Marked Price and Discount

Marked Price, Discount and Selling Price

The price printed on a tag is the marked price (MP). Questions also call it the list price, printed price, tag price or catalogue price. A discount is the amount taken off the marked price, so the customer pays

$$\text{SP} = \text{MP} - \text{Discount}$$

Discount % is always worked on the MP unless the question clearly says something else:

$$\text{Discount}\% = \frac{\text{Discount}}{\text{MP}} \times 100, \qquad \text{SP} = \text{MP} \times \frac{100 - d}{100}$$

• Finding the SP. A jacket marked ₹4,500 is sold at an 18% discount. The discount is 18% of 4,500 = ₹810, so SP = ₹3,690. Faster: 4,500 × 0.82 = ₹3,690.
• Finding the discount %. A lamp marked ₹750 is sold for ₹630. The discount is ₹120, and 120 ÷ 750 = 0.16, so the discount is 16%.
• Going back to the MP. A customer pays ₹1,071 after a 15% discount. That ₹1,071 is 85% of the MP, so MP = 1,071 × $\frac{100}{85}$ = ₹1,260. Adding 15% of ₹1,071 gives ₹1,231.65, which is wrong, because the 15% was taken from the MP and not from what the customer paid.

Discount as a fraction or in rupees. "One-fifth off" is a 20% discount. "₹1 off on every ₹8" is $\frac{1}{8}$ = 12.5%. "₹90 off on a ₹600 item" is 15%. Turn every form into a % of the MP before you compare offers.

Two discount rates on the same MP. "A customer saves ₹54 more if the discount is 20% instead of 15%. Find the MP." The extra 5% of the MP is ₹54, so MP = 54 × 20 = ₹1,080. (Check: ₹216 off against ₹162 off.)

Where CP fits. The discount links MP and SP; by itself it says nothing about the cost price. Profit or loss still comes from comparing SP with CP, as in Profit and Loss Basics. So a full question runs on three prices:

CP → (marked up) → MP → (discounted) → SP

The rest of this section joins these steps.

[SSC Tip] Words like "off", "rebate", "reduction" and "concession" all mean a discount on the MP. When an option equals the discount worked on the SP (₹120 ÷ ₹630 = 19.05% in the lamp case), it is the trap.

Markup: Marking the Price Above Cost

A shopkeeper first decides how far above cost to print the price. The gap MP − CP is the markup, and the markup % is worked on the CP unless the question says otherwise.

$$\text{MP} = \text{CP} \times \frac{100 + m}{100}$$

• Forward. A bag costs ₹1,600 and is marked 35% above cost: MP = 1,600 × 1.35 = ₹2,160.
• Backward. A watch is marked ₹3,150, which is 40% above cost: CP = 3,150 × $\frac{100}{140}$ = ₹2,250. Taking 40% off ₹3,150 gives ₹1,890, the usual wrong option.
• From the markup amount. "The MP is ₹420 more than the CP, and the markup is 28%." Then 28% of CP = ₹420, so CP = 420 ÷ 0.28 = ₹1,500 and MP = ₹1,920.

Markup as a ratio. A markup that is a neat fraction gives a ready MP : CP ratio:

CP below MP is a different percentage. If MP is 25% above CP, then CP is $\frac{1}{5}$ = 20% below MP. In general, a markup of m% on CP is $\frac{m}{100 + m} \times 100$ % of the MP. Going the other way, if the markup is 20% of the MP, then CP = 80% of MP and the markup on cost is $\frac{20}{80}$ = 25%.

Markup is not profit. If the goods are sold at the MP, the profit % equals the markup %. Once a discount is given, the profit is smaller than the markup, and it can even turn into a loss. Profit is always found from SP and CP.

CP of some articles = MP of others. "The CP of 12 pens equals the MP of 9 pens" means 12 CP = 9 MP, so MP : CP = 12 : 9 = 4 : 3, a markup of $33\frac{1}{3}\%$. The article counts flip over into the price ratio.

