True Discount Questions and Answers

Answer: B

Let the original price be 100. After the rise it is 140, and the discount \( d \) must bring it back to 100 exactly.

\[ 140(1 - d) = 100 \quad\Longrightarrow\quad 1 - d = \frac{100}{140} = \frac{5}{7} \]

\[ d = \frac{2}{7} = 0.2857 = 28.57\% \approx 28.5\% \]

A 40% rise is not undone by a 40% cut, because the cut is applied to the larger figure. The general result is

\[ d = \frac{x}{100 + x} \times 100 \]

which for \( x = 40 \) gives \( \tfrac{40}{140} \times 100 = 28.57\% \).

Answer: B

Single 30% discount:

\[ 2000 \times 0.30 = \text{Rs. } 600 \]

Successive 25% and 5%:

\[ 2000 \times 0.75 \times 0.95 = 2000 \times 0.7125 = \text{Rs. } 1425 \]

\[ \text{discount} = 2000 - 1425 = \text{Rs. } 575 \]

\[ \text{difference} = 600 - 575 = \text{Rs. } 25 \]

The single discount is the better one. The shortfall equals \( 2000 \times 0.25 \times 0.05 = 25 \) - the product of the two rates, which is exactly what successive discounting loses.

Answer: C

\[ (1 - 0.15)(1 - 0.10) = 0.85 \times 0.90 = 0.765 \]

\[ \text{single equivalent discount} = 1 - 0.765 = 0.235 = 23.5\% \]

Or by the formula

\[ a + b - \frac{ab}{100} = 15 + 10 - \frac{150}{100} = 23.5\% \]

The naive sum of 25% overstates the discount by exactly the correction term \( \tfrac{ab}{100} = 1.5\% \).

Answer: B

A 5% discount leaves 95% of the list price.

\[ \text{SP} = 20 \times (1 - 0.05) = 20 \times 0.95 = \text{Rs. } 19 \]

Equivalently, the discount is \( 0.05 \times 20 = \text{Rs. } 1 \), and \( 20 - 1 = 19 \). A discount always lowers the price, so any option above Rs. 20 can be discarded without calculating.

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.

\[ k = \frac{\text{TD}}{\text{PW}} = \frac{160}{1600} = 0.1 \]

\[ \text{BG} = \text{TD} \times k = 160 \times 0.1 = \text{Rs. } 16 \]

Equivalently \( \text{BG} = \dfrac{\text{TD}^{2}}{\text{PW}} = \dfrac{160^{2}}{1600} = 16 \) - a form worth remembering, since it needs neither the rate nor the time.

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

Revenue is price multiplied by volume, so scale each by its own factor.

\[ \text{price factor} = 1 - 0.25 = 0.75, \qquad \text{volume factor} = 1 + 0.40 = 1.40 \]

\[ \text{revenue factor} = 0.75 \times 1.40 = 1.05 \]

\[ \text{change} = 1.05 - 1 = 0.05 = 5\% \text{ increase} \]

Subtracting the percentages (\( 40\% - 25\% = 15\% \)) is wrong because they act on different bases; the product is what matters. The revenue would be unchanged only if volume rose by \( \tfrac{1}{0.75} - 1 = 33.3\% \).

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: 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

Apply the first discount, then find what fraction of that remains.

\[ 6400 \times (1 - 0.20) = \text{Rs. } 5120 \]

\[ \frac{4608}{5120} = 0.90 \quad\Longrightarrow\quad \text{second discount} = 10\% \]

Checking: \( 5120 \times 0.90 = 4608 \).

The overall reduction is \( \dfrac{6400 - 4608}{6400} = 28\% \), which matches \( 1 - 0.80 \times 0.90 \) - but 28% is the combined figure, not the second discount.