Dishonest Dealers and Successive Transactions

False Weights: Selling at Cost Price with a Light Weight

A dishonest shopkeeper can sell "at cost price" and still gain, because he hands over less than he charges for. If he uses a 900 g weight in place of 1 kg, he takes the price of 1,000 g but gives only 900 g.

His cost is the cost of the 900 g he gives; his takings are the price of 1,000 g. So he gains 100 g worth on 900 g worth:

$$\text{Gain}\% = \frac{\text{Error}}{\text{True value} - \text{Error}} \times 100 = \frac{100}{900} \times 100 = 11\frac{1}{9}\%$$

The error is the shortfall (100 g here). The base is what he actually gives (900 g), not the 1,000 g he claims. Writing $\frac{100}{1000}$ = 10% is the trap in almost every false-weight question.

Check with money. Say sugar costs ₹40 a kg. The 900 g he gives costs him ₹36, and he collects ₹40. Gain = ₹4 on ₹36 = $11\frac{1}{9}\%$.

Common weights.

Weight from the gain. Turn the formula round: true weight ÷ weight used = 1 + gain. "A grocer sells at cost price and gains 25%. What weight does he use for 1 kg?" 1,000 ÷ 1.25 = 800 g.

"x% less" in words. "He gives 10% less than he claims" is the 900 g case, a gain of $\frac{10}{90}$ = $11\frac{1}{9}\%$. In general, giving x% less at cost price gains $\frac{x}{100 - x} \times 100$ %.

Other measures. The same rule works for litres and metres. A milkman whose "1 litre" measure holds only 920 ml gains $\frac{80}{920}$ = $8\frac{16}{23}\%$ while charging the cost price.

False Weight with a Profit, a Loss or a Discount

Now the dealer also changes the price. Treat the price and the weight as two separate multipliers and multiply them:

$$\frac{\text{Money in}}{\text{Money out}} = \frac{100 + p}{100} \times \frac{\text{True weight}}{\text{Weight given}}$$

Here p is the profit % he takes on the price (use a minus sign for a loss).

Profit on the price plus a light weight. "A trader adds 5% to the cost price and also uses 900 g for 1 kg." Ratio = 1.05 × $\frac{1000}{900}$ = $\frac{7}{6}$, so he really gains $\frac{1}{6}$ = $16\frac{2}{3}\%$. Adding 5% and $11\frac{1}{9}\%$ gives $16\frac{1}{9}\%$, which is wrong.

A loss on the price that turns into a gain. "A shopkeeper sells at 9% below cost but uses a 700 g weight for 1 kg." Ratio = 0.91 × $\frac{1000}{700}$ = 1.3, a 30% gain.

Weight needed for a target gain. "A dealer sells at 10% below cost but wants an overall gain of 12.5%. What weight should he use for 1 kg?" Weight = 1,000 × $\frac{0.9}{1.125}$ = 800 g.

When the two cancel. Selling at a 4% loss with a 960 g weight gives 0.96 × $\frac{1000}{960}$ = 1: no gain, no loss.

Markup, discount and a false weight. All the multipliers line up. "Goods are marked 25% above cost, sold at a 20% discount, and weighed with a 900 g weight for 1 kg." The price part is 1.25 × 0.8 = 1, so the price alone gives nothing. The weight gives $\frac{10}{9}$, so the overall gain is $11\frac{1}{9}\%$.

[Tier 2] Finding the weight when there is a discount too. "A shop marks goods 25% above cost, gives a 4% discount, and gains 60% overall by using a false weight. What weight does it use for 1 kg?" The price part is 1.25 × 0.96 = 1.2. So 1.2 × $\frac{1000}{w}$ = 1.6, and w = 750 g.

A forward check. Put CP = ₹100 for 1 kg and work out how many grams he really gives, what they cost him and what he collects for them.

Cheating While Buying and While Selling

Some dealers cheat their suppliers too. While buying, they take more than they pay for; while selling, they give less than they charge for. Each trick is one more multiplier.