Markup and Discount Together: Profit or Loss

Most discount questions in SSC give a markup and a discount and ask for the profit or loss. Chain the two multipliers:

$$\text{SP} = \text{CP} \times \frac{100 + m}{100} \times \frac{100 - d}{100}$$

Net % in one line. The markup is a rise of m% and the discount is a fall of d% on the raised price, so the successive change rule (Chapter 3) gives

$$\text{Profit}\% = m - d - \frac{m d}{100}$$

A negative answer is a loss.

• Marked 40% above cost and sold at a 15% discount: 40 − 15 − $\frac{40 \times 15}{100}$ = 40 − 15 − 6 = 19% profit. On a CP of ₹2,000: MP = ₹2,800, SP = ₹2,380 and the profit is ₹380, which is 19%.
• Marked 20% above cost and sold at a 25% discount: 20 − 25 − 5 = −10, a 10% loss.
• Marked 30% above cost and sold at a 30% discount: 30 − 30 − 9 = −9, a 9% loss. The same rate up and then down always gives a loss of $\frac{x^2}{100}$ %.

The fraction way. Write each step as a fraction and multiply. A 50% markup is $\frac{3}{2}$ and a 20% discount is $\frac{4}{5}$, so SP ÷ CP = $\frac{6}{5}$, a 20% profit.

Profit in rupees to CP. "Goods are marked 50% above cost and sold at a 16% discount, giving a profit of ₹312. Find the CP." SP ÷ CP = 1.5 × 0.84 = 1.26, so the profit is 26% of CP and CP = 312 ÷ 0.26 = ₹1,200. (MP ₹1,800, SP ₹1,512.)

Price raised, then discounted. The same rule handles a shop that first raises its MP and then announces a discount. Raising the MP by 25% and then giving 20% off brings the price back to the old MP, because $\frac{5}{4} \times \frac{4}{5} = 1$. Raising it by 40% and giving 30% off gives 1.4 × 0.7 = 0.98, so the customer pays 2% less than the old MP.

[SSC Tip] The trap option is plain m − d (40 − 15 = 25% in the first case). The discount is taken on the MP, which is bigger than the CP, so it eats more than its face value. Always take off the $\frac{md}{100}$ term too.

The MP : CP Ratio Route

When a question gives two of markup, discount and profit and asks for the third, write the SP in two ways. From the MP it is MP × (100 − d)%, and from the CP it is CP × (100 + p)%. So

$$\text{MP} \times (100 - d) = \text{CP} \times (100 + p) \quad \Rightarrow \quad \frac{\text{MP}}{\text{CP}} = \frac{100 + p}{100 - d}$$

For a loss of l%, use 100 − l in place of 100 + p. Every type below is this one line.

1. MP for a target profit. "An item costs ₹840. What should its MP be so that the shopkeeper gains 25% even after a 16% discount?" MP = 840 × $\frac{125}{84}$ = ₹1,250. Check: 1,250 × 0.84 = ₹1,050, and 1,050 ÷ 840 = 1.25.

2. Markup % needed. "How far above cost should a trader mark his goods to gain 17% after a 10% discount?" $\frac{\text{MP}}{\text{CP}} = \frac{117}{90} = \frac{13}{10}$, so he marks them 30% above cost. As one formula: markup % = $\frac{p + d}{100 - d} \times 100 = \frac{27}{90} \times 100$ = 30%. Adding 17 + 10 = 27% is the trap.

3. Discount % allowed. "Goods are marked 50% above cost. What discount still leaves a 20% profit?" With CP = 100 the MP is 150 and the SP must be 120, so the discount is 30 on 150 = 20%. As a formula: d = $\frac{m - p}{100 + m} \times 100$.

4. CP from the MP. "An article marked ₹1,500 is sold at a 12% discount, at a profit of 10%. Find the CP." CP = 1,500 × $\frac{88}{110}$ = ₹1,200.

5. Ratio given. "MP : CP = 7 : 5 and the discount is 10%. Find the profit %." Take MP = 7 and CP = 5. SP = 0.9 × 7 = 6.3, a profit of 1.3 on 5 = 26%.

6. Profit at MP to the discount needed. "Sold at the MP, an article gives a 35% profit. What discount gives only an 8% profit?" With CP = 100, MP = 135 and the SP must be 108. Discount = 27 on 135 = 20%. Answering 35 − 8 = 27% is wrong because that 27 is on the CP, not on the MP.

