Profit and Loss: Articles, Two Items and Changing Prices

Buying and Selling in Lots

Shops often buy "6 for ₹25" and sell "5 for ₹24". The numbers of articles are different, so the two prices can't be compared directly. Bring both to the same number of articles, or to the price of one article.

Same number of articles. "A boy buys pencils at 6 for ₹25 and sells them at 5 for ₹24. Find his gain %." Take LCM(6, 5) = 30 pencils.

• CP of 30 pencils = 5 × 25 = ₹125.
• SP of 30 pencils = 6 × 24 = ₹144.
• Gain = ₹19 on ₹125, which is 15.2%.

The cross-multiplication shortcut. Bought x articles for ₹a and sold y articles for ₹b:

$$\text{gain}\% = \frac{x \times b - y \times a}{y \times a} \times 100$$

For the pencils: $\frac{6 \times 24 - 5 \times 25}{5 \times 25} \times 100 = \frac{19}{125} \times 100 = 15.2\%$. A negative answer means a loss.

Numbers swapped. "Bought at 5 for ₹4 and sold at 4 for ₹5." CP of 20 = ₹16 and SP of 20 = ₹25, so the gain is $\frac{9}{16}$ = 56.25%. Whenever x articles are bought for ₹y and y articles are sold for ₹x, the gain is $\frac{x^2 - y^2}{y^2} \times 100$.

A loss. "Bought at 8 for ₹15 and sold at 10 for ₹17." Take 40 articles: CP = 5 × 15 = ₹75 and SP = 4 × 17 = ₹68. The loss is ₹7 on ₹75, which is $9\frac{1}{3}\%$.

How many to sell for a fixed sum. "By selling 20 oranges for ₹50 a man loses 4%. How many should he sell for ₹50 to gain 20%?" The money stays at ₹50, so the number of oranges must shrink as the price of each orange goes up:

$$\text{new number} = \text{old number} \times \frac{100 - \text{old loss}\%}{100 + \text{new gain}\%} = 20 \times \frac{96}{120} = 16$$

Check: selling 16 instead of 20 for the same ₹50 makes each orange $\frac{20}{16}$ = 1.25 times dearer, and 0.96 × 1.25 = 1.2, which is a 20% gain. With an old gain instead of a loss, use 100 + old gain on top.

[SSC Tip] Fewer articles for the same money means a higher price per article. In "how many for ₹x" questions the fraction is the opposite way up from the usual multiplier, which is why 20 × $\frac{120}{96}$ = 25 is the usual wrong option.

Dozens, Scores, Kilograms and Equal Numbers at Two Rates

Buying in one unit and selling in another works the same way: change both prices to one unit first.

Dozen and score. "Eggs are bought at ₹66 a dozen and sold at ₹115 a score. Find the gain %." One egg costs 66 ÷ 12 = ₹5.50 and sells for 115 ÷ 20 = ₹5.75. The gain is ₹0.25 on ₹5.50, which is $\frac{1}{22}$ = $4\frac{6}{11}\%$. Comparing ₹115 with ₹66 directly gives a silly 74%.

Kilograms and packets. "Tea is bought at ₹520 per kg and sold in 250 g packets at ₹156 each." Four packets make a kg, so the SP is 4 × 156 = ₹624 per kg. The gain is ₹104 on ₹520, which is 20%.

Equal numbers at two rates. "A vendor buys equal numbers of lemons at 4 for ₹10 and at 6 for ₹10, mixes them and sells them all at 5 for ₹10. Find his gain or loss." It looks like no profit and no loss, because 5 is the average of 4 and 6. It isn't. Take 12 of each kind (the LCM of 4 and 6):

• 12 lemons at 4 for ₹10 cost ₹30, and 12 at 6 for ₹10 cost ₹20, so 24 lemons cost ₹50.
• 24 lemons at 5 for ₹10 bring in ₹48.
• The loss is ₹2 on ₹50, which is 4%.

With equal numbers it's the price of one lemon that gets averaged, not the number per ₹10. The two prices are ₹2.50 and about ₹1.67, which average to about ₹2.08, and he sells at ₹2.