• Taking a% extra while buying: for the price of 1,000 g he takes (1,000 + 10a) g.
• Giving b% less while selling: for the price of 1,000 g he gives (1,000 − 10b) g.

$$\text{Overall ratio} = \frac{100 + a}{100 - b}$$

Forward check. "A trader takes 20% extra while buying and gives 25% less while selling, at the same price per kg." For ₹100 he receives 1,200 g. He sells 750 g as "1 kg" for ₹100, so the 1,200 g makes 1.6 such lots and fetches ₹160. Gain = 60% = $\frac{1.2}{0.75}$ − 1.

Read the wording. "Cheats by 8% while selling" can mean two things:

1. He gives 8% less than he charges for (920 g sold as 1 kg). Then you divide by 0.92.
2. He charges for 8% more than he gives (1,000 g billed as 1,080 g). Then you multiply by 1.08.

So "cheats by 8% while buying and 8% while selling" gives 1.08 ÷ 0.92 − 1 = $17\frac{9}{23}\%$ on the first reading and 1.08 × 1.08 − 1 = 16.64% on the second. The short wording "cheats to the extent of x% while buying and selling" is usually solved the second way, as two successive gains of x%. When the question names the actual weights ("uses 920 g for 1 kg"), use them and divide.

Cheating on both sides plus a profit on the price. Multiply in the third factor. "He takes 20% extra while buying, gives 25% less while selling, and also adds 5% to the cost price." 1.2 ÷ 0.75 × 1.05 = 1.68, a 68% gain.

Cheating only while buying. If he takes 25% extra while buying and then sells honestly at cost price, he gains exactly 25%. The base is the money he paid, so there is no "divide by less" effect on this side.

Less Weight with a Higher Price; Faulty Scales and Impure Goods

Giving x% less weight and charging y% more. Both tricks work against the buyer. For the cost price of 1 kg plus y%, he gives only (100 − x)% of 1 kg:

$$\text{Gain}\% = \left(\frac{100 + y}{100 - x} - 1\right) \times 100$$

"A seller gives 20% less weight and charges 10% more than the cost price." Gain = $\frac{110}{80}$ − 1 = $\frac{3}{8}$ = 37.5%. Adding 20 + 10 = 30% is the trap.

Going backwards: "A seller gives 12.5% less weight. How far above cost must he price his goods to gain 20%?" 1 + y = 1.2 × 0.875 = 1.05, so he charges 5% above cost.

Faulty measuring rods. A cloth seller who uses a 75 cm rod as a metre gains $\frac{25}{75}$ = $33\frac{1}{3}\%$ at cost price, by the same rule as for weights. If he buys with a rod that is too long (110 cm for each metre he pays for) and sells with one that is too short (88 cm), his gain is $\frac{110}{88}$ − 1 = 25%.

A balance that reads wrong. If a balance shows 1 kg when only 875 g is on it, a seller who weighs on it at cost price gains $\frac{125}{875}$ = $14\frac{2}{7}\%$. A balance that shows 1 kg for 1,050 g cheats the seller instead: he loses $\frac{50}{1050}$ = $4\frac{16}{21}\%$.

Impure goods. Mixing a free filler into the goods works like a light weight, but read the base.

• "Adds sand equal to 10% of the sugar" and sells the mix at the cost price of sugar: 1,100 g is sold for what 1,000 g cost, a 10% gain.
• "The mix contains 10% sand": every 1 kg sold has only 900 g of sugar in it, a gain of $11\frac{1}{9}\%$.

[SSC Tip] "x% less weight" and "x% impurity in the mix" both divide by (100 − x). "Adds x% impurity" multiplies by (100 + x). One word in the question decides which.

Successive Transactions: A Sells to B, B Sells to C

When an article passes through several hands, each sale is one multiplier on the price before it:

$$\text{Last price} = \text{First CP} \times \frac{100 \pm x}{100} \times \frac{100 \pm y}{100} \times \cdots$$

Finding the last price. "A buys a scooter for ₹50,000 and sells it to B at 12% profit. B sells it to C at 5% profit. What does C pay?" 50,000 × 1.12 × 1.05 = ₹58,800.

Finding the first CP. "A sells to B at 30% profit, and B sells to C at 10% profit. C pays ₹28,600. What did A pay?" Divide back: 28,600 ÷ (1.3 × 1.1) = 28,600 ÷ 1.43 = ₹20,000. The traps are taking 40% off ₹28,600 (₹17,160) and dividing by 1.4 (about ₹20,429).