Two Discounts, Two Results

Some questions give the result at two different discounts and ask for the MP, the CP or a third result. The key: a change in the discount % is a slice of the MP, and it equals the change in the SP.

Profit at one discount, loss at another. "At an 8% discount a shopkeeper gains ₹114. At a 20% discount he loses ₹30. Find the MP and the CP." Moving from 8% to 20% cuts the SP by 12% of the MP, and that cut turns a ₹114 gain into a ₹30 loss, a swing of ₹144. So 12% of MP = 144 and MP = ₹1,200. At an 8% discount SP = ₹1,104, so CP = 1,104 − 114 = ₹990. Check: at a 20% discount SP = ₹960, a loss of ₹30.

Two profits. "At a 15% discount the profit is ₹125, and at a 20% discount it is ₹80." The extra 5% of the MP is 125 − 80 = ₹45, so MP = ₹900. The SP at a 15% discount is ₹765, and CP = 765 − 125 = ₹640.

Add the two amounts when one is a profit and the other a loss; subtract them when both are profits (or both are losses).

Profit % at a new discount. "At a 12% discount the profit is 21%. What is the profit at a 20% discount?" MP : CP = 121 : 88 = 11 : 8, so MP = 137.5 when CP = 100. At a 20% discount SP = 110, a 10% profit.

Quick way: every extra 1% of discount takes 1% of the MP off the SP. Here MP = 1.375 × CP, so 8% more discount removes 8 × 1.375 = 11% of CP from the profit: 21 − 11 = 10%. Answering 21 − 8 = 13% forgets that the discount is on the MP.

[Tier 2] Buying at a discount, selling above the list price. A dealer who buys at a discount on the list price and sells above the list price has both prices tied to the same list price. "A dealer buys goods at 25% off the list price and sells them at 5% above the list price." With list price 100, CP = 75 and SP = 105, so the profit is $\frac{30}{75}$ = 40%, not 30%.

Discount Equal to Profit and Other Linked Conditions

Some questions do not give the markup at all. Instead they tie two amounts together. Put MP = 100 (or CP = 100), turn the condition into an equation, and the rest follows.

Discount (in ₹) equals profit (in ₹). MP − SP = SP − CP, so the SP sits exactly halfway between the CP and the MP. "A shopkeeper gives a 20% discount, and the discount equals his profit." With MP = 100, SP = 80 and the discount is 20. The profit is also 20, so CP = 60. Profit % = $\frac{20}{60}$ = $33\frac{1}{3}\%$, and the goods were marked $66\frac{2}{3}\%$ above cost.

Discount % equals profit %. "Goods marked 50% above cost are sold at a discount, and the profit % equals the discount %." With CP = 100, MP = 150. If both are d%, then 150 × $\frac{100 - d}{100}$ = 100 + d, so 150 − 1.5d = 100 + d and d = 20%. Check: 150 × 0.8 = 120, a 20% profit. In general, d = $\frac{m}{200 + m} \times 100$.

Discount equals the markup. If the rupee discount equals the rupee markup, the SP comes back to the CP: no profit, no loss, whatever the rates are.

Discount given as a % of CP. "The discount is 15% of the CP, and the profit is 20%." With CP = 100, SP = 120 and the discount is 15, so MP = 135. The markup is 35%, and the discount as a % of the MP is $\frac{15}{135}$ = $11\frac{1}{9}\%$.

Articles in the condition. "The discount on 6 articles equals the SP of 2 articles." So 6 × (MP − SP) = 2 × SP, which gives 6 MP = 8 SP and SP = $\frac{3}{4}$ of MP: a 25% discount. In general, if the discount on n articles equals the SP of k articles, the discount is $\frac{k}{n + k} \times 100$ %.

"The MP of 21 articles equals the SP of 25 articles" gives SP ÷ MP = $\frac{21}{25}$, a discount of $\frac{4}{25}$ = 16%.

Same rate up and down. A trader who marks up by x% and then gives x% off loses $\frac{x^2}{100}$ %. If the loss is 6.25%, then x² = 625 and x = 25%.