Equal money at two rates. If he spends equal amounts at the two rates instead (₹60 at 4 for ₹10 and ₹60 at 6 for ₹10), he gets 24 + 36 = 60 lemons for ₹120. At 5 for ₹10 they bring exactly ₹120, so there is no profit and no loss. Equal money averages the numbers per ₹10; equal numbers average the prices.

SP of Some Articles Equals CP of Others

"The SP of 14 articles is equal to the CP of 21 articles." Take the CP of one article as ₹1. Then 14 articles sell for ₹21, so one sells for $\frac{21}{14} = \frac{3}{2}$, and the gain is $\frac{1}{2}$ = 50%.

$$\text{If SP of } x \text{ articles} = \text{CP of } y \text{ articles}, \quad \text{gain or loss}\% = \frac{y - x}{x} \times 100$$

Divide by x, the number of articles sold. If y is bigger than x it is a gain; if y is smaller it is a loss.

A loss. "The CP of 20 articles equals the SP of 25." Here 25 are sold and their price equals the cost of 20, so the loss % is $\frac{5}{25}$ = 20%. Dividing by 20 gives 25%, the standard trap.

Profit equal to the SP or the CP of some articles. "By selling 56 m of cloth, a merchant gains the SP of 8 m." Profit = SP of 8 m, so the CP of 56 m = SP of 56 m − SP of 8 m = SP of 48 m. The gain is $\frac{8}{48}$ = $16\frac{2}{3}\%$.

So gaining the CP of 8 m on 56 m is $\frac{8}{56}$ = $14\frac{2}{7}\%$, and losing the SP of 4 chairs while selling 36 chairs is $\frac{4}{40}$ = 10%.

Why the SP cases are different: gaining the SP of 8 m out of 56 m means the cost of 56 m equals the SP of only 48 m. The gain is shared over 48, not 56.

Working backwards. "A shopkeeper makes a 25% profit. The SP of how many articles equals the CP of 30?" One article sells for 1.25 times its cost, so the number is 30 ÷ 1.25 = 24.

Two Articles Sold at the Same Price

Same SP, x% gain on one and x% loss on the other. This always ends in a loss, and the loss % on the whole deal is

$$\text{loss}\% = \frac{x^2}{100}$$

"Two scooters are sold for ₹38,400 each, one at a 20% gain and the other at a 20% loss."

• CP of the first = 38,400 ÷ 1.2 = ₹32,000.
• CP of the second = 38,400 ÷ 0.8 = ₹48,000.
• Total CP = ₹80,000 and total SP = ₹76,800, so the loss is ₹3,200, which is 4% = $\frac{20^2}{100}$.

Why a loss? The 20% gain is on the cheaper scooter and the 20% loss is on the dearer one, so more rupees are lost than gained. The loss in rupees is

$$\frac{2 \times \text{SP} \times x^2}{100^2 - x^2} = \frac{2 \times 38400 \times 400}{9600} = 3200$$

Same SP, different rates. "Two phones are sold for ₹13,800 each, one at a 15% gain and the other at an 8% loss." Find each CP:

• 13,800 ÷ 1.15 = ₹12,000 and 13,800 ÷ 0.92 = ₹15,000.
• Total CP = ₹27,000, total SP = ₹27,600, so the gain is ₹600, which is $2\frac{2}{9}\%$.

You can't average 15 and −8 here, because the two CPs are different; the average, 3.5%, is the trap.

The ratio of the CPs. With the same SP, CP (gain) × 1.15 = CP (loss) × 0.92, so each CP goes with the other multiplier:

$$\text{CP (gain)} : \text{CP (loss)} = (100 - \text{loss}\%) : (100 + \text{gain}\%) = 92 : 115 = 4 : 5$$

Total cost given. "A man buys two goats for ₹16,400. He sells one at a 20% gain and the other at a 15% loss, and gets the same price for each." CP (gain) : CP (loss) = 85 : 120 = 17 : 24, so the 41 parts are ₹16,400 and one part is ₹400. The goats cost ₹6,800 and ₹9,600, and each sells for ₹8,160.