A loss in the chain. "A sells to B at 30% profit, and B sells to C at 20% loss. C pays ₹10,400." A's CP = 10,400 ÷ (1.3 × 0.8) = 10,400 ÷ 1.04 = ₹10,000. A 30% gain followed by a 20% loss is a net 4% rise on the first price, not 10%.

Overall % along the chain. Two steps of x% and y% combine as x + y + $\frac{xy}{100}$ (put a minus sign on a loss). For three steps, multiply three factors. A manufacturer, a wholesaler and a retailer adding 8%, 25% and 20% give 1.08 × 1.25 × 1.2 = 1.62, so the customer pays 62% over the factory cost.

Somebody in the middle. "A buys a fridge for ₹8,000 and sells it to B at 15% profit. C pays B ₹10,120. What is B's profit %?" B paid ₹9,200, so B's profit is 920 on 9,200 = 10%. Find each person's CP from the step before; never take B's profit on A's cost.

Selling Back and Longer Chains

Selling back to the first owner. "A buys a phone for ₹20,000 and sells it to B at 10% profit. Later B sells it back to A at a 10% loss. What does A gain from the two deals?" A gets ₹22,000 from B. B's loss is on his own CP, so B sells back for 22,000 × 0.9 = ₹19,800. A has the phone again and 22,000 − 19,800 = ₹2,200 more cash than before.

The usual slip is to take B's 10% loss on A's ₹20,000 (₹18,000) and answer ₹4,000.

If A then sells the phone again for ₹21,000, count every payment: −20,000 + 22,000 − 19,800 + 21,000 = ₹3,200 in all. In questions like this, write every payment as cash in or cash out for the person asked about. The phone itself drops out when he ends up holding it again.

Rupee gap along the chain. "P sells a machine to Q at a 10% loss, and Q sells it to R at a 25% gain. R pays ₹1,000 more than P paid. Find P's CP." The net factor is 0.9 × 1.25 = 1.125, so ₹1,000 is 12.5% of P's CP and the CP is ₹8,000.

Rates given as fractions. A gain of $\frac{1}{3}$ ($33\frac{1}{3}\%$) followed by a loss of $\frac{1}{4}$ (25%) gives $\frac{4}{3} \times \frac{3}{4} = 1$: the last buyer pays exactly what the first seller paid.

Finding a missing rate. "A sells to B at a 20% gain, B sells to C at some gain, and C pays 50% more than A's CP. What was B's gain %?" 1.5 ÷ 1.2 = 1.25, so 25%. Subtracting 50 − 20 = 30% is the trap.

[Tier 2] Costs added in the middle. "A sells a car to B at 10% profit. B spends ₹5,000 on repairs and sells it to C at 20% profit for ₹1,38,000. What did A pay?" B's total cost = 1,38,000 ÷ 1.2 = ₹1,15,000. B paid A 1,15,000 − 5,000 = ₹1,10,000, so A's CP = 1,10,000 ÷ 1.1 = ₹1,00,000. Work back one step at a time and take off the repair cost at the right step.

Commission, Brokerage and an Agent's Cut

When a middleman takes a share, the seller's real takings or the buyer's real cost change. Put the commission on the right side.

Commission paid by the seller (on the SP). The seller keeps only (100 − c)% of the SP. "A dealer pays his agent 8% commission on the selling price and still gains 15%. Find the SP of an article costing ₹2,300." SP × 0.92 = 2,300 × 1.15 = 2,645, so SP = ₹2,875.

Brokerage paid by the buyer (on the purchase). The buyer's CP goes up. A trader who buys goods for ₹40,000 and pays 2.5% brokerage has a CP of 40,000 × 1.025 = ₹41,000. To gain 10% he must take in ₹45,100.

Both sides. "A man buys a plot for ₹5,00,000 and pays 2% brokerage. He sells it for ₹6,37,500 and again pays 2% brokerage. Find his profit %." His CP = ₹5,10,000. He keeps 6,37,500 × 0.98 = ₹6,24,750. Profit = ₹1,14,750 on ₹5,10,000 = 22.5%.