Read which two things are equal. "Discount equals profit" compares rupee amounts. "Discount % equals profit %" compares two percentages that sit on different bases (the discount on the MP, the profit on the CP). Set up the equation for the one the question states; mixing them is the usual way to land on a wrong option.

Successive Discounts and the Single Equivalent Discount

In a sale, a second discount is taken on the price left after the first one. So a 20% discount followed by a 10% discount is not 30% off.

MP ₹9,500 with discounts of 20% and then 10%: after the first, the price is ₹7,600; the second takes 10% of ₹7,600 and leaves ₹6,840. The total discount is ₹2,660, which is 28% of the MP.

Formula for two discounts.

$$\text{Single equivalent discount} = d_1 + d_2 - \frac{d_1 d_2}{100}$$

20 + 10 − 2 = 28%. The minus term is always there, so successive discounts always add up to less than their sum.

Multipliers. Write each discount as the part you pay and multiply: 20% then 10% is 0.8 × 0.9 = 0.72, so you pay 72% and the discount is 28%. Handy fractions:

Three discounts. Multiply three factors, or use the formula twice. For 25%, 10% and 4%: 0.75 × 0.9 × 0.96 = 0.648, so the single discount is 35.2%. With the formula: 25 and 10 give 32.5, then 32.5 and 4 give 32.5 + 4 − 1.3 = 35.2%.

Order does not matter. 10% then 20% is 0.9 × 0.8, the same 0.72. When options list the same discounts in different orders, they are equal.

Equal discounts. Two discounts of 10% make 19%, two of 20% make 36%, and three of 10% make 1 − 0.729 = 27.1%.

Working back to the MP. "After discounts of 15% and 10%, a customer pays ₹2,754. Find the MP." MP = 2,754 ÷ (0.85 × 0.9) = 2,754 ÷ 0.765 = ₹3,600.

Finding the missing discount. "An article marked ₹1,800 is sold for ₹1,224 after two discounts, the first being 20%." After the first discount the price is ₹1,440, so the second discount is 216 on 1,440 = 15%. Working ₹216 out of ₹1,800 gives 12%, the trap.

Single against successive. A single 35% discount and successive discounts of 20% and 15% (which make 32%) differ by 3% of the MP. If that gap is ₹96, the MP is 96 ÷ 0.03 = ₹3,200.

Buy x Get y Free

A "buy x, get y free" offer is a discount in disguise. The customer pays for x articles and takes home x + y, so

$$\text{Effective discount} = \frac{y}{x + y} \times 100$$

• Buy 4, get 1 free: $\frac{1}{5}$ = 20% (not $\frac{1}{4}$ = 25%).
• Buy 3, get 2 free: $\frac{2}{5}$ = 40%.
• Buy 5, get 3 free: $\frac{3}{8}$ = 37.5%.
• Buy 2, get 1 free: $\frac{1}{3}$ = $33\frac{1}{3}\%$.

The base is the number of articles taken home, because the saving is spread over all of them. "1 free in every 5" is the same as buy 4, get 1 free.

Free articles plus a discount. "Buy 3, get 1 free, and a further 10% off the bill." The buyer pays $\frac{3}{4} \times \frac{9}{10} = \frac{27}{40}$ of the MP of what he takes home, so the effective discount is $\frac{13}{40}$ = 32.5%. This is just a successive discount of 25% and 10%.

Profit with free articles. "Goods are marked 50% above cost, and the offer is buy 4, get 1 free. Find the profit %." For 5 articles the shop spends 5 CP and collects 4 MP = 4 × 1.5 CP = 6 CP. Profit = 1 CP on 5 CP = 20%. In one line:

$$\frac{\text{SP}}{\text{CP}} = \frac{100 + m}{100} \times \frac{x}{x + y}$$

Markup needed with a free offer. Turn the line round. "A shop gives 1 article free with every 4 bought and wants a 20% profit. How far above cost should it mark the goods?" $\frac{\text{MP}}{\text{CP}} = 1.2 \times \frac{5}{4} = 1.5$, so 50% above cost.