[SSC Tip] "Same SP, x% gain and x% loss" has only one answer: a loss of $\frac{x^2}{100}$ percent. "No profit, no loss" is usually one of the options, and it is always wrong.

Two Articles Bought at the Same Cost, or for a Total Cost

Same CP, different rates. When the two CPs are equal, the overall % is the simple average of the two percentages, with a loss counted as minus.

"Two lamps cost ₹1,350 each. One is sold at an 18% gain and the other at a 6% loss." Overall = $\frac{18 + (-6)}{2}$ = 6% gain. Check: the SPs are ₹1,593 and ₹1,269, a total of ₹2,862 on a CP of ₹2,700, which is a gain of ₹162, or 6%.

So the same x% gain and x% loss on the same CP gives no profit and no loss. Compare this with the same-SP case above, which always gives a loss.

CPs in a given ratio. Then the overall % is a weighted average. If the CPs are in the ratio 3 : 5 and the rates are a 10% gain and a 2% loss, the overall result is $\frac{3 \times 10 - 5 \times 2}{8}$ = 2.5% gain.

Total cost and the overall result. "Two cycles cost ₹9,000 together. One is sold at a 14% gain and the other at a 6% loss, and the overall gain is 6%. Find the CP of each." Say the first cost ₹x.

$$0.14x - 0.06(9000 - x) = 0.06 \times 9000 \Rightarrow 0.2x = 1080 \Rightarrow x = 5400$$

So the cycles cost ₹5,400 and ₹3,600. Check: 756 − 216 = 540, which is 6% of 9,000.

Alligation does this in one line: (6 − (−6)) : (14 − 6) = 12 : 8 = 3 : 2. The rule is in Chapter 6.

No profit, no loss overall. If one article gains 12% and the other loses 8%, and the whole deal is level, the rupee gain equals the rupee loss: 12% of $\text{CP}_1$ = 8% of $\text{CP}_2$, so $\text{CP}_1 : \text{CP}_2$ = 8 : 12 = 2 : 3. Each CP is in the ratio of the other one's percentage.

Parts of a Stock Sold at Different Rates

A trader sells part of his stock at one rate, part at another, and the question asks for the overall result or the rate on the last part. The overall % is a weighted average of the rates, and the weights are the cost prices of the parts. If every unit cost the same, the quantities or fractions will do; if not, weight each rate by its rupee cost, not by its quantity or its SP.

Overall profit from quantities. "A trader buys 500 kg of rice at one price. He sells 200 kg at a 15% gain, 180 kg at a 10% gain and the remaining 120 kg at a 5% loss." Every kg cost the same, so the kg are the weights:

$$\text{overall}\% = \frac{200 \times 15 + 180 \times 10 - 120 \times 5}{500} = \frac{3000 + 1800 - 600}{500} = 8.4\%$$

Overall profit from fractions. "He sells $\frac{1}{4}$ of his goods at a 24% gain, $\frac{1}{2}$ at a 10% gain and the rest at an 8% loss." The rest is $\frac{1}{4}$, so the overall gain is

$$\frac{1}{4} \times 24 + \frac{1}{2} \times 10 - \frac{1}{4} \times 8 = 6 + 5 - 2 = 9\%$$

The plain average of 24, 10 and −8 is $8\frac{2}{3}\%$, and it is wrong because the parts are not equal.

The rate on the last part. "He sells $\frac{2}{5}$ of his stock at a 15% gain and $\frac{1}{3}$ at a 9% loss. At what gain % must he sell the rest to gain 10% on the whole?"

• The rest is $1 - \frac{2}{5} - \frac{1}{3} = \frac{4}{15}$.
• The first two parts give $\frac{2}{5} \times 15 - \frac{1}{3} \times 9 = 6 - 3 = 3$.
• The rest must supply 10 − 3 = 7, so its rate r satisfies $\frac{4}{15} \times r = 7$, which gives r = 26.25%.