Markup, discount and commission together. All three are multipliers. "Goods are marked 50% above cost, sold at a 20% discount, and the agent gets 2.5% of the SP." The seller keeps 1.5 × 0.8 × 0.975 = 1.17 of the CP: a profit of 17%.

Finding the MP. Reverse it. "A firm allows a 10% discount, pays 4% commission on the SP, and wants an 8% profit." $\frac{\text{MP}}{\text{CP}} = \frac{1.08}{0.9 \times 0.96} = \frac{1.08}{0.864}$ = 1.25, so the goods must be marked 25% above cost.

Who pays the agent? If the buyer pays the agent, the buyer's cost rises and the seller's takings do not change. If the seller pays, his takings fall. Read who pays whom before you multiply.

[SSC Tip] A commission on the SP is a % of the SP, so you cannot just add it to the profit %. In the first case, taking 15% + 8% = 23% on the CP gives SP = ₹2,829, not ₹2,875.

Goods Lost, Stolen or Damaged

When part of the stock is lost, the dealer has paid for all of it but can sell only the rest. Take the CP of the whole stock as 100 and count takings against that 100.

Selling the rest at a profit. "A trader loses 20% of his goods in a flood and sells the rest at 25% above cost. What is the overall result?" Takings = 80 × 1.25 = 100, the same as the cost of the whole stock: no profit, no loss.

Price needed on the rest. "A fruit seller finds that 10% of his stock has gone bad. How far above cost must he sell the rest to gain 17% overall?" He needs 117 from 90 units of stock: 117 ÷ 90 = 1.3, so 30% above cost.

Damaged goods sold cheaply. "Part of a shopkeeper's stock is damaged in a fire. He sells $\frac{1}{4}$ of the stock at half its cost and the rest at x% profit, and still gains 20% overall." The damaged quarter brings 25 × 0.5 = 12.5. He needs 120 − 12.5 = 107.5 from the 75 good units, so x = $\frac{107.5}{75}$ − 1 = $43\frac{1}{3}\%$.

[Tier 2] Insurance. "A trader's stock cost ₹2,00,000. Goods costing ₹60,000 are stolen, and the insurance company pays 75% of their cost. He sells the remaining goods at 20% profit. Find his overall profit %." The insurance brings ₹45,000. The remaining goods cost ₹1,40,000 and sell for ₹1,68,000. Takings = 1,68,000 + 45,000 = ₹2,13,000, so the profit is ₹13,000 on ₹2,00,000 = 6.5%.

Counted items. Broken eggs or rotten fruit counted one by one, with a price per dozen, are worked in Profit and Loss: Articles, Two Items and Changing Prices. Here the loss is a % or a fraction of the stock, and the cost of the whole stock is the base.

[SSC Tip] The profit % is always on the cost of the whole stock, including the goods that were lost. Dividing by the cost of the goods that were sold (80 in the first case) is the trap.

Clearance Sales on Part of the Stock

At the end of a season a shop sells what is left at a loss or at a big discount. Find the takings from each part on a CP of 100 and compare the total with 100.

Part at a profit, the rest at a loss. "A shopkeeper sells 75% of his stock at 20% profit and clears the rest at a 20% loss." Takings = 75 × 1.2 + 25 × 0.8 = 90 + 20 = 110, so he gains 10%. (The general case of a stock sold in parts at different rates is in Profit and Loss: Articles, Two Items and Changing Prices; the method is the same.)

The loss the clearance can bear. "A trader sells 80% of his stock at 25% profit. What is the largest loss % he can take on the rest and still make 15% overall?" The 80% brings 80 × 1.25 = 100. He needs 115, so the last 20 units must bring 15, a loss of 5 on 20 = 25%.

Clearance as a discount on the MP. "Goods are marked 40% above cost. Three-fourths of the stock sells at the MP, and the rest is cleared at 30% off the MP." Takings = 75 × 1.4 + 25 × 1.4 × 0.7 = 105 + 24.5 = 129.5: a 29.5% profit.

Discount needed in the clearance. "Goods are marked 60% above cost, and 60% of the stock sells at the MP. What discount on the MP can be given on the rest so that the overall profit is 36%?" The first part brings 60 × 1.6 = 96. The other 40 units must bring 136 − 96 = 40, though at the MP they would bring 64. So the discount is $\frac{24}{64}$ = 37.5%.

