True Discount Questions and Answers

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.

Answer: D

The cash discount applies to the already-reduced price, so multiply the two factors.

\[ 3600 \times (1 - 0.25) = 3600 \times 0.75 = \text{Rs. } 2700 \]

\[ 2700 \times (1 - 0.02) = 2700 \times 0.98 = \text{Rs. } 2646 \]

Adding the rates to 27% would give Rs. 2628 - Rs. 18 too little, because the 2% is charged on Rs. 2700, not on the full Rs. 3600.

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:

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

Rearranging gives the standard relation

\[ \text{TD} = \frac{\text{Amount} \times \frac{RT}{100}}{1 + \frac{RT}{100}} \]

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

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

\[ 189 = \frac{A \times 0.12}{1.12} \quad\Longrightarrow\quad A = \frac{189 \times 1.12}{0.12} = 1764 \]

As a check, PW = 1764 − 189 = 1575, and 12% of 1575 is indeed 189.

Answer: D

T = 5/2 year
Let the banker’s discount, BD = Rs.1
Then, banker’s gain, BG = (9/25)*1 = Rs. 9/25
BG = BD-TD
=> 9/25 = 1-TD
=> TD = 1-9/25 = 16/25
F = BD*TD/(BD-TD)
= (1*16/25)/(1-(16/25)) (16/25)/(9/25) = Rs. 16/9
BD = simple interest on the face value of the bill for unexpired time = FTR/100
=> 1 = (16/9)*(5/2)*R/100
=> 100 = (16/9)*(5/2)*R
=> 100 = 16*5*R/9*2
=> 100 = 8*5*R/9
=> R = 100*9/8*5
= 100*9/40 = 5*9/2 = 45/2 = 22 1/2 %

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

Work through the first discount, then compare what is left with the selling price.

\[ 800 \times (1 - 0.10) = \text{Rs. } 720 \]

\[ \frac{612}{720} = 0.85 \quad\Longrightarrow\quad \text{second discount} = 15\% \]

Checking: \( 720 \times 0.85 = 612 \).

The combined discount here is \( 1 - 0.90 \times 0.85 = 23.5\% \), which is \( \dfrac{800-612}{800} \) - but the question asks only for the second step.

Answer: C

Let the bill be \( A \), due in 1 year, with money worth \( 12.5\% \).

Original arrangement - the whole bill in 1 year:

\[ \text{PW}_{\text{old}} = \frac{A}{1 + 0.125} = \frac{A}{1.125} = 0.8889\,A \]

New arrangement - half now (no discount) and half in 2 years:

\[ \text{PW}_{\text{new}} = \frac{A}{2} + \frac{A/2}{1 + 0.25} = 0.5A + 0.4A = 0.9A \]

The creditor gains the difference:

\[ 0.9A - 0.8889A = 0.01111\,A = 40 \]

\[ A = \frac{40}{0.01111} = \text{Rs. } 3600 \]

Deferring half by a further year costs less than being paid the other half a year early gains - because discounting bites harder on the nearer payment.

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:

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

True discount is simple interest on the present worth, so start there:

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

\[ \text{TD} = \text{PW} \times 0.28 \quad\Longrightarrow\quad 168 = 0.28\,\text{PW} \quad\Longrightarrow\quad \text{PW} = 600 \]

\[ \text{Sum due} = \text{PW} + \text{TD} = 600 + 168 = \text{Rs. } 768 \]

Dividing 168 by 0.28 gives the present worth, not the answer - the discount still has to be added back to reach the sum due.

Answer: B

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.

\[ k = \frac{\text{TD}}{\text{PW}} = \frac{36}{800} = 0.045 \]

\[ A = \text{PW} + \text{TD} = 800 + 36 = \text{Rs. } 836 \]

\[ \text{BD} = A\,k = 836 \times 0.045 = \text{Rs. } 37.62 \]

Equivalently \( \text{BD} = \text{TD}(1+k) = 36 \times 1.045 = 37.62 \). The banker's gain is the 62 paise difference.

Answer: B

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.

Let \( k \) be the rate–time product. The banker's discount on Rs. 1600 equals the true discount on Rs. 1680:

\[ 1600\,k = \frac{1680\,k}{1+k} \]

Cancelling \( k \):

\[ 1600(1+k) = 1680 \quad\Longrightarrow\quad 1+k = 1.05 \quad\Longrightarrow\quad k = 0.05 \]

\[ \frac{15\,T}{100} = 0.05 \quad\Longrightarrow\quad T = \frac{1}{3} \text{ year} = 4 \text{ months} \]

The rate enters only at the final step; everything before it depends on the ratio \( 1680/1600 \) alone.