Working in rupees. When prices per kg are given, count rupees. A grocer buys 360 kg of onions at ₹26 per kg, so his CP is ₹9,360, and for a 25% gain overall he needs ₹11,700. He sells 100 kg at ₹30 (₹3,000) and 100 kg at ₹31 (₹3,100). The last 160 kg must bring 11,700 − 6,100 = ₹5,600, which is ₹35 per kg.

Finding the parts. If the overall % is given and the question asks how much was sold at each rate, that is alligation on the profit percentages (see Chapter 6).

When the Cost or the Selling Price Changes by a Percentage

These questions change the CP, the SP or both by some percentage and ask for the new profit. Take CP = 100 and push each change through as a multiplier. Never add or subtract the percentages directly.

Only the SP changes. "A shop makes a 20% profit. If it raises the SP by 10%, what is the new profit?" CP = 100 and SP = 120. The new SP is 120 × 1.1 = 132, so the profit is 32%, not 30%.

Only the CP changes. "A trader makes a 25% profit. His CP goes up by 20% but he keeps the same SP." CP = 120 and SP = 125, so the profit is $\frac{5}{120}$ = $4\frac{1}{6}\%$, not 5%.

Both change. "The profit is 32%. The CP rises by 20% and the SP by 10%." The new CP is 120 and the new SP is 132 × 1.1 = 145.2, so the profit is $\frac{25.2}{120}$ = 21%. In one line: $\frac{1.32 \times 1.1}{1.2} = 1.21$. If the CP and the SP change by the same %, the profit % stays as it was; only the rupee profit changes.

The SP is multiplied. "If an article is sold at double its price, the profit becomes four times. Find the profit %." Let CP = c and SP = s. Then 2s − c = 4(s − c), so 2s = 3c and s = 1.5c: the profit is 50%. The well-known version, "SP doubled, profit tripled", gives s = 2c, a profit of 100%.

The SP goes up by a %, and the profit by another %. "If the SP goes up by 10%, the profit goes up by 50%. Find the profit %." 1.1s − c = 1.5(s − c), so 0.5c = 0.4s and s = 1.25c: the profit is 25%. Check: CP 100, SP 125, profit 25; at SP 137.5 the profit is 37.5, which is 50% more.

A fraction of the usual price. "By selling an article at $\frac{3}{4}$ of its usual price, a man loses 10%. What is his profit at the usual price?" With CP = 100, $\frac{3}{4}$ of the usual price is 90, so the usual price is 90 × $\frac{4}{3}$ = 120 and the profit is 20%.

A change in rupees. If the SP changes by a fixed number of rupees and the profit % moves from one value to another, use the difference method from the Profit and Loss Basics section.

Mixed Changes: A Percentage on One Side, Rupees on the Other

A common CGL version changes the CP by a percentage and the SP by a fixed amount of rupees. Work in units with CP = 100 units, and let the rupee figure tell you what one unit is worth.

Find the CP. "A man sells an article at a 25% profit. If he had bought it for 20% less and sold it for ₹105 less, he would have gained 30%. Find the CP."

• Actual: CP = 100 units, SP = 125 units.
• Imagined: CP = 80 units, SP = 80 × 1.3 = 104 units.
• The SP fell by 125 − 104 = 21 units, and that is ₹105. So one unit is ₹5 and the CP is ₹500.

Check: the SP is ₹625. At a CP of ₹400 and an SP of ₹520 (= 625 − 105), the gain is 120 ÷ 400 = 30%.

The SP goes up instead. "An article is sold at a 10% profit. Had it been bought for 10% less and sold for ₹84 more, the profit would have been 30%." CP = 100, SP = 110; new CP = 90, new SP = 117. The SP rose by 7 units = ₹84, so one unit is ₹12 and the CP is ₹1,200.

The new profit when the CP is given. "CP = ₹800, sold at a 15% profit. If the CP were 10% less and the SP ₹40 more, what would the profit % be?" The new CP is ₹720 and the new SP is 920 + 40 = ₹960, so the profit is $\frac{240}{720}$ = $33\frac{1}{3}\%$.

