True Discount Questions and Answers

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

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

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

With \( T = \dfrac{6}{12} = 0.5 \) year at 6%:

\[ k = \frac{6 \times 0.5}{100} = 0.03 \]

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

\[ \text{TD} = \frac{\text{BD}}{1+k} = \frac{18.54}{1.03} = \text{Rs. } 18 \]

The banker's gain is the 54 paise difference - small here because the period is short and the rate low.

Answer: A

Successive discounts multiply the retained fractions.

\[ 10 \times (1 - 0.20) \times (1 - 0.10) = 10 \times 0.80 \times 0.90 \]

\[ = 10 \times 0.72 = \text{Rs. } 7.20 \]

The single equivalent discount is 28%, not 30% - the 10% is charged on Rs. 8 rather than on Rs. 10, so it takes off 80 paise instead of one rupee.

Answer: B

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

Here \( T = 2\tfrac{1}{2} = 2.5 \) years at 15%:

\[ \frac{RT}{100} = \frac{15 \times 2.5}{100} = 0.375 \]

\[ \text{PW} = \frac{2310}{1.375} = \text{Rs. } 1680 \]

Checking: the true discount is \( 2310 - 1680 = 630 \), and 37.5% of 1680 is indeed 630. Note the interest is reckoned on the present worth, not on the Rs. 2310 face value.

Answer: C

Simple interest is reckoned on the amount, true discount on the present worth. Both use the same \( \dfrac{RT}{100} \), so

\[ \text{SI} = A \times \frac{RT}{100}, \qquad \text{TD} = \text{PW} \times \frac{RT}{100} = (A - \text{TD}) \times \frac{RT}{100} \]

Dividing one by the other:

\[ \frac{\text{SI}}{\text{TD}} = \frac{A}{A - \text{TD}} \quad\Longrightarrow\quad \frac{85}{80} = \frac{A}{A - 80} \]

\[ 85A - 6800 = 80A \quad\Longrightarrow\quad 5A = 6800 \quad\Longrightarrow\quad A = 1360 \]

The shortcut worth remembering is

\[ \text{Sum} = \frac{\text{SI} \times \text{TD}}{\text{SI} - \text{TD}} = \frac{85 \times 80}{5} = 1360 \]

Answer: C

Apply the known discount first, then see what fraction of the remainder she actually paid.

\[ 560 \times (1 - 0.20) = \text{Rs. } 448 \]

\[ \frac{336}{448} = 0.75 \quad\Longrightarrow\quad \text{second discount} = 1 - 0.75 = 25\% \]

Checking: \( 448 \times 0.75 = 336 \).

Taking \( \dfrac{560 - 336}{560} = 40\% \) gives the combined discount, not the second one - and indeed \( 1 - 0.80 \times 0.75 = 0.40 \).

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.

Here \( k = \dfrac{10 \times 2}{100} = 0.2 \).

\[ \text{BG} = \text{TD} \times k \quad\Longrightarrow\quad \text{TD} = \frac{24}{0.2} = \text{Rs. } 120 \]

\[ \text{TD} = \text{PW} \times k \quad\Longrightarrow\quad \text{PW} = \frac{120}{0.2} = \text{Rs. } 600 \]

The chain runs BG → TD → PW, dividing by \( k \) at each step. Stopping one step early leaves Rs. 120, and the face value would be Rs. 720.

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:

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

\[ \text{PW} = 260 - 20 = 240 \quad\Longrightarrow\quad 240 \times \frac{RT}{100} = 20 \quad\Longrightarrow\quad \frac{RT}{100} = \frac{1}{12} \]

Halving the time halves that product to \( \dfrac{1}{24} \), so for the same Rs. 260:

\[ \text{PW}' = \frac{260}{1 + \frac{1}{24}} = \frac{260 \times 24}{25} = 249.60 \]

\[ \text{TD}' = 260 - 249.60 = \text{Rs. } 10.40 \]

Halving the time does not simply halve the discount to Rs. 10 - that is the first distractor, and it is slightly too small.

Answer: B

Take the cost price as 100, so the marked price is 120.

\[ \text{SP} = 120 \times (1 - 0.30) = 120 \times 0.70 = 84 \]

\[ \text{loss} = 100 - 84 = 16 \quad\Longrightarrow\quad \text{loss}\% = \frac{16}{100} \times 100 = 16\% \]

Loss percentage is always measured against the cost price, not the marked price. Against the marked price the fall would read \( \tfrac{36}{120} = 30\% \), which is just the discount again.

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.