True Discount Questions and Answers

Answer: A

A "double discount" applies the second rate to the already-reduced price.

\[ 1000 \times (1 - 0.20) = \text{Rs. } 800 \]

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

The single equivalent discount is

\[ 1 - 0.80 \times 0.90 = 0.28 = 28\% \]

not 30%, so he pays Rs. 720 rather than Rs. 700. Rs. 800 is the price after the first discount only.

Answer: B

Correct answer: Rs. 200

True discount is the interest on the present worth, not on the amount, so it is found from

\[TD = \frac{A \times R \times T}{100 + RT} = \frac{2200 \times 10 \times 1}{100 + 10} = \frac{22000}{110} = 200\]

The present worth is Rs. 2000, and the interest on that for one year at 10% is indeed Rs. 200.

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: 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: B

Apply the two discounts in turn.

\[ 400 \times (1 - 0.10) = \text{Rs. } 360 \]

\[ 360 \times (1 - 0.05) = \text{Rs. } 342 \]

The single equivalent discount is

\[ 1 - 0.90 \times 0.95 = 0.145 = 14.5\% \]

slightly less than the 15% the rates suggest, because the 5% comes off Rs. 360 rather than Rs. 400.

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:

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

Bring the credit offer back to today's money and then compare like with like.

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

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

\[ 12000 > 11500 \]

The cash offer is worth Rs. 500 more in today's money, so Rs. 12,000 in cash is the better deal. The larger headline figure loses because eight months at 18% costs more than the extra Rs. 880 is worth.

Answer: B

The same bill \( A \) is being discounted over two different periods, so write each present worth in terms of \( A \) and the rate \( r \) (as a decimal).

\[ A = 1250(1 + 2r) \]

\[ A = 1200\left(1 + \tfrac{35}{12}r\right) \qquad \left(2 \text{ years } 11 \text{ months} = \tfrac{35}{12} \text{ years}\right) \]

Equating the two:

\[ 1250 + 2500r = 1200 + 3500r \]

\[ 50 = 1000r \quad\Longrightarrow\quad r = 0.05 = 5\% \]

\[ A = 1250(1 + 0.10) = \text{Rs. } 1375 \]

The longer the wait, the smaller the present worth - and the size of that drop over the extra 11 months is exactly what pins the rate down.

Answer: A

Single 30% discount:

\[ 1000 \times 0.70 = \text{Rs. } 700 \]

Successive 15% and 15%:

\[ 1000 \times 0.85 \times 0.85 = 1000 \times 0.7225 = \text{Rs. } 722.50 \]

\[ \text{difference} = 722.50 - 700 = \text{Rs. } 22.50 \]

The single discount is the better one for the buyer. The gap equals \( 1000 \times 0.15 \times 0.15 = 22.50 \) - the product term that successive discounting always forfeits.

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.

\[ k = \frac{\text{TD}}{\text{PW}} = \frac{110}{1100} = 0.1 \]

\[ A = \text{PW} + \text{TD} = 1100 + 110 = \text{Rs. } 1210 \]

\[ \text{BD} = A\,k = 1210 \times 0.1 = \text{Rs. } 121 \]

The banker's gain is \( 121 - 110 = \text{Rs. } 11 \), which is also \( \text{TD} \times k = 110 \times 0.1 \) - a useful cross-check.

Answer: C

Chain the retained fractions.

\[ (1-0.20)(1-0.10)(1-0.25) = 0.80 \times 0.90 \times 0.75 \]

\[ 0.80 \times 0.90 = 0.72, \qquad 0.72 \times 0.75 = 0.54 \]

\[ \text{single equivalent discount} = 1 - 0.54 = 0.46 = 46\% \]

The rates add to 55%, which is offered as the first option; the true figure is always lower, because each successive discount acts on a smaller base.