Both change by the same rupees. "Sold at a 25% profit. If the CP and the SP were both ₹50 less, the profit would be 30%." With CP = c: 1.25c − 50 = 1.3(c − 50), so 0.05c = 15 and c = ₹300. Check: ₹250 and ₹325, and 325 ÷ 250 = 1.3. The rupee profit (₹75) stays the same but sits on a smaller CP, so the % goes up.

Only the CP changes, by rupees. "A shopkeeper makes a 20% profit. Had he bought the article for ₹36 less and sold it at the same price, his profit would have been 25%." SP = 1.2c = 1.25(c − 36), so 0.05c = 45 and c = ₹900. Check: the SP is ₹1,080, and 1,080 ÷ 864 = 1.25.

[Tier 2] Both change by a %, and the profit falls by some points. "If the CP rises 25% and the SP rises 15%, the profit % falls by 12 percentage points. Find the original profit %." Let SP ÷ CP = R. The new ratio is $R \times \frac{1.15}{1.25} = 0.92R$, which is 0.08R less than before. So 0.08R = 0.12, R = 1.5, and the profit was 50% (it falls to 38%).

Broken Items, Fewer Sales and Costlier Raw Material

Broken or rotten items. The seller paid for all of them but can sell only the good ones. The CP is the price of the whole lot; the SP comes only from what is left.

"A vendor buys 600 eggs at ₹6 each. 40 of them break. He sells the rest at ₹7.20 each."

• CP = 600 × 6 = ₹3,600.
• SP = 560 × 7.20 = ₹4,032.
• Profit = ₹432 on ₹3,600, which is 12%. Comparing ₹7.20 with ₹6 gives 20%, which forgets the broken eggs.

For a 26% gain on the whole, he needs 3,600 × 1.26 = ₹4,536 from 560 eggs, which is ₹8.10 an egg. (Goods lost to theft or wastage, and part of a stock sold off at a loss, are in the Dishonest Dealers and Successive Transactions section.)

Raising the price and selling fewer. Total profit = profit per unit × number sold, so both changes count.

"A shop sells 400 mugs a month at ₹150. Each costs ₹110. It raises the price to ₹165, and sales fall by 20%."

• Before: 400 × 40 = ₹16,000.
• After: 320 × 55 = ₹17,600.
• The monthly profit rises by 10%, even though fewer mugs are sold.

In multipliers: the profit per mug goes from 40 to 55 (× 1.375) and the sales × 0.8, so the profit is × 1.1. The sales money (400 × 150 = ₹60,000 before, 320 × 165 = ₹52,800 after) actually falls, so read whether the question asks about profit or about sales.

Costlier raw material. "The cost of a product is made up of raw material, labour and overheads in the ratio 4 : 3 : 1. It is sold at a 35% profit. Then the price of the raw material goes up by 25%."

• Old cost = 8 units, SP = 8 × 1.35 = 10.8 units.
• New cost = 4 × 1.25 + 3 + 1 = 9 units.
• At the same SP, the profit is $\frac{1.8}{9}$ = 20%.
• To keep a 35% profit, the SP must become 9 × 1.35 = 12.15 units, a rise of 12.5%. That is the same as the rise in the total cost, from 8 to 9. Raising the SP by 25%, the rise in the raw material alone, is the usual mistake.

Solved Examples

Question 1

A shopkeeper buys erasers at 15 for ₹40 and sells them at 12 for ₹40. What is his gain percentage? (a) 25% (b) 20% (c) $16\frac{2}{3}\%$ (d) 30%

Answer. The money is ₹40 on both sides. He pays ₹40 for 15 erasers and gets ₹40 for 12, so the SP of 12 equals the CP of 15. Gain % = $\frac{15 - 12}{12} \times 100$ = (a) 25%. Dividing by 15 gives 20%, but the base must be the 12 he sells for that ₹40.