A free gift with a different price. "With a ₹1,200 mixer, a ₹300 kettle is given free." The customer pays ₹1,200 for goods marked ₹1,500, a discount of $\frac{300}{1500}$ = 20%. Divide the free item's price by the total MP, not by the price paid (300 ÷ 1,200 = 25% is the trap).

Comparing Discount Schemes

To compare offers, turn each one into a single equivalent discount, then compare those numbers. The bigger discount is better for the buyer; the smaller one is better for the seller.

Single against successive. "Which is better on an item marked ₹4,500: a single 32% discount, or 20% followed by 12%?" The second is only 20 + 12 − 2.4 = 29.6%, so the single 32% is better for the buyer, by 2.4% of ₹4,500 = ₹108.

In general, a single discount equal to the sum of two successive ones is always better for the buyer, by $\frac{d_1 d_2}{100}$ % of the MP, so an option that calls them equal is wrong.

Two pairs with the same total. 25% + 10% gives 35 − 2.5 = 32.5%, while 20% + 15% gives 35 − 3 = 32%. When the totals match, the pair with the smaller product $d_1 d_2$ gives the bigger discount, so the more uneven split wins.

Many small against one big. Three discounts of 10% make 27.1%, which is just above a single 27% but below a single 28%. Close calls like this are where SSC places its options, so work them out and do not guess.

Free articles against a % discount.

So "buy 3, get 1 free" and "25% off" are equal for a buyer who takes all 4 articles, and "buy 2, get 1 free" beats 30% off.

From the seller's side. "A shop marks its goods 40% above cost. Which offer leaves it the larger profit: 25% off, or buy 4, get 1 free?" With 25% off: 1.4 × 0.75 = 1.05, a 5% profit. With buy 4, get 1 free: 1.4 × 0.8 = 1.12, a 12% profit. The free offer leaves more profit because its discount is smaller (20%).

When two offers are equal. A single 40% discount and successive discounts of 25% and 20% are the same, because 25 + 20 − 5 = 40. The difference in what the buyer pays is zero on any MP.

Tax After a Discount, Trade and Cash Discounts, and Sale Conditions

GST after a discount. Tax is charged on the price after the discount, so the bill is a discount followed by a rise:

$$\text{Bill} = \text{MP} \times \frac{100 - d}{100} \times \frac{100 + t}{100}$$

MP ₹3,500, discount 12%, GST 18%: the discounted price is ₹3,080 and the bill is 3,080 × 1.18 = ₹3,634.40. Taking a net 18 − 12 = 6% rise gives ₹3,710, which is wrong.

Backwards: a bill of ₹4,248 after a 10% discount and 18% GST means MP = 4,248 ÷ (0.9 × 1.18) = 4,248 ÷ 1.062 = ₹4,000.

Trade discount and cash discount. A wholesaler gives retailers a trade discount on the list price. A retailer who pays in cash, or pays at once, gets a further cash discount on the reduced amount. The two are successive discounts. List price ₹12,000, trade discount 20%, cash discount 5%: 12,000 × 0.8 × 0.95 = ₹9,120.

[Tier 2] "A wholesaler allows a trade discount of 20% and a cash discount of 6.25% and still makes a 20% profit. How far above cost is the list price?" The buyer pays 0.8 × 0.9375 = 0.75 of the list price, so $\frac{\text{MP}}{\text{CP}} = \frac{1.2}{0.75}$ = 1.6, and the list price is 60% above cost.

Discounts with a condition. Read who qualifies and on which part of the bill.

• "10% off the whole bill if it is above ₹3,000": a ₹2,950 bill gets nothing, so check the condition before you discount.
• "5% off the first ₹2,000 and 15% off the amount above ₹2,000" on a bill of ₹6,400: discount = 100 + 660 = ₹760, so the customer pays ₹5,640.
• "50% off the second shirt" when two shirts of the same price are bought is a 25% discount on the pair.

Seasonal sales. "Sale: 30% off, plus an extra 10% at the counter" is a successive discount: 0.7 × 0.9 = 0.63, so 37% in all, not 40%.

[SSC Tip] A discount and a tax at the same rate do not cancel. A 12% discount followed by 12% GST makes the bill 0.88 × 1.12 = 0.9856 of the MP, which is 1.44% below it.