A fixed clearance price. "A dealer buys 240 shirts at ₹450 each and sells 180 of them at ₹600 each. At what price must he clear each of the rest to make 20% profit overall?" Cost = ₹1,08,000, so he needs ₹1,29,600. The 180 shirts bring ₹1,08,000, so the last 60 must bring ₹21,600: ₹360 each, a 20% loss on each of them.

Solved Examples

Question 1

A shopkeeper sells rice at its cost price but uses a weight of 940 g in place of 1 kg. What is his gain %? (a) 6% (b) $6\frac{18}{47}\%$ (c) $5\frac{35}{53}\%$ (d) $6\frac{3}{8}\%$

Answer. He charges for 1,000 g but gives 940 g, so he gains 60 g worth on the 940 g he gives. Gain = $\frac{60}{940} \times 100 = \frac{300}{47}$ = (b) $6\frac{18}{47}\%$. Check: if rice costs ₹47 a kg, 940 g costs him ₹44.18 and he collects ₹47. 6% divides 60 by the 1,000 g he claims, which is the trap.

Question 2

A dishonest dealer claims to sell pulses at cost price, but he gains $42\frac{6}{7}\%$ by using a false weight. What weight does he use for 1 kg? (a) 700 g (b) $571\frac{3}{7}$ g (c) 750 g (d) 725 g

Answer. $42\frac{6}{7}\%$ is $\frac{3}{7}$, so true weight ÷ weight used = 1 + $\frac{3}{7}$ = $\frac{10}{7}$. The weight used = 1,000 × $\frac{7}{10}$ = (a) 700 g. Check: he gains 300 g worth on the 700 g he gives, and $\frac{300}{700} = \frac{3}{7}$.

Question 3

A trader sells goods at 15% above the cost price and also uses a weight of 920 g for 1 kg. What is his total gain %? (a) 23% (b) 24.2% (c) 25% (d) 26%

Answer. Multiply the two parts: price 1.15 and weight $\frac{1000}{920}$. 1.15 × $\frac{1000}{920}$ = $\frac{1150}{920}$ = 1.25, so the gain is (c) 25%. Check: on a CP of ₹100 a kg, the 920 g costs him ₹92 and he collects ₹115, a gain of ₹23 on ₹92.

Question 4

A shopkeeper sells sugar at 12% below the cost price, but uses an 800 g weight in place of 1 kg. What is his gain or loss %? (a) 13% gain (b) 10% gain (c) 12% loss (d) 8% gain

Answer. Price part 0.88, weight part $\frac{1000}{800}$ = 1.25. Overall 0.88 × 1.25 = 1.1, which is (b) 10% gain. Check: 800 g costs him ₹80 (CP ₹100 a kg) and he collects ₹88.

Question 5

A dealer sells his goods at 5% above the cost price and wants to make an overall gain of 40%. What weight should he use in place of 1 kg? (a) 750 g (b) 650 g (c) 700 g (d) 720 g

Answer. He needs 1.05 × $\frac{1000}{w}$ = 1.4, so w = 1,000 × $\frac{1.05}{1.4}$ = (a) 750 g. Check: 750 g costs him ₹75 (CP ₹100 a kg) and he collects ₹105, which is 40% more.

Question 6

A shopkeeper marks his goods 30% above the cost price, gives a discount of 10% and also uses a 900 g weight for 1 kg. What is his gain %? (a) 17% (b) $28\frac{1}{9}\%$ (c) 31% (d) 30%

Answer. Price part: 1.3 × 0.9 = 1.17. Weight part: $\frac{1000}{900} = \frac{10}{9}$. Overall 1.17 × $\frac{10}{9}$ = 1.3, so he gains (d) 30%. The 0.9 from the discount and the $\frac{10}{9}$ from the weight cancel.

