True Discount Questions and Answers

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: 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: A

The Rs. 224 he did not have to pay is exactly the 35% discount.

\[ 0.35 \times \text{MP} = 224 \quad\Longrightarrow\quad \text{MP} = \frac{224}{0.35} = \text{Rs. } 640 \]

What he actually paid is the remaining 65%:

\[ 0.65 \times 640 = \text{Rs. } 416 \]

Equivalently, \( 640 - 224 = 416 \). Both Rs. 640 and Rs. 224 appear as options because they are the two intermediate values, not the amount paid.

Answer: C

The extra saving comes from the extra 5 percentage points of discount, and that difference is measured on the marked price.

\[ (25\% - 20\%) \times \text{MP} = 1000 \]

\[ 0.05 \times \text{MP} = 1000 \quad\Longrightarrow\quad \text{MP} = \frac{1000}{0.05} = \text{Rs. } 20{,}000 \]

As a check, a 20% discount leaves Rs. 16,000 and a 25% discount leaves Rs. 15,000 - a difference of exactly Rs. 1000. Only the gap between the two discount rates matters, never either rate on its own.

Answer: B

Successive discounts multiply rather than add. For discounts \( a \) and \( b \) the price retained is \( (1-a)(1-b) \).

1) 5% then 5%:

\[ 0.95 \times 0.95 = 0.9025 \quad\Longrightarrow\quad \text{discount } 9.75\% \]

2) a single 10%:

\[ 0.90 \quad\Longrightarrow\quad \text{discount } 10\% \]

3) 8% then 2%:

\[ 0.92 \times 0.98 = 0.9016 \quad\Longrightarrow\quad \text{discount } 9.84\% \]

The single 10% is largest, so option 2 wins. Two discounts summing to 10% always fall short of a single 10%, because the second is applied to an already reduced price - the shortfall is the product \( ab \), here 0.25% and 0.16%.

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: A

Work out what one trouser saves, then divide.

\[ \text{saving per trouser} = 0.25 \times 320 = \text{Rs. } 80 \]

\[ n = \frac{400}{80} = 5 \]

So he must buy 5 trousers. Dividing the Rs. 400 target by the full Rs. 320 price instead gives 1.25 and tempts a rounding to 4 - but the saving, not the price, is what accumulates toward the target.

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: 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

Successive discounts multiply; they do not add to 30%.

\[ \text{SP} = 500 \times (1 - 0.20) \times (1 - 0.10) \]

\[ = 500 \times 0.80 \times 0.90 = 500 \times 0.72 = \text{Rs. } 360 \]

The equivalent single discount is 28%, not 30%: the second 10% is charged on Rs. 400 rather than on Rs. 500, so it saves Rs. 40 instead of Rs. 50. Adding the discounts would give Rs. 350, which is the listed trap.