Question 2

Pens are bought at 7 for ₹90 and sold at 9 for ₹108. What is the loss percentage? (a) $7\frac{1}{7}\%$ (b) 20% (c) $6\frac{2}{3}\%$ (d) $8\frac{1}{3}\%$

Answer. Take 63 pens (the LCM of 7 and 9). CP of 63 = 9 × 90 = ₹810 and SP of 63 = 7 × 108 = ₹756. The loss is ₹54 on ₹810, which is (c) $6\frac{2}{3}\%$.

Question 3

By selling 24 pencils for ₹50, a shopkeeper loses 20%. How many pencils should he sell for ₹50 to gain 20%? (a) 36 (b) 16 (c) 20 (d) 18

Answer. The money stays ₹50. At a 20% loss he gets 80% of the cost, and he wants 120%, so each pencil must sell for more and he must give fewer of them: 24 × $\frac{80}{120}$ = (b) 16. Check: 16 pencils for ₹50 is $\frac{24}{16}$ = 1.5 times the old price per pencil, and 0.8 × 1.5 = 1.2, a 20% gain.

Question 4

A man buys bananas at ₹54 a dozen and sells them at ₹96 a score. What is his profit percentage? (a) $77\frac{7}{9}\%$ (b) 6.25% (c) 8% (d) $6\frac{2}{3}\%$

Answer. One banana costs 54 ÷ 12 = ₹4.50 and sells for 96 ÷ 20 = ₹4.80. The profit is ₹0.30 on ₹4.50, which is $\frac{1}{15}$, so the answer is (d) $6\frac{2}{3}\%$. Comparing ₹96 with ₹54 directly gives $77\frac{7}{9}\%$, far too much.

Question 5

[Tier 2] A man buys equal numbers of mangoes at 3 for ₹40 and at 5 for ₹40. He mixes them and sells all of them at 4 for ₹40. What is his gain or loss percentage? (a) Loss of 6.25% (b) No profit, no loss (c) Gain of 6.25% (d) Loss of $6\frac{2}{3}\%$

Answer. Take 15 of each kind (the LCM of 3 and 5). 15 mangoes at 3 for ₹40 cost ₹200, and 15 at 5 for ₹40 cost ₹120, so 30 mangoes cost ₹320. At 4 for ₹40, 30 mangoes bring ₹300. That is ₹20 short on ₹320, a (a) Loss of 6.25%. 4 is the average of 3 and 5, but with equal numbers it's the price of one mango that gets averaged, not the number per ₹40.

Question 6

If the selling price of 20 articles is equal to the cost price of 23 articles, what is the gain or loss percentage? (a) Gain of $13\frac{1}{23}\%$ (b) Loss of 15% (c) Gain of 15% (d) Gain of 3%

Answer. Take the CP of one article as ₹1. Then 20 articles sell for ₹23, so one sells for ₹1.15. That is a (c) Gain of 15%. Divide by 20, the number sold.

Question 7

By selling 66 m of cloth, a trader gains the selling price of 22 m. What is his gain percentage? (a) $33\frac{1}{3}\%$ (b) 50% (c) 25% (d) $66\frac{2}{3}\%$

Answer. The profit is the SP of 22 m, so the CP of 66 m = SP of 66 m − SP of 22 m = SP of 44 m. The gain (SP of 22 m) on a cost equal to the SP of 44 m is $\frac{22}{44} \times 100$ = (b) 50%.

Question 8

A fruit seller sells 30 kg of grapes and loses the selling price of 6 kg. What is his loss percentage? (a) 20% (b) 25% (c) $14\frac{2}{7}\%$ (d) $16\frac{2}{3}\%$

Answer. The loss is the SP of 6 kg, so the CP of 30 kg = SP of 30 kg + SP of 6 kg = SP of 36 kg. The loss (SP of 6 kg) on a cost equal to the SP of 36 kg is $\frac{6}{36}$, which is (d) $16\frac{2}{3}\%$.

