to get the amount.
Rate trap. Read R as percent per year. Multiply by T before using the formula.
Gap trap. SI − TD is the interest on TD. It is not the TD itself.
MCQ on True Discount
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