Question 7

A trader takes 20% more than he pays for while buying, and gives 10% less than he charges for while selling. He buys and sells at the same price per kg. What is his gain %? (a) 30% (b) 32% (c) $33\frac{1}{3}\%$ (d) 28%

Answer. For ₹100 he gets 1,200 g. He sells 900 g for ₹100, so 1,200 g fetches 100 × $\frac{1200}{900}$ = ₹133.33. Gain = $\frac{1.2}{0.9}$ − 1 = $\frac{1}{3}$ = (c) $33\frac{1}{3}\%$. 32% multiplies 1.2 by 1.1, but he gives 10% less, so you divide.

Question 8

While buying, a dishonest trader takes 15% more goods than he pays for. While selling, he charges for 15% more goods than he actually gives. What is his gain %? (a) 30% (b) 32.25% (c) 31% (d) 33%

Answer. For ₹100 he gets 115 units of goods. When selling, every 100 units he gives are billed as 115, so the 115 units bring 115 × 1.15 = ₹132.25. Gain = 1.15 × 1.15 − 1 = (b) 32.25%. This is how the short wording "cheats by 15% while buying and selling" is usually solved. If the question instead says he gives 15% less while selling, it is 1.15 ÷ 0.85 − 1 = $35\frac{5}{17}\%$, so read the wording.

Question 9

[Tier 2] A trader uses a weight 10% heavier than the true weight while buying and a weight 10% lighter than the true weight while selling. He also sells at 10% above his cost price. What is his total gain %? (a) 30% (b) 33.1% (c) $34\frac{4}{9}\%$ (d) $32\frac{2}{9}\%$

Answer. Buying: for the price of 1 kg he gets 1.1 kg. Selling: he gives 0.9 kg as "1 kg" and charges 1.1 times the CP. Ratio = $\frac{1.1}{0.9}$ × 1.1 = $\frac{121}{90}$, so the gain is $\frac{31}{90}$ = (c) $34\frac{4}{9}\%$.

Question 10

A shopkeeper gives 15% less weight than he should and also charges 2% more than the cost price. What is his gain %? (a) 20% (b) 17% (c) 17.3% (d) 18%

Answer. For the price of 1 kg plus 2%, he gives only 85% of 1 kg. Ratio = $\frac{102}{85}$ = 1.2, so the gain is (a) 20%. Check: 850 g costs him ₹85 (CP ₹100 a kg) and he collects ₹102.

Question 11

A cloth merchant uses a rod 95 cm long as his metre, and sells cloth at 5% above the cost price. What is his gain %? (a) 10% (b) 10.25% (c) $10\frac{5}{19}\%$ (d) $10\frac{10}{19}\%$

Answer. For the price of a metre plus 5%, he gives only 95 cm. Ratio = $\frac{105}{95} = \frac{21}{19}$, so the gain is $\frac{2}{19}$ = (d) $10\frac{10}{19}\%$.

Question 12

A grocer mixes 1 kg of sand with every 9 kg of sugar and sells the mixture at 8% above the cost price of sugar. What is his gain %? (a) 20% (b) 18% (c) 18.8% (d) $19\frac{1}{9}\%$

Answer. Every 10 kg of mixture costs him only 9 kg of sugar. With sugar at ₹100 a kg, 10 kg costs ₹900 and sells for 10 × 108 = ₹1,080. Gain = 180 on 900 = (a) 20%. Adding 8% and $11\frac{1}{9}\%$ gives $19\frac{1}{9}\%$; the two parts have to be multiplied.

Question 13

A sells a laptop to B at a 25% profit, and B sells it to C at an 8% profit. If C pays ₹40,500, how much did A pay for it? (a) ₹37,500 (b) ₹27,135 (c) ₹30,451.13 (d) ₹30,000

Answer. Go back one sale at a time. B's CP = 40,500 ÷ 1.08 = ₹37,500, and A's CP = 37,500 ÷ 1.25 = (d) ₹30,000. In one step: 40,500 ÷ 1.35.

Question 14

A sells a sofa to B at a 40% gain, and B sells it to C at a 15% loss. C pays ₹9,520. How much did B pay for the sofa? (a) ₹11,200 (b) ₹8,000 (c) ₹10,948 (d) ₹13,328

Answer. B's loss is on B's own CP, so 85% of B's CP = ₹9,520. B paid 9,520 ÷ 0.85 = (a) ₹11,200. (A had paid 11,200 ÷ 1.4 = ₹8,000.)

