True Discount Questions and Answers

Answer: C

The true discount is simple interest on the present worth, so find that first.

\[ \text{PW} = 2562 - 122 = \text{Rs. } 2440 \]

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

\[ 122 = \frac{2440 \times R \times \tfrac{1}{3}}{100} \quad\Longrightarrow\quad R = \frac{122 \times 300}{2440} = 15\% \]

Charging the interest on the Rs. 2562 face value instead gives about 14.3%, which is why 14% sits among the options.

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 \( k = \dfrac{12 \times 3}{100} = 0.36 \):

\[ \text{BG} = \text{TD} \times k \quad\Longrightarrow\quad \text{TD} = \frac{270}{0.36} = \text{Rs. } 750 \]

\[ \text{BD} = \text{TD} + \text{BG} = 750 + 270 = \text{Rs. } 1020 \]

Rs. 750 is the true discount, an intermediate step; the banker's discount is always the larger of the two, exceeding it by exactly the banker's gain.

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: C

Simple Interest on Rs. (260-20) for a gven time= Rs.20 

Simple Interest on Rs. (260-20) for a half time  = Rs.10 

True Discount on Rs.250 = Rs.10  

True Discount on Rs.260 = Rs.[(10/250) x 260] = Rs. 10.40

Answer: A

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 Rs. 3600 is not received today but in 2 years, so compare its present worth with the Rs. 3000 he spent.

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

The present worth exactly equals the cost price, so

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

Comparing Rs. 3600 with Rs. 3000 directly suggests a 20% profit, but that ignores two years of waiting - which is precisely what true discount measures.

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 Rs. 2200 arrives a year late, so discount it back to today.

\[ \text{PW} = \frac{2200}{1 + \dfrac{10 \times 1}{100}} = \frac{2200}{1.1} = 2000 \]

\[ \text{gain} = 2000 - 1950 = \text{Rs. } 50 \]

Comparing 2200 with 1950 suggests a Rs. 250 profit, but Rs. 200 of that is merely compensation for waiting a year - only the remaining Rs. 50 is real gain.

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: C

The Rs. 1404 comes as two equal installments of Rs. 702, one due in 6 months and one in 12, so each is discounted for its own period.

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

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

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

Discounting the full Rs. 1404 for an averaged 9 months gives Rs. 1324.5 - close, but not equal, because discounting is not linear in time.

Answer: D

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.

Neither the rate nor the time is given, so use the form that needs neither:

\[ \text{BG} = \frac{\text{TD}^{2}}{\text{PW}} \]

\[ 16 = \frac{\text{TD}^{2}}{576} \quad\Longrightarrow\quad \text{TD}^{2} = 9216 \]

\[ \text{TD} = \sqrt{9216} = \text{Rs. } 96 \]

This identity follows from \( \text{BG} = \text{TD}\,k \) and \( \text{TD} = \text{PW}\,k \): substituting \( k = \text{TD}/\text{PW} \) into the first gives it directly.

Answer: D

Successive discounts multiply the retained fractions.

\[ (1 - 0.20)(1 - 0.15) = 0.80 \times 0.85 = 0.68 \]

\[ \text{single equivalent discount} = 1 - 0.68 = 0.32 = 32\% \]

The general formula is

\[ a + b - \frac{ab}{100} = 20 + 15 - \frac{300}{100} = 32\% \]

Simply adding the two gives 35%, which is why that value is offered; 68% is the fraction retained, not the discount.