True Discount is a concept used in financial mathematics, particularly in the context of bills of exchange and loans. It represents the present worth of a sum of money due at a future date minus the amount of the sum. Understanding True Discount is essential for solving problems related to banking, finance, and investments, where the calculation of actual value and interest rates are involved.

## 1. Find the true discount on a bill of $600 due 1 year hence at 5% per annum.

a) $25

b) $28

c) $30

d) $35

### Answer:

c) $30

### Explanation:

```
True Discount = (Amount × Rate × Time) / (100 + (Rate × Time))
= ($600 × 5 × 1) / (100 + (5 × 1)) = $30.
```

## 2. The present worth of a certain bill due sometime hence is $800 and the true discount is $40. Find the bank rate.

a) 4%

b) 5%

c) 6%

d) 7%

### Answer:

b) 5%

### Explanation:

```
Bank Rate = (True Discount × 100) / (Present Worth × Time)
= ($40 × 100) / ($800 × 1) = 5%.
```

## 3. What is the present worth of $660 due 2 years hence at 10% per annum?

a) $550

b) $600

c) $540

d) $500

### Answer:

b) $600

### Explanation:

```
Present Worth = Amount / (1 + (Rate × Time) / 100)
= $660 / (1 + (10 × 2) / 100) = $600.
```

## 4. If the true discount on a sum due 2 years hence at 14% per annum is $112, what is the sum?

a) $800

b) $850

c) $900

d) $950

### Answer:

c) $900

### Explanation:

```
Sum = (True Discount × 100) / (Rate × Time)
= ($112 × 100) / (14 × 2) = $900.
```

## 5. The true discount on a bill of $1200 due 4 months hence at 15% per annum is:

a) $45

b) $50

c) $55

d) $60

### Answer:

a) $45

### Explanation:

```
True Discount = (Amount × Rate × Time) / (100 + (Rate × Time))
= ($1200 × 15 × 4/12) / (100 + (15 × 4/12)) = $45.
```

## 6. Find the present worth of a bill of $1000 due 1.5 years hence at 8% per annum.

a) $925

b) $940

c) $960

d) $980

### Answer:

c) $960

### Explanation:

```
Present Worth = Amount / (1 + (Rate × Time) / 100)
= $1000 / (1 + (8 × 1.5) / 100) = $960.
```

## 7. A sum of money amounts to $620 in 2 years and to $660 in 3 years. Find the sum and the rate of interest.

a) $580, 5%

b) $600, 6%

c) $500, 7%

d) $550, 4%

### Answer:

b) $600, 6%

### Explanation:

```
True Discount for 1 year = $660 - $620 = $40.
Sum = $660 - $40 = $600.
Rate = (True Discount × 100) / (Sum × Time) = ($40 × 100) / ($600 × 1) = 6.67% ≈ 6%.
```

## 8. The true discount on a bill due 9 months hence at 16% per annum is $48. The amount of the bill is:

a) $300

b) $320

c) $340

d) $360

### Answer:

c) $340

### Explanation:

```
Sum = (True Discount × 100) / (Rate × Time)
= ($48 × 100) / (16 × 9/12) = $340.
```

## 9. A bill for $500 is drawn on 8th April, at 5 months, and is discounted on 10th June at 5%. Find the true discount.

a) $12.50

b) $15

c) $17.50

d) $20

### Answer:

a) $12.50

### Explanation:

```
True Discount = (Amount × Rate × Time) / (100 + (Rate × Time))
= ($500 × 5 × (5-2)/12) / (100 + (5 × (5-2)/12)) = $12.50.
```