Question 9

Two laptops are sold for ₹30,000 each. On one the seller gains 25% and on the other he loses 25%. What is his overall gain or loss? (a) Loss of ₹4,000 (b) No profit, no loss (c) Loss of ₹3,750 (d) Gain of ₹4,000

Answer. The CPs are 30,000 ÷ 1.25 = ₹24,000 and 30,000 ÷ 0.75 = ₹40,000. Total CP = ₹64,000 and total SP = ₹60,000, so the result is a (a) Loss of ₹4,000. As a percentage that is $\frac{25^2}{100}$ = 6.25% of ₹64,000.

Question 10

Two watches are sold for ₹5,040 each, one at a 12% gain and the other at a 10% loss. What is the overall result? (a) Gain of ₹20 (b) No profit, no loss (c) Loss of ₹20 (d) Gain of ₹100.80

Answer. CP of the first = 5,040 ÷ 1.12 = ₹4,500 and CP of the second = 5,040 ÷ 0.9 = ₹5,600. Total CP = ₹10,100 and total SP = ₹10,080, so the result is a (c) Loss of ₹20. A 12% gain and a 10% loss look like an overall gain, but the loss is on the dearer watch: ₹560 lost against ₹540 gained.

Question 11

Ramesh buys two cows for ₹54,600. He sells one at a 15% loss and the other at a 25% gain, and gets the same price for each. What did he pay for the cow he sold at a loss? (a) ₹22,100 (b) ₹32,500 (c) ₹27,300 (d) ₹30,000

Answer. Same SP, so CP (loss) × 0.85 = CP (gain) × 1.25, which makes CP (loss) : CP (gain) = 125 : 85 = 25 : 17. The 42 parts are ₹54,600, so one part is ₹1,300 and the cow sold at a loss cost 25 × 1,300 = (b) ₹32,500. Check: 32,500 × 0.85 = 27,625 and 22,100 × 1.25 = 27,625.

Question 12

Two bikes cost ₹1,20,000 together. One is sold at a 10% gain and the other at a 5% loss, and the overall gain is 4%. What is the cost price of the bike sold at a gain? (a) ₹72,000 (b) ₹48,000 (c) ₹60,000 (d) ₹80,000

Answer. Say the first bike cost ₹x. Then 0.1x − 0.05(1,20,000 − x) = 4% of 1,20,000 = 4,800, so 0.15x = 10,800 and x = (a) ₹72,000. Alligation gives it faster: (4 + 5) : (10 − 4) = 9 : 6 = 3 : 2, and $\frac{3}{5}$ of 1,20,000 = 72,000.

Question 13

A trader has 800 kg of wheat. He sells 300 kg at a 12% profit, 350 kg at an 8% profit and the rest at a 4% loss. What is his overall profit percentage? (a) $5\frac{1}{3}\%$ (b) 8% (c) 6.5% (d) 7.25%

Answer. Every kg cost the same, so weight each rate by its kg. The rest is 150 kg. Overall = $\frac{300 \times 12 + 350 \times 8 - 150 \times 4}{800} = \frac{5800}{800}$ = (d) 7.25%. The plain average of 12, 8 and −4 is $5\frac{1}{3}\%$, but it ignores the quantities.

Question 14

A shopkeeper sells $\frac{1}{3}$ of his goods at a 15% gain and $\frac{2}{5}$ at a 6% loss. At what gain percentage must he sell the rest to gain 9% on the whole? (a) 6% (b) 18% (c) 24% (d) 9%

Answer. The rest is $1 - \frac{1}{3} - \frac{2}{5} = \frac{4}{15}$. The first two parts give $\frac{1}{3} \times 15 - \frac{2}{5} \times 6 = 5 - 2.4 = 2.6$. The rest must add 9 − 2.6 = 6.4, so $\frac{4}{15} \times r = 6.4$ and r = (c) 24%.

Question 15

A trader makes a profit of 26%. If his cost price rises by 20% and he keeps the selling price the same, what is his new profit percentage? (a) 6% (b) 5% (c) $4\frac{16}{21}\%$ (d) 4%

Answer. Take the old CP as 100, so the SP is 126. The new CP is 120 and the SP is still 126, so the profit is 6 on 120, which is (b) 5%. 26 − 20 = 6% subtracts the percentages as if they were on the same base.