Question 15

A buys a motorbike for ₹40,000 and sells it to B at a 15% profit. Later B sells it back to A at a 10% loss. What does A gain from the two deals? (a) ₹6,000 (b) ₹4,600 (c) ₹10,000 (d) ₹2,000

Answer. A gets ₹46,000 from B. B's 10% loss is on his own CP of ₹46,000, so B sells it back for ₹41,400. A has the bike again and 46,000 − 41,400 = (b) ₹4,600 more than he started with. Taking B's loss on A's ₹40,000 (selling back at ₹36,000) leads to ₹10,000.

Question 16

A buys a cooler for ₹6,250 and sells it to B at a 12% profit. B sells it to C for ₹7,700. What is B's profit %? (a) 11.2% (b) 23.2% (c) 10% (d) $9\frac{1}{11}\%$

Answer. B paid 6,250 × 1.12 = ₹7,000. B's profit = 7,700 − 7,000 = ₹700 on ₹7,000, which is (c) 10%.

Question 17

A dealer pays his agent a commission of 10% on the selling price and still makes a profit of 17%. If an article costs him ₹2,000, what is its selling price? (a) ₹2,540 (b) ₹2,600 (c) ₹2,574 (d) ₹2,340

Answer. He must keep 2,000 × 1.17 = ₹2,340. The agent takes 10% of the SP, so he keeps 90% of it: SP = 2,340 ÷ 0.9 = (b) ₹2,600. Check: 10% of 2,600 is ₹260, and 2,600 − 260 = 2,340.

Question 18

A firm gives a 12% discount on the marked price and pays its agent 5% commission on the selling price, and still gains 25.4%. By what % above the cost price are its goods marked? (a) 42.4% (b) 50% (c) 45% (d) 48%

Answer. After the discount and the commission, the firm keeps 0.88 × 0.95 = 0.836 of the MP. This must be 1.254 times the CP, so $\frac{\text{MP}}{\text{CP}} = \frac{1.254}{0.836}$ = 1.5, and the goods are marked (b) 50% above cost.

Question 19

A dealer buys goods for ₹2,50,000 and pays 2% brokerage on the purchase. On the sale he must pay a 4% commission on the selling price. At what price must he sell the goods to gain 20% on his total cost? (a) ₹3,15,000 (b) ₹3,06,000 (c) ₹3,18,750 (d) ₹3,18,240

Answer. His total cost is 2,50,000 × 1.02 = ₹2,55,000, and a 20% gain means he must keep ₹3,06,000. He keeps 96% of the selling price, so the price = 3,06,000 ÷ 0.96 = (c) ₹3,18,750. ₹3,06,000 leaves out the commission.

Question 20

A trader finds that 25% of his goods have been stolen. He sells the rest at 40% above the cost price. What is his overall profit or loss %? (a) 15% profit (b) 5% loss (c) 10% profit (d) 5% profit

Answer. Take the cost of the whole stock as 100. The 75 units left sell for 75 × 1.4 = 105. So he collects 105 on a cost of 100, a (d) 5% profit. Don't divide 105 by the 75 sold: he paid for the stolen goods too.

Question 21

One-fifth of a shopkeeper's stock is damaged and is sold at 40% of its cost. At what % profit must he sell the rest so that he gains 20% on the whole stock? (a) 30% (b) 37.5% (c) 25% (d) 40%

Answer. Take the cost of the whole stock as 100. The damaged 20 units bring 20 × 0.4 = 8. He needs 120 in all, so the other 80 units must bring 112. That is 112 ÷ 80 = 1.4 times their cost, a profit of (d) 40%.

Question 22

A shop marks its goods 50% above the cost price. It sells 70% of the stock at the marked price and clears the rest at 40% off the marked price. What is its overall profit %? (a) 32% (b) 38% (c) 20% (d) 29%

Answer. Take the cost of the whole stock as 100. The 70 units at the MP bring 70 × 1.5 = 105. The other 30 units sell at 1.5 × 0.6 = 0.9 of cost, bringing 27. Total 132 on a cost of 100, a profit of (a) 32%. 38% takes the clearance part as a 10% gain (50 − 40), but 40% off the MP leaves a 10% loss.