True Discount Questions and Answers

Answer: A

Writing \( k = \dfrac{RT}{100} \), the four quantities are related by

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

Both discounts use the same \( k \), one on the amount and one on the present worth:

\[ \text{BD} = A\,k, \qquad \text{TD} = (A - \text{TD})\,k \]

Dividing one by the other:

\[ \frac{72}{60} = \frac{A}{A - 60} \quad\Longrightarrow\quad 72A - 4320 = 60A \]

\[ 12A = 4320 \quad\Longrightarrow\quad A = \text{Rs. } 360 \]

The standard shortcut is

\[ \text{Sum} = \frac{\text{BD} \times \text{TD}}{\text{BD} - \text{TD}} = \frac{72 \times 60}{12} = 360 \]

Answer: C

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.

With \( k = \dfrac{10 \times 3}{100} = 0.3 \):

\[ \text{TD} = \frac{\text{BG}}{k} = \frac{27}{0.3} = \text{Rs. } 90 \]

\[ \text{PW} = \frac{\text{TD}}{k} = \frac{90}{0.3} = \text{Rs. } 300 \]

Two divisions by the same \( k \), because the banker's gain is interest on the true discount just as the true discount is interest on the present worth.

Answer: A

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.

With \( T = \dfrac{4}{12} = \dfrac{1}{3} \) year at 15%:

\[ k = \frac{15 \times 1/3}{100} = 0.05 \]

Since \( \text{BD} = A\,k \) and \( \text{TD} = \dfrac{A\,k}{1+k} \), the two differ only by the factor \( 1+k \):

\[ \text{TD} = \frac{\text{BD}}{1+k} = \frac{420}{1.05} = \text{Rs. } 400 \]

The face value is \( 420/0.05 = \text{Rs. } 8400 \), and the banker's gain is Rs. 20 - but neither is needed once the \( 1+k \) relation is used.

Answer: A

Amount = Rs. 1860.
True discount = Rs. 60
∴ Present worth = Rs. 1860 - Rs. 60 = Rs. 1800
S.I. on Rs. 1800 for 8 months = Rs. 60
∴ Rate = [(100 * 60) / (1800 * 2/3)]% = 5%
∴ Banker’s discount = (1860 * 5 * 2/3)/100 = Rs. 62

Answer: C

Start from the profit, which is measured on the cost price.

\[ 0.10 \times \text{CP} = 63 \quad\Longrightarrow\quad \text{CP} = \text{Rs. } 630 \]

\[ \text{SP} = 630 + 63 = \text{Rs. } 693 \]

That selling price is the list price after a 23% discount:

\[ \text{LP} \times (1 - 0.23) = 693 \quad\Longrightarrow\quad \text{LP} = \frac{693}{0.77} = \text{Rs. } 900 \]

The chain runs cost → selling price → list price; jumping straight from cost to list price by combining 10% and 23% mixes the two different bases.

Answer: B

Writing \( k = \dfrac{RT}{100} \), the four quantities are related by

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

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

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

The banker's discount charges interest on the Rs. 540 face value, the true discount only on the Rs. 450 present worth - hence the Rs. 18 gap.

Answer: C

The commission is deducted from the marked price, so the Rs. 9595 is 95% of it.

\[ \text{MP} \times (1 - 0.05) = 9595 \]

\[ \text{MP} = \frac{9595}{0.95} = \text{Rs. } 10100 \]

Checking: 5% of 10100 is 505, and \( 10100 - 505 = 9595 \).

Adding 5% to Rs. 9595 gives Rs. 10,074.75, which is not the answer - a percentage taken off a larger base cannot be undone by adding the same percentage to the smaller one.

Answer: B

The present worth (PW) is what the sum is really worth today; the true discount (TD) is the interest that PW would earn over the remaining time:

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

The debt falls due in 1 year and is being settled after 3 months, so it is being paid 9 months early. Only that unexpired period is discounted.

\[ T = \frac{9}{12} = \frac{3}{4} \text{ year}, \qquad \frac{RT}{100} = \frac{12 \times 3/4}{100} = 0.09 \]

\[ \text{PW} = \frac{10028}{1.09} = 9200 \]

So he should pay Rs. 9200. The commonest error is discounting for the full year rather than for the nine months actually saved.

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

Simple interest is charged on the whole sum, true discount only on the present worth, so

\[ \frac{\text{SI}}{\text{TD}} = \frac{A}{A - \text{TD}} \quad\Longrightarrow\quad \frac{120}{100} = \frac{A}{A - 100} \]

\[ 120A - 12000 = 100A \quad\Longrightarrow\quad A = \text{Rs. } 600 \]

Now use the simple-interest formula on that sum over 3 years:

\[ 120 = \frac{600 \times R \times 3}{100} = 18R \]

\[ R = \frac{120}{18} = \frac{20}{3} = 6\tfrac{2}{3}\% \]

The present worth is \( 600 - 100 = \text{Rs. } 500 \), and 20% of 500 is indeed Rs. 100 - a useful check on both answers at once.