Solved Examples

Question 1

An electric kettle marked ₹1,875 is sold for ₹1,500. What is the rate of discount? (a) 25% (b) 20% (c) 18.75% (d) 22.5%

Answer. The discount is 1,875 − 1,500 = ₹375. Discount % is always on the MP, so it is 375 ÷ 1,875 = $\frac{1}{5}$ = (b) 20%. Working ₹375 out of the ₹1,500 paid gives 25%, which is the trap.

Question 2

After a discount of 15%, a customer pays ₹1,904 for a mixer grinder. What is its marked price? (a) ₹2,189.60 (b) ₹2,284.80 (c) ₹2,200 (d) ₹2,240

Answer. The ₹1,904 is 85% of the MP, so MP = 1,904 × $\frac{100}{85}$ = (d) ₹2,240. Check: 15% of 2,240 is ₹336, and 2,240 − 336 = 1,904.

Question 3

A customer who gets a 22% discount on an article instead of a 15% discount saves ₹63 more. What is the marked price of the article? (a) ₹900 (b) ₹630 (c) ₹420 (d) ₹1,260

Answer. Both discounts are on the same MP, so the extra saving is (22 − 15)% = 7% of the MP. So 7% of MP = ₹63, and MP = 63 × $\frac{100}{7}$ = (a) ₹900. Check: 22% of 900 = ₹198 and 15% of 900 = ₹135, a gap of ₹63.

Question 4

A shopkeeper marks a bag 75% above its cost price and sells it for ₹1,820 after giving a 20% discount. What is the cost price of the bag? (a) ₹1,040 (b) ₹1,456 (c) ₹1,300 (d) ₹1,274

Answer. I go back one step at a time. The ₹1,820 is 80% of the MP, so MP = 1,820 ÷ 0.8 = ₹2,275. The MP is 175% of the CP, so CP = 2,275 ÷ 1.75 = (c) ₹1,300. Check: 1,300 × 1.75 × 0.8 = 1,820.

Question 5

A trader marks his goods 45% above the cost price and allows a discount of 20%. What is his profit %? (a) 16% (b) 25% (c) 9% (d) 20%

Answer. Net effect = 45 − 20 − $\frac{45 \times 20}{100}$ = 45 − 20 − 9 = (a) 16%. On a CP of ₹100: MP = ₹145 and SP = 145 × 0.8 = ₹116, so the profit is ₹16. Just subtracting (45 − 20 = 25%) misses that the discount is taken on ₹145, not on ₹100.

Question 6

An article is marked 15% above its cost price and sold at a discount of 20%. What is the gain or loss %? (a) 5% loss (b) 8% loss (c) 8% gain (d) 3% loss

Answer. SP ÷ CP = 1.15 × 0.8 = 0.92. So the SP is 92% of the CP, which is (b) 8% loss. With the formula: 15 − 20 − 3 = −8, and the minus sign means a loss.

Question 7

Goods are marked 35% above cost and sold at a 12% discount. The profit on one article is ₹470. What is its cost price? (a) ₹2,970 (b) ₹3,375 (c) ₹2,000 (d) ₹2,500

Answer. SP ÷ CP = 1.35 × 0.88 = 1.188, so the profit is 18.8% of the CP. So CP = 470 ÷ 0.188 = (d) ₹2,500. Check: MP = ₹3,375 and SP = 3,375 × 0.88 = ₹2,970, which is ₹470 more than ₹2,500.

Question 8

A study table costs a shopkeeper ₹1,560. At what price should he mark it so that he earns a 20% profit even after allowing a 22% discount? (a) ₹2,215.20 (b) ₹2,283.84 (c) ₹2,400 (d) ₹1,872

Answer. He wants SP = 1,560 × 1.2 = ₹1,872. After a 22% discount the buyer pays 78% of the MP, so 78% of MP = 1,872 and MP = 1,872 ÷ 0.78 = (c) ₹2,400. In ratio form, $\frac{\text{MP}}{\text{CP}} = \frac{120}{78}$.

