True Discount Questions and Answers

Answer: C

Compare the two schemes by the fraction of the price that survives them.

20% then 20%:

\[ 800 \times 0.80 \times 0.80 = 800 \times 0.64 = \text{Rs. } 512 \]

30% then 10%:

\[ 800 \times 0.70 \times 0.90 = 800 \times 0.63 = \text{Rs. } 504 \]

\[ \text{extra saving} = 512 - 504 = \text{Rs. } 8 \]

Both schemes total 40% on paper, yet the more lopsided split wins. In general \( (1-a)(1-b) = 1 - a - b + ab \), so for a fixed sum \( a+b \) the retained fraction shrinks as the product \( ab \) shrinks - and \( 0.30 \times 0.10 = 0.03 \) is smaller than \( 0.20 \times 0.20 = 0.04 \).

Answer: C

Find the discounted price of the dozen first, then the price of one pair.

\[ \text{price of 12 pairs} = 8000 \times 0.90 = \text{Rs. } 7200 \]

\[ \text{price of 1 pair} = \frac{7200}{12} = \text{Rs. } 600 \]

\[ n = \frac{2400}{600} = 4 \]

Dividing Rs. 2400 by the undiscounted unit price of \( 8000/12 = 666.67 \) gives 3.6, which rounds down to 3 - the discount must be applied before the division, not after.

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.

With \( k = \dfrac{12 \times 1}{100} = 0.12 \):

\[ \text{BG} = \text{TD} \times k \quad\Longrightarrow\quad \text{TD} = \frac{6}{0.12} = \text{Rs. } 50 \]

Checking the whole chain: PW = \( 50/0.12 = 416.67 \), face value \( 466.67 \), and BD = \( 466.67 \times 0.12 = 56 \), which is indeed Rs. 6 more than the true discount.

Answer: B

Take the cost price as 100. He first marks the article up by 30%, then takes 20% off that inflated figure.

\[ \text{marked price} = 100 \times 1.30 = 130 \]

\[ \text{selling price} = 130 \times 0.80 = 104 \]

\[ \text{gain} = 104 - 100 = 4 \quad\Longrightarrow\quad 4\% \]

The two percentages do not cancel: 20% of 130 is 26, while 30% of 100 is only 30 - so a Rs. 4 margin survives. In general the net factor is \( 1.30 \times 0.80 = 1.04 \).

Answer: A

Rs. 1404 is paid in two equal half-yearly installments, so each installment is Rs. 702 - one due in 6 months, the other in 12 months. Discount each separately.

First installment (\( T = 0.5 \) year at 8%):

\[ \text{PW}_{1} = \frac{702}{1 + 0.04} = \frac{702}{1.04} = 675 \]

Second installment (\( T = 1 \) year):

\[ \text{PW}_{2} = \frac{702}{1 + 0.08} = \frac{702}{1.08} = 650 \]

\[ \text{Total PW} = 675 + 650 = \text{Rs. } 1325 \]

Discounting the whole Rs. 1404 for a single average period gives a different (and wrong) figure - each installment must be discounted for its own waiting time.

Answer: D

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:

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

\[ \text{PW} = 1760 - 160 = 1600 \]

The true discount is simple interest on the present worth, so

\[ 160 = \frac{1600 \times 12 \times T}{100} = 192\,T \]

\[ T = \frac{160}{192} = \frac{5}{6} \text{ year} = 10 \text{ months} \]

Note the interest is charged on the present worth of Rs. 1600, not on the Rs. 1760 face value - using the face value gives about 9 months, which is why that option is listed.

Answer: A

The two schemes differ only in the discount rate, so the shortfall is the extra discount measured on the marked price.

\[ (8\% - 5\%) \times \text{MP} = 45 \]

\[ 0.03 \times \text{MP} = 45 \quad\Longrightarrow\quad \text{MP} = \frac{45}{0.03} = \text{Rs. } 1500 \]

Checking: a 5% discount gives Rs. 1425 and an 8% discount Rs. 1380 - exactly Rs. 45 apart. The cost price never enters the calculation, because it is the same under both schemes and cancels out of the profit difference.

Answer: A

\[ \text{PW} = 5125 - 125 = \text{Rs. } 5000 \]

The true discount is simple interest on that present worth, over \( T = \dfrac{6}{12} = 0.5 \) year:

\[ 125 = \frac{5000 \times R \times 0.5}{100} = 25R \]

\[ R = \frac{125}{25} = 5\% \text{ per annum} \]

The half-year must be converted before dividing; leaving \( T = 6 \) would give a rate about twelve times too small.

Answer: B

Work forwards from cost to selling price, then backwards from selling price to the advertised (marked) price.

\[ \text{SP} = 900 \times 1.10 = \text{Rs. } 990 \]

The 12% discount is taken off the advertised price, so

\[ \text{MP} \times (1 - 0.12) = 990 \]

\[ \text{MP} = \frac{990}{0.88} = \text{Rs. } 1125 \]

Profit is reckoned on the cost price and discount on the marked price - two different bases, which is why they cannot be netted off against each other.

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.