True Discount Questions and Answers

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: A

Successive discounts multiply the retained fractions.

\[ 10 \times (1 - 0.20) \times (1 - 0.10) = 10 \times 0.80 \times 0.90 \]

\[ = 10 \times 0.72 = \text{Rs. } 7.20 \]

The single equivalent discount is 28%, not 30% - the 10% is charged on Rs. 8 rather than on Rs. 10, so it takes off 80 paise instead of one rupee.

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.

Answer: C

Convert the order to dozens first, since both the price and the discount threshold are quoted per dozen.

\[ \frac{3000}{12} = 250 \text{ dozen} \]

That exceeds 12 dozen, so the 15% discount applies to the whole order.

\[ \text{list price} = 250 \times 100 = \text{Rs. } 25000 \]

\[ \text{payable} = 25000 \times 0.85 = \text{Rs. } 21250 \]

The discount applies to the entire order once the threshold is crossed, not only to the dozens above 12 - charging full price for the first 12 dozen would give Rs. 21,430.

Answer: D

Take the cost price as 100 and let the labelled price be \( M \). After a 10% discount the seller must still clear a 20% profit.

\[ M \times (1 - 0.10) = 100 \times 1.20 \]

\[ 0.9M = 120 \quad\Longrightarrow\quad M = \frac{120}{0.9} = 133.33 \]

\[ \text{increase over cost} = 133.33 - 100 = 33\tfrac{1}{3}\% \]

Adding 20% and 10% to get 30% is the trap: the profit is measured on the cost price while the discount comes off the labelled price, so the two act on different bases.

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: B

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

Here \( T = 2\tfrac{1}{2} = 2.5 \) 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 true discount is \( 2310 - 1680 = 630 \), and 37.5% of 1680 is indeed 630. Note the interest is reckoned on the present worth, not on the Rs. 2310 face value.

Answer: C

Write \( k = \dfrac{R \times 2}{100} \) for the rate–time product over the two years. Simple interest is charged on the full sum, true discount on the present worth:

\[ \text{SI} = 750\,k, \qquad \text{TD} = \frac{960\,k}{1+k} \]

Setting them equal and cancelling \( k \):

\[ 750(1+k) = 960 \quad\Longrightarrow\quad 1 + k = 1.28 \quad\Longrightarrow\quad k = 0.28 \]

\[ \frac{2R}{100} = 0.28 \quad\Longrightarrow\quad R = 14\% \]

Cancelling \( k \) from both sides is what makes the problem solvable in one line; the two years then divide out at the very end.

Answer: C

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

\[ \text{PW} = 2562 - 122 = \text{Rs. } 2440 \]

With \( T = \dfrac{4}{12} = \dfrac{1}{3} \) year:

\[ 122 = \frac{2440 \times R \times \tfrac{1}{3}}{100} \quad\Longrightarrow\quad R = \frac{122 \times 300}{2440} = 15\% \]

Charging the interest on the Rs. 2562 face value instead gives about 14.3%, which is why 14% sits among the options.

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 \( 1.30 \times 0.80 = 1.04 \).