About True Discount
True discount is what you save by paying a bill before it falls due. The idea is simple: the present worth is the money which, with interest added, would grow into the amount of the bill. The discount is the amount minus that present worth.
What you need to understand
- The amount of a bill is what is payable in the future; the present worth is what it is worth today.
- Present worth plus true discount equals the amount of the bill.
- The true discount is the interest on the present worth, not on the amount.
- Simple interest on the amount is always larger than the true discount, because the amount is larger than the present worth.
- The difference between the simple interest on the amount and the true discount is the interest on the discount itself.
- Rate multiplied by time is the key quantity in every formula, so calculate it first.
Formulas to remember
How to work through these questions
- Work out rate multiplied by time as a single number before substituting anything.
- Decide whether the question asks for the discount or the present worth, because the two are easily confused.
- If the present worth is asked for, remember it is always smaller than the amount of the bill.
- For the difference between simple interest and the true discount, use the true discount as the base.
- Check that the present worth plus the discount returns the amount of the bill.
Mistakes that cost marks
- Calculating simple interest on the amount and calling it the true discount.
- Giving the present worth when the discount was asked for, or the other way round.
- Using the amount instead of the present worth as the base for the discount.
- Forgetting that the true discount is always smaller than simple interest on the same amount.
- Leaving the time in months while the rate is per annum, which throws the answer out by a factor of twelve.
Worked example
What is the present worth of a bill of Rs. 1540 due 4 years hence at 10% per annum?
- Rate multiplied by time is 10 x 4 = 40.
- Present worth = amount x 100 / (100 + rate x time).
- That is 1540 x 100 / 140 = 1100.
- Check: the true discount is 1540 - 1100 = 440, and 10% of 1100 for four years is indeed 440.
Answer: Rs. 1100
Practice questions with answers
A few True Discount questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Find the present worth of Rs. 600 due 5 years at 10% per annum.
Answer: Option C — with explanation
Present worth is the money which, with simple interest at 10% for 5 years, would amount to Rs. 600.
Present worth = amount x 100 / (100 + rate x time) = 600 x 100/150 = 400.
The true discount is 600 - 400 = 200.
Common mistakes
- gave the true discount instead of the present worth: 200
- repeated the amount due: 600
- gave the simple interest on the amount: 300
Question 2
For a bill of Rs. 1050 due 2 years at 25% per annum, find the excess of the banker's discount over the true discount.
Answer: Option A — with explanation
The banker's discount is the simple interest on the amount due, while the true discount is the interest on the present worth, so the banker's discount is the larger of the two.
Simple interest on Rs. 1050 for 2 years at 25% = 525.
True discount = 350, so the difference is 175.
Common mistakes
- gave the true discount instead of the difference: 350
- halved the difference: 87
- added one times the rate-and-time product: 357
Question 3
What is the true discount on a bill of Rs. 260 due 6 years at 5% per annum?
Answer: Option C — with explanation
Here rate x time is 5 x 6 = 30.
True discount = amount x (rate x time) / (100 + rate x time) = 260 x 30/130 = 60.
The present worth is 260 - 60 = 200.
Common mistakes
- gave the present worth instead of the discount: 200
- gave the excess of the simple interest over the discount: 18
- used simple interest on the amount instead of the true discount: 78
Question 4
Find the present worth of Rs. 1440 due 1 year at 20% per annum.
-
A
288
-
B
1440
-
C
240
-
D
1200
Answer: Option D — with explanation
Present worth is the money which, with simple interest at 20% for 1 years, would amount to Rs. 1440.
Present worth = amount x 100 / (100 + rate x time) = 1440 x 100/120 = 1200.
The true discount is 1440 - 1200 = 240.
Common mistakes
- repeated the amount due: 1440
- gave the simple interest on the amount: 288
- gave the true discount instead of the present worth: 240
Question 5
What is the difference between the banker's discount and the true discount on a bill of Rs. 980 payable 4 years at 10%?
Answer: Option B — with explanation
The banker's discount is the simple interest on the amount due, while the true discount is the interest on the present worth, so the banker's discount is the larger of the two.
Simple interest on Rs. 980 for 4 years at 10% = 392.
True discount = 280, so the difference is 112.
Common mistakes
- gave the true discount instead of the difference: 280
- added one times the rate-and-time product: 287
- halved the difference: 56
Frequently asked questions
What is the difference between true discount and simple interest?
Simple interest is calculated on the amount of the bill, while the true discount is calculated on the present worth, which is smaller. So the true discount is always the smaller figure.
What is present worth?
It is the money which, if invested today at the given rate, would grow to exactly the amount of the bill by the time the bill falls due.
Why do we multiply rate by time first?
Because it appears in both the numerator and the denominator of every formula in this topic. Working it out once keeps the substitution short and reduces the chance of a slip.
Is the true discount larger when the time is longer?
Yes. The longer the bill has to run, the more interest the present worth would earn, so the present worth is smaller and the discount is larger.