Question 9

By what percentage above the cost price must a shopkeeper mark his goods so that he gains 12% after allowing a discount of 4%? (a) 16% (b) 16.48% (c) $16\frac{2}{3}\%$ (d) 15%

Answer. $\frac{\text{MP}}{\text{CP}} = \frac{100 + 12}{100 - 4} = \frac{112}{96} = \frac{7}{6}$. So the MP is $\frac{1}{6}$ more than the CP, which is (c) $16\frac{2}{3}\%$ above cost. Check: CP 6, MP 7, SP = 7 × 0.96 = 6.72, and 6.72 ÷ 6 = 1.12.

Question 10

A shopkeeper marks his goods 75% above the cost price. What discount % should he allow so that he still gains 40%? (a) 35% (b) 25% (c) 22% (d) 20%

Answer. Take CP = 100. Then MP = 175 and the SP must be 140. The discount is 175 − 140 = 35 on an MP of 175, which is $\frac{35}{175}$ = (d) 20%. Taking the 35 as a % of the CP gives 35%, but the discount must be worked on the MP.

Question 11

The ratio of the cost price to the marked price of an article is 16 : 25. If a discount of 12% is allowed, what is the profit %? (a) 37.5% (b) 44.25% (c) 36% (d) 32%

Answer. Take CP = 16 and MP = 25. SP = 25 × 0.88 = 22. Profit = 22 − 16 = 6 on 16, which is $\frac{6}{16}$ = (a) 37.5%.

Question 12

When an article is sold at a 15% discount, the shopkeeper gains 19%. What will be his gain or loss % if he sells it at a 25% discount? (a) 9% gain (b) 5% loss (c) 5% gain (d) 6% gain

Answer. $\frac{\text{MP}}{\text{CP}} = \frac{119}{85} = \frac{7}{5}$, so with CP = 100 the MP is 140. At a 25% discount SP = 140 × 0.75 = 105, which is (c) 5% gain. Quick way: the extra 10% discount is 10% of the MP = 14% of the CP, so the gain drops from 19% to 5%.

Question 13

If a sewing machine is sold at a 12% discount, the shopkeeper makes a profit of ₹200. If it is sold at a 24% discount, he makes a loss of ₹100. What is its cost price? (a) ₹2,000 (b) ₹2,500 (c) ₹2,200 (d) ₹1,900

Answer. Going from a 12% to a 24% discount cuts the SP by 12% of the MP. That cut turns a ₹200 profit into a ₹100 loss, a drop of ₹300. So 12% of MP = ₹300 and MP = ₹2,500. At a 12% discount SP = 2,500 × 0.88 = ₹2,200, so CP = 2,200 − 200 = (a) ₹2,000. Check: at a 24% discount SP = ₹1,900, a loss of ₹100.

Question 14

A shopkeeper allows a 25% discount on an article and finds that the discount is equal to his profit in rupees. By what % above the cost price had he marked it? (a) 50% (b) 100% (c) 75% (d) $33\frac{1}{3}\%$

Answer. Take MP = 100. The discount is 25, so SP = 75. The profit equals the discount, 25, so CP = 75 − 25 = 50. The MP (100) is double the CP (50), so it was marked (b) 100% above cost. 50% is his profit % (25 on 50), not the markup.

Question 15

A trader marks his goods 40% above the cost price. He then gives a discount and finds that his profit % is equal to the discount %. What is the discount %? (a) 20% (b) $16\frac{2}{3}\%$ (c) $14\frac{2}{7}\%$ (d) 18%

Answer. Take CP = 100, so MP = 140. If both are d%, the SP is 140 × $\frac{100 - d}{100}$ and it is also 100 + d. So 140 − 1.4d = 100 + d, which gives 2.4d = 40 and d = (b) $16\frac{2}{3}\%$. Check: 140 × $\frac{5}{6}$ = $116\frac{2}{3}$, a profit of $16\frac{2}{3}\%$.

Question 16

What single discount is equivalent to two successive discounts of 22% and 15%? (a) 33.7% (b) 37% (c) 34.3% (d) 33%

Answer. Single discount = 22 + 15 − $\frac{22 \times 15}{100}$ = 37 − 3.3 = (a) 33.7%. With multipliers: 0.78 × 0.85 = 0.663, so the buyer pays 66.3% and saves 33.7%.

