True Discount Questions and Answers

Answer: C

\[ \text{PW} = 3520 - 320 = \text{Rs. } 3200 \]

True discount is simple interest on the present worth:

\[ 320 = \frac{3200 \times 18 \times T}{100} = 576\,T \]

\[ T = \frac{320}{576} = \frac{5}{9} \text{ year} \]

That is about 6 months and 20 days. Using the Rs. 3520 face value in place of the present worth gives \( \tfrac{320}{633.6} = \tfrac{4}{9} \) - exactly the distractor offered.

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.

Here \( k = \dfrac{6 \times 3}{100} = 0.18 \), and the banker's gain is interest on the true discount:

\[ \text{BG} = \text{TD} \times k \quad\Longrightarrow\quad 18 = \text{TD} \times 0.18 \]

\[ \text{TD} = \frac{18}{0.18} = \text{Rs. } 100 \]

The banker's gain is always the smallest of the four quantities, being interest on the discount rather than on either principal.

Answer: B

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

The Rs. 3300 arrives a year late, so bring it back to today's money before comparing.

\[ \text{PW} = \frac{3300}{1 + \dfrac{10 \times 1}{100}} = \frac{3300}{1.1} = \text{Rs. } 3000 \]

\[ \text{gain} = 3000 - 2825 = \text{Rs. } 175 \]

The raw difference of Rs. 475 looks like the profit, but Rs. 300 of it is merely the price of waiting a year - which is why Rs. 475 is offered as a distractor.

Answer: C

Multiply the three retained fractions.

\[ (1-0.15)(1-0.16)(1-0.20) = 0.85 \times 0.84 \times 0.80 \]

\[ 0.85 \times 0.84 = 0.714, \qquad 0.714 \times 0.80 = 0.5712 \]

\[ \text{single equivalent discount} = 1 - 0.5712 = 0.4288 = 42.88\% \]

The rates sum to 51%, far above the truth. The order of the three discounts makes no difference, since multiplication is commutative.

Answer: A

\[ \text{PW} = 5125 - 125 = \text{Rs. } 5000 \]

The true discount is simple interest on that present worth, over \( T = \dfrac{6}{12} = 0.5 \) year:

\[ 125 = \frac{5000 \times R \times 0.5}{100} = 25R \]

\[ R = \frac{125}{25} = 5\% \text{ per annum} \]

The half-year must be converted before dividing; leaving \( T = 6 \) would give a rate about twelve times too small.

Answer: A

Simple interest is reckoned on the amount, true discount on the present worth, and both use the same rate–time product \( k \):

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

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

\[ 85A - 6800 = 80A \quad\Longrightarrow\quad 5A = 6800 \quad\Longrightarrow\quad A = \text{Rs. } 1360 \]

The standard shortcut is

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

Answer: D

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

\[ \text{PW} = \frac{3660}{1 + \dfrac{10 \times 2}{100}} = \frac{3660}{1.2} = \text{Rs. } 3050 \]

\[ \text{gain} = 3050 - 3000 = \text{Rs. } 50 \]

\[ \text{gain}\% = \frac{50}{3000} \times 100 = 1.67\% \]

That figure appears nowhere on the list, so the answer is none of these. It is worth contrasting this with the Rs. 3600 version of the same problem, where the present worth comes to exactly Rs. 3000 and the gain is 0%.

Answer: C

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

The Rs. 1296 is payable after 8 months, so \( T = \dfrac{8}{12} = \dfrac{2}{3} \) year at 12%:

\[ \frac{RT}{100} = \frac{12 \times 2/3}{100} = 0.08 \]

\[ \text{PW} = \frac{1296}{1.08} = \text{Rs. } 1200 \]

\[ \text{gain} = 1200 - 1056 = \text{Rs. } 144 \]

\[ \text{gain}\% = \frac{144}{1056} \times 100 = 13.64\% \approx 13.6\% \]

Comparing Rs. 1296 with Rs. 1056 directly suggests 22.7% - the last distractor - but that ignores the eight months of waiting.

Answer: A

Write \( k = \dfrac{R \times 2}{100} \). Simple interest is on the full sum; true discount is on the present worth:

\[ \text{SI} = 750\,k, \qquad \text{TD} = \frac{810\,k}{1+k} \]

Setting them equal and cancelling \( k \):

\[ 750(1+k) = 810 \quad\Longrightarrow\quad 1+k = 1.08 \quad\Longrightarrow\quad k = 0.08 \]

\[ \frac{2R}{100} = 0.08 \quad\Longrightarrow\quad R = 4\% \]

The two years enter only at the last step; everything before it is independent of the period.

Answer: D

First recover the rate–time product from the original bill.

\[ \text{PW} = 110 - 10 = 100 \quad\Longrightarrow\quad 100k = 10 \quad\Longrightarrow\quad k = 0.1 \]

Doubling the time doubles \( k \) to 0.2, so for the same Rs. 110:

\[ \text{PW}' = \frac{110}{1.2} = 91.67 \quad\Longrightarrow\quad \text{TD}' = 110 - 91.67 = \text{Rs. } 18.33 \]

True discount does not double when the time doubles - the Rs. 20 option is exactly that assumption. It grows more slowly, because the present worth it is computed on shrinks at the same time.