True Discount Questions and Answers

Answer: B

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

Convert the credit offer to today's money before comparing.

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

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

\[ 12000 > 11500 \]

The cash offer is worth Rs. 500 more today, so Rs. 12,000 cash wins. The Rs. 880 premium fails to cover eight months of interest at 18%.

Answer: A

Simple interest is charged on the amount and true discount on the present worth, so with \( k = \dfrac{RT}{100} \):

\[ \text{SI} - \text{TD} = A\,k - \frac{A\,k}{1+k} = \frac{A\,k^{2}}{1+k} \]

Here \( T = 0.5 \) year at \( 12\tfrac{1}{2}\% \):

\[ k = \frac{12.5 \times 0.5}{100} = 0.0625 \]

\[ 25 = \frac{A \times 0.0625^{2}}{1.0625} = \frac{A \times 0.00390625}{1.0625} \]

\[ A = \frac{25 \times 1.0625}{0.00390625} = \text{Rs. } 6800 \]

The gap between simple interest and true discount is itself the banker's gain, which is why it depends on \( k^{2} \) and stays small for short periods.

Answer: C

Correct answer: The cash offer, by Rs. 500

Compare the two at the same date by reducing the credit offer to its present worth. With \(RT = 18 \times \tfrac{2}{3} = 12\):

\[PW = \frac{12880 \times 100}{112} = 11500\]

The cash offer of Rs. 12,000 is worth more today, so it is better by \(12000 - 11500 = \) Rs. 500.

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: B

With \( T = \dfrac{9}{12} = 0.75 \) year at 16%:

\[ k = \frac{16 \times 0.75}{100} = 0.12 \]

\[ \text{TD} = \frac{A\,k}{1+k} \quad\Longrightarrow\quad 189 = \frac{0.12\,A}{1.12} \]

\[ A = \frac{189 \times 1.12}{0.12} = \text{Rs. } 1764 \]

Checking: the present worth is \( 1764 - 189 = 1575 \), and 12% of 1575 is exactly 189.

Answer: B

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

With \( T = 3 \) years at 8%:

\[ \frac{RT}{100} = \frac{8 \times 3}{100} = 0.24 \]

\[ \text{PW} = \frac{930}{1.24} = \text{Rs. } 750 \]

\[ \text{TD} = 930 - 750 = \text{Rs. } 180 \]

Checking: 24% of Rs. 750 is Rs. 180, confirming that the discount is interest on the present worth. Taking 24% of the Rs. 930 face value would give Rs. 223.20 - that is the banker's discount, not the true one.

Answer: B

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

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

The discount grows, but by less than double: the present worth it is charged on falls at the same time, so the two effects partly cancel.

Answer: C

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

With \( T = 2\tfrac{1}{2} = 2.5 \) 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 discount is \( 2310 - 1680 = 630 \), and 37.5% of 1680 is 630 - interest on the present worth, exactly as the definition requires.

Answer: B

S.I on Rs.750= T.D on Rs. 960 

This means P.W of Rs. 960 due 2 years hence is Rs.750 

Therefore , T.D= Rs. (960 - 750) = Rs. 210 

Thus, S.I on Rs. 750 for 2 years is Rs. 210 

Rate =[ (100 x 210) / (750 x 2) ] % = 14%

Answer: C

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

\[ k = \frac{14 \times 2}{100} = 0.28 \]

\[ \text{PW} = \frac{\text{TD}}{k} = \frac{168}{0.28} = \text{Rs. } 600 \]

\[ \text{sum due} = 600 + 168 = \text{Rs. } 768 \]

The division gives only the present worth; the discount still has to be added back to reach the sum due.