True Discount Questions and Answers

Answer: A

\[ \text{Amount} = \text{PW}\left(1 + \frac{RT}{100}\right), \qquad \text{TD} = \text{Amount} - \text{PW} \]

Bring the credit offer back to today's money and then compare.

\[ T = \frac{8}{12} = \frac{2}{3} \text{ year}, \qquad \frac{RT}{100} = \frac{18 \times 2/3}{100} = 0.12 \]

\[ \text{PW of credit offer} = \frac{12880}{1.12} = \text{Rs. } 11500 \]

\[ 12000 > 11500 \]

So Rs. 12,000 in cash is worth Rs. 500 more. Eight months of interest at 18% costs more than the extra Rs. 880 the credit offer promises.

Answer: A

\[ \text{Amount} = \text{PW}\left(1 + \frac{RT}{100}\right), \qquad \text{TD} = \text{Amount} - \text{PW} \]

\[ \text{PW} = \frac{3600}{1 + \dfrac{10 \times 2}{100}} = \frac{3600}{1.2} = \text{Rs. } 3000 \]

The present worth exactly matches what he paid, so

\[ \text{gain} = 3000 - 3000 = 0 \quad\Longrightarrow\quad 0\% \]

The apparent Rs. 600 profit is exactly the interest the money would have earned over the two years, so in real terms he is no better off.

Answer: A

Correct answer: Rs. 100

Simple interest is reckoned on the amount and true discount on the present worth, and the standard relation between them is

\[\text{Sum} = \frac{SI \times TD}{SI - TD} = \frac{25 \times 20}{25 - 20} = \frac{500}{5} = 100\]

The gap between the two figures is exactly the interest on the true discount itself.

Answer: A

Two equal half-yearly installments of Rs. 702 each - one due in 6 months, the other in 12 - must be discounted separately for their own waiting times.

\[ \text{PW}_{1} = \frac{702}{1 + \dfrac{8 \times 0.5}{100}} = \frac{702}{1.04} = \text{Rs. } 675 \]

\[ \text{PW}_{2} = \frac{702}{1 + \dfrac{8 \times 1}{100}} = \frac{702}{1.08} = \text{Rs. } 650 \]

\[ \text{total} = 675 + 650 = \text{Rs. } 1325 \]

Discounting the whole Rs. 1404 for a single averaged period does not give the same answer, because discounting is not linear in time.

Answer: C

Correct answer: 8 months

\[60 = \frac{1860 \times 5T}{100 + 5T} \quad\Longrightarrow\quad 60(100 + 5T) = 9300T\]

\[6000 + 300T = 9300T \quad\Longrightarrow\quad 9000T = 6000 \quad\Longrightarrow\quad T = \tfrac{2}{3} \text{ year}\]

Two-thirds of a year is 8 months. The present worth is Rs. 1800, and 8 months of interest on it at 5% is Rs. 60.

Answer: D

Correct answer: Rs. 600

True discount is simply the simple interest on the present worth.

\[TD = \frac{PW \times R \times T}{100} = \frac{2500 \times 8 \times 3}{100} = 600\]

The bill itself is therefore worth \(2500 + 600 = \) Rs. 3100 at maturity.

Answer: B

Correct answer: Rs. 200

True discount is the interest on the present worth, not on the amount, so it is found from

\[TD = \frac{A \times R \times T}{100 + RT} = \frac{2200 \times 10 \times 1}{100 + 10} = \frac{22000}{110} = 200\]

The present worth is Rs. 2000, and the interest on that for one year at 10% is indeed Rs. 200.

Answer: B

Correct answer: Rs. 1590

Here \(RT = 12 \times \tfrac{1}{2} = 6\), so

\[TD = \frac{A \times RT}{100 + RT} \quad\Longrightarrow\quad 90 = \frac{6A}{106}\]

\[A = \frac{90 \times 106}{6} = 1590\]

The present worth is Rs. 1500, on which six months of interest at 12% is Rs. 90.

Answer: B

Four quantities are related throughout this topic. Writing \( k = \dfrac{RT}{100} \):

\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{BD} - \text{TD} = \text{TD}\,k \]

The banker's discount is interest on the face value; the true discount is interest on the present worth; the banker's gain is the interest on the true discount.

\[ \text{PW} = 540 - 90 = \text{Rs. } 450 \quad\Longrightarrow\quad k = \frac{\text{TD}}{\text{PW}} = \frac{90}{450} = 0.2 \]

\[ \text{BD} = A\,k = 540 \times 0.2 = \text{Rs. } 108 \]

The banker's discount always exceeds the true discount, here by \( 108 - 90 = \text{Rs. } 18 \), because it charges interest on the larger face value.

Answer: B

Four quantities are related throughout this topic. Writing \( k = \dfrac{RT}{100} \):

\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{BD} - \text{TD} = \text{TD}\,k \]

The banker's discount is interest on the face value; the true discount is interest on the present worth; the banker's gain is the interest on the true discount.

\[ k = \frac{\text{TD}}{\text{PW}} = \frac{36}{800} = 0.045 \]

\[ A = \text{PW} + \text{TD} = 800 + 36 = \text{Rs. } 836 \]

\[ \text{BD} = A\,k = 836 \times 0.045 = \text{Rs. } 37.62 \]

Equivalently \( \text{BD} = \text{TD}(1+k) = 36 \times 1.045 = 37.62 \). The banker's gain is the 62 paise difference.