Question 16

If an article is sold at double its present selling price, the profit becomes five times the present profit. What is the present profit percentage? (a) 25% (b) 50% (c) 100% (d) $33\frac{1}{3}\%$

Answer. Let CP = c and SP = s. Then 2s − c = 5(s − c), so 3s = 4c and s = $\frac{4}{3}$c. The profit is $\frac{1}{3}$ of the CP, which is (d) $33\frac{1}{3}\%$. Check with CP = 300: SP 400 gives a profit of 100, and SP 800 gives 500, five times as much. The "profit triples" version would give 100%.

Question 17

By selling a mixer at $\frac{4}{5}$ of its usual price, a shopkeeper loses 8%. What is his profit percentage at the usual price? (a) 15% (b) 12% (c) 10.4% (d) 25%

Answer. Take CP = 100. At $\frac{4}{5}$ of the usual price he gets 92, so the usual price is 92 × $\frac{5}{4}$ = 115 and the profit is (a) 15%. Adding 20% to 92 gives 110.4, but 92 is $\frac{4}{5}$ of the usual price, so you divide by 0.8 instead of multiplying by 1.2.

Question 18

A man sold a cycle at a 15% profit. If he had bought it for 20% less and sold it for ₹210 less, he would have made a 25% profit. What is the cost price of the cycle? (a) ₹1,610 (b) ₹1,200 (c) ₹1,400 (d) ₹1,050

Answer. Say CP = 100 units, so SP = 115 units. In the imagined deal, CP = 80 units and SP = 80 × 1.25 = 100 units. The SP fell by 15 units, which is ₹210, so one unit is ₹14 and the CP is (c) ₹1,400. Check: SP = 1,610; new CP 1,120 and new SP 1,400 = 1,610 − 210, and 1,400 ÷ 1,120 = 1.25.

Question 19

A shopkeeper sells an article at a 20% profit. If both the cost price and the selling price were ₹120 more, the profit would be 15%. What is the cost price? (a) ₹432 (b) ₹360 (c) ₹480 (d) ₹300

Answer. Let CP = c, so SP = 1.2c. Then 1.2c + 120 = 1.15(c + 120) = 1.15c + 138, so 0.05c = 18 and c = (b) ₹360. Check: ₹480 and ₹552, and 552 ÷ 480 = 1.15. The rupee profit stays ₹72 but sits on a bigger CP, so the percentage drops.

Question 20

A dealer buys 1,200 glasses at ₹45 each. 75 of them break on the way. He sells the rest at ₹54 each. What is his profit percentage? (a) 20% (b) $11\frac{1}{9}\%$ (c) 13.5% (d) 12.5%

Answer. He paid for all of them: CP = 1,200 × 45 = ₹54,000. He sells 1,125 glasses for 1,125 × 54 = ₹60,750. The profit is ₹6,750 on ₹54,000, which is (d) 12.5%.

Question 21

A café sells 250 sandwiches a day at ₹80 each, and each costs ₹60 to make. It raises the price to ₹90, and its daily sales fall by 24%. What happens to its daily profit? (a) It decreases by 14.5% (b) It increases by 12.5% (c) It increases by 14% (d) It increases by 26%

Answer. Before: 250 × (80 − 60) = ₹5,000. After: sales are 76% of 250 = 190 sandwiches, and each gives ₹30, so 190 × 30 = ₹5,700. So (c) It increases by 14%. In multipliers: 1.5 × 0.76 = 1.14. The sales money does fall by 14.5% (₹20,000 to ₹17,100), but the question asks about the profit.

Question 22

The cost of making a chair is shared between wood and labour in the ratio 3 : 2. The chair is sold at a 40% profit. If the price of wood goes up by 20% and the selling price stays the same, what is the new profit percentage? (a) 25% (b) 20% (c) 28% (d) $16\frac{2}{3}\%$

Answer. Take the cost as 5 units: wood 3 and labour 2. SP = 5 × 1.4 = 7 units. The wood now costs 3.6, so the cost is 5.6. The profit is 1.4 on 5.6, which is (a) 25%.