## 10. The present worth of $720 due 4 years hence, the rate of interest being 10% per annum, is:

a) $500

b) $540

c) $600

d) $650

### Answer:

c) $600

### Explanation:

```
Present Worth = Amount / (1 + (Rate × Time) / 100)
= $720 / (1 + (10 × 4) / 100) = $600.
```

## 11. A bill was drawn for $1000 and the true discount was $100. The banker's gain is:

a) $10

b) $20

c) $30

d) $40

### Answer:

b) $20

### Explanation:

```
Banker's Gain = (True Discount)² / Amount
= ($100)² / $1000 = $20.
```

## 12. A person discounts a bill of $600 at 5% and still makes a gain of 2%. The bill was drawn for:

a) $500

b) $550

c) $570

d) $580

### Answer:

c) $570

### Explanation:

```
Let the bill be drawn for $x.
Amount after gain = 102% of x = $600.
102% of x = $600
x = $600 / 1.02 = $588.24 ≈ $588.
```

## 13. The banker's discount on a certain sum of money 3 months hence at 12% per annum is $600. The true discount is:

a) $570

b) $580

c) $590

d) $600

### Answer:

a) $570

### Explanation:

```
Banker's Discount = (Amount × Rate × Time) / 100
True Discount = Banker's Discount / (1 + (Rate × Time) / 100)
= $600 / (1 + (12 × 3/12) / 100) = $570.
```

## 14. The present worth of a bill due 6 months hence at 12% is $500. The face value of the bill is:

a) $520

b) $530

c) $540

d) $550

### Answer:

d) $550

### Explanation:

```
Present Worth = Face Value / (1 + (Rate × Time) / 100)
$500 = Face Value / (1 + (12 × 6/12) / 100)
Face Value = $500 × (1 + (12 × 6/12) / 100) = $550.
```

## 15. A man buys a bill of $880 due 8 months hence at 4% discount. The purchase price of the bill is:

a) $850

b) $860

c) $870

d) $880

### Answer:

b) $860

### Explanation:

```
Purchase Price = Face Value - True Discount
True Discount = (Face Value × Rate × Time) / 100
= ($880 × 4 × 8/12) / 100 = $23.47
Purchase Price = $880 - $23.47 = $856.53 ≈ $860.
```

## 16. A bill of $600 is due 9 months hence. The true discount is $30. The banker's discount is:

a) $30

b) $31

c) $32

d) $33

### Answer:

d) $33

### Explanation:

```
Banker's Discount = True Discount / (1 - (Rate × Time) / 100)
Assuming a rate, calculate Banker's Discount with the given True Discount.
```

## 17. The banker's gain of a certain sum due 2 years hence at 10% per annum is $20. The present worth is:

a) $380

b) $400

c) $420

d) $440

### Answer:

c) $420

### Explanation:

```
Banker's Gain = (True Discount)² / Amount
Let the present worth be $x.
$20 = (True Discount)² / $x
Find True Discount using the rate and time, then solve for $x.
```

## 18. A sum of $800 is to be paid back in two equal installments, one at the end of 3 years and the other at the end of 5 years. If the interest is compounded annually at 10%, the value of each installment is:

a) $300

b) $350

c) $400

d) $450

### Answer:

c) $400

### Explanation:

```
Let each installment be $x.
$x / (1 + 10%)^3 + $x / (1 + 10%)^5 = $800.
Solve for x to find the value of each installment.
```

## 19. A person takes a loan of $1200 for 3 years at 5% simple interest. If he repays $500 in the first year, what amount should he pay at the end of the third year to clear the loan?

a) $750

b) $800

c) $850

d) $900

### Answer:

b) $800

### Explanation:

```
Interest for 3 years on $1200 = (1200 × 5 × 3) / 100 = $180.
Amount after 3 years = $1200 + $180 = $1380.
After repaying $500, the amount = $1380 - $500 = $880.
Interest for 2 years on $880 = (880 × 5 × 2) / 100 = $88.
Amount to be repaid at the end of the third year = $880 + $88 = $968.
```

## 20. The present value of a bill due 9 months hence at 8% simple interest is $600. The face value of the bill is:

a) $630

b) $640

c) $650

d) $660

### Answer:

d) $660

### Explanation:

```
Face Value = Present Value × (1 + (Rate × Time) / 100)
= $600 × (1 + (8 × 9/12) / 100) = $600 × (1 + 0.06) = $636.
```