Question 17

A television marked ₹8,000 is sold after three successive discounts of 25%, 15% and 8%. How much does the customer pay? (a) ₹4,160 (b) ₹4,692 (c) ₹4,416 (d) ₹5,100

Answer. Price = 8,000 × 0.75 × 0.85 × 0.92. Step by step: ₹8,000 → ₹6,000 → ₹5,100 → (b) ₹4,692. The single discount here is 41.35%.

Question 18

An article marked ₹3,500 is sold for ₹2,646 after two successive discounts. If the first discount is 10%, what is the second discount? (a) 14.4% (b) 16% (c) 15% (d) 18%

Answer. After the first discount the price is 3,500 × 0.9 = ₹3,150. The second discount brings it to ₹2,646, a cut of ₹504. So the second discount is 504 ÷ 3,150 = (b) 16%. Working ₹504 out of the MP gives 14.4%, but the second discount works on ₹3,150.

Question 19

The difference between a single discount of 36% and two successive discounts of 24% and 12% on an article is ₹144. What is the marked price of the article? (a) ₹4,800 (b) ₹6,000 (c) ₹5,000 (d) ₹4,000

Answer. 24% and 12% make 24 + 12 − 2.88 = 33.12%. The gap from 36% is 2.88% of the MP, so MP = 144 ÷ 0.0288 = (c) ₹5,000. Check: 36% of 5,000 is ₹1,800, while 5,000 × 0.76 × 0.88 = ₹3,344 means a discount of ₹1,656. The gap is ₹144.

Question 20

Which of these offers is the best for a buyer? (a) A single discount of 35% (b) Successive discounts of 20% and 15% (c) Buy 2, get 1 free (d) Successive discounts of 25%, 10% and 5%

Answer. I turn each offer into one discount. 20% and 15% make 32%. Buy 2, get 1 free is $\frac{1}{3}$ = $33\frac{1}{3}\%$. For 25%, 10% and 5%: 0.75 × 0.9 × 0.95 = 0.64125, so the discount is 35.875%. That just beats the flat 35%, so the best offer is (d) Successive discounts of 25%, 10% and 5%.

Question 21

A shopkeeper marks his goods 75% above the cost price and offers 'buy 5, get 2 free'. What is his profit %? (a) 25% (b) 5% (c) 35% (d) 20%

Answer. For 7 articles he spends 7 CP and collects the MP of 5, which is 5 × 1.75 = 8.75 CP. Profit = 1.75 on 7, which is (a) 25%. Treating the offer as a 40% discount ($\frac{2}{5}$) gives 5%, but the 2 free articles are spread over all 7, so the discount is only $\frac{2}{7}$.

Question 22

A shop offers 'buy 4, get 1 free' and a further 10% off the bill, and still makes a profit of 17%. By what % above the cost price are the goods marked? (a) 47% (b) $73\frac{1}{3}\%$ (c) 46.25% (d) 62.5%

Answer. The free article is a 20% discount ($\frac{1}{5}$), and then 10% more comes off, so the buyer pays 0.8 × 0.9 = 0.72 of the MP. MP ÷ CP = 1.17 ÷ 0.72 = 1.625, so the goods are marked (d) 62.5% above cost.

Question 23

A water purifier marked ₹4,800 is sold at a 25% discount, and 5% GST is charged on the discounted price. How much does the buyer pay? (a) ₹3,840 (b) ₹3,600 (c) ₹3,780 (d) ₹3,960

Answer. Discounted price = 4,800 × 0.75 = ₹3,600. GST is charged on this: 3,600 × 1.05 = (c) ₹3,780. ₹3,840 takes a net 20% off the MP (25 − 5), but the tax is on the smaller, discounted price.

Question 24

[Tier 2] A wholesaler allows a trade discount of 25% on the list price and a further cash discount of 4% on the remaining amount, and still gains 8%. By what % is the list price above the cost price? (a) 37% (b) 44% (c) 40% (d) 50%

Answer. The trade and cash discounts are successive, so the buyer pays 0.75 × 0.96 = 0.72 of the list price. List price ÷ CP = 1.08 ÷ 0.72 = 1.5, so the list price is (d) 50% above cost.