Commercial arithmetic — SSC CGL Quant Notes

Concepts and rules

Cost price (CP) is what the seller pays, including overheads such as transport or repairs. Selling price (SP) is what the buyer pays. Profit = SP − CP when SP > CP; Loss = CP − SP when CP > SP.

Profit% = Profit × 100/CP and Loss% = Loss × 100/CP. They are on CP unless the question says 'on the selling price'.

SP = CP × (100 + P%)/100 and CP = SP × 100/(100 + P%); for a loss replace + by −. As fractions: 20% profit means SP = 6/5 of CP, 25% loss means SP = 3/4 of CP.

If the CP of x articles equals the SP of y articles, Profit% = (x − y) × 100/y (a loss if y > x, with |x − y| in the numerator).

Two articles sold at the SAME selling price, one at x% profit and one at x% loss, always give an overall loss of x²/100 per cent.

False weights: a seller who gives F grams while charging for T grams gains (T − F) × 100/F per cent even when he sells at cost price.

Profit as a per cent of SP, p%, equals p × 100/(100 − p) per cent of CP. For example 20% on SP = 25% on CP.

Chains of sales (A to B to C): multiply the factors. A 20% profit followed by a 10% loss gives 6/5 × 9/10 = 27/25 of the first CP.

Method and shortcuts

Step 1: decide which price is the base (normally CP = 100). Step 2: write every percentage as a multiplying factor. Step 3: set the known rupee value equal to its share and scale.

Two selling prices, two results: if SP1 gives p1% and SP2 gives p2%, then (SP2 − SP1) corresponds to (p2 − p1)% of CP, so CP = (SP2 − SP1) × 100/(p2 − p1). A loss counts as a negative profit.

Per-unit method for 'buy a for ₹m, sell b for ₹n': compare the price of one unit, or take the LCM of a and b as the number of units.

False weight with a stated profit or loss: gain factor = (SP factor) × (true weight/false weight). Subtract 1 and convert to per cent.

When it fails: the x²/100 loss rule needs equal SELLING prices. With equal cost prices, x% profit and x% loss cancel to no profit no loss.

Example 1: Profit per cent

Question: Arjun buys a fan for ₹640 and sells it for ₹800. Find his profit per cent.

Solution: Profit = 800 − 640 = ₹160. Profit% = 160 × 100/640 = 25%. Dividing by SP would wrongly give 20%.

Answer: 25%

Example 2: From loss to gain

Question: By selling a watch for ₹1,140 a man loses 5%. At what price should he sell it to gain 5%?

Solution: 95% of CP = 1,140, so CP = 1,140 × 100/95 = ₹1,200. Required SP = 105% of 1,200 = ₹1,260. Shortcut: 1,140 × 105/95 = 1,260.

Answer: ₹1,260

Example 3: CP of x equals SP of y

Question: The cost price of 20 pens equals the selling price of 16 pens. Find the profit or loss per cent.

Solution: Let each pen cost ₹1, so 20 pens cost ₹20 and 16 pens sell for ₹20; one pen sells for 20/16 = ₹1.25. Profit = 25%. Formula: (20 − 16) × 100/16 = 25%.

Answer: 25% profit

Example 4: Same SP, equal profit and loss

Question: Two mobile phones are sold for ₹9,900 each, one at 10% profit and the other at 10% loss. What is the overall result? (a) No profit no loss (b) Loss of ₹200 (c) Gain of ₹200 (d) Loss of ₹198

Solution: CP1 = 9,900 × 100/110 = ₹9,000 and CP2 = 9,900 × 100/90 = ₹11,000. Total CP = ₹20,000, total SP = ₹19,800, so loss = ₹200, which is 100/100 = 1% of CP as the rule x²/100 predicts.

Answer: (b) Loss of ₹200

Example 5: False weight with a declared loss

Question: A grocer claims to sell rice at 10% loss, but he uses an 800 g weight in place of 1 kg. What is his actual gain per cent? (a) 10% (b) 15% (c) 8% (d) 12.5%

Solution: Let 1 kg cost ₹100, so 800 g costs ₹80. He charges the price of 1 kg less 10%, that is ₹90, but hands over goods costing ₹80.

Gain = 90 − 80 = ₹10 on a cost of ₹80, so gain% = 10 × 100/80 = 12.5%. Factor form: 0.9 × 1000/800 = 1.125.

Answer: (d) 12.5%

Example 6: A chain of sales

Question: A sells a scooter to B at 20% profit, B sells it to C at 10% loss, and C sells it to D at 25% profit for ₹6,750. What did A pay for it? (a) ₹5,400 (b) ₹4,500 (c) ₹5,000 (d) ₹5,200

Solution: Overall factor = 6/5 × 9/10 × 5/4 = 270/200 = 27/20 = 1.35. So A's CP × 1.35 = 6,750.

A's CP = 6,750 ÷ 1.35 = ₹5,000. Check: 5,000 → 6,000 (B pays) → 5,400 (C pays) → 6,750 (D pays).

Answer: (c) ₹5,000

Common mistakes

Calculating profit per cent on the selling price. Divide by CP unless the question says otherwise.

Adding percentages in a chain: 20% profit then 10% loss is not 10% profit; it is 6/5 × 9/10 = 1.08, an 8% profit.

Assuming equal profit and loss cancel. With equal SPs there is always a loss of x²/100 per cent.

In false-weight questions dividing by the true weight. Gain is (T − F)/F: 900 g for 1 kg gives 100/900 = 11.11%, not 10%.

Ignoring overheads. Repair, transport and tax paid by the seller are added to CP before finding profit.

Practice questions

Q1. A vendor buys bananas at 6 for ₹10 and sells them at 5 for ₹10. What is his profit per cent? (a) 16.67% (b) 20% (c) 25% (d) 18%

Q2. Selling a chair for ₹2,016 gives a profit of 12%. For what price should it be sold to earn a profit of 20%? (a) ₹2,160 (b) ₹2,180 (c) ₹2,140 (d) ₹2,200

Q3. A trader's profit is 20% of his selling price. What is his profit as a per cent of cost price? (a) 20% (b) 22.5% (c) 16.67% (d) 25%

Answers

Q1. (b) Take 30 bananas: CP = ₹50, SP = ₹60, profit = 10 × 100/50 = 20%.

Q2. (a) CP = 2,016 × 100/112 = ₹1,800; SP = 1,800 × 120/100 = ₹2,160.

Q3. (d) Let SP = 100: profit 20, CP 80, profit% = 20 × 100/80 = 25%.

Quick revision

Profit% and loss% are on CP; SP = CP × (100 ± P%)/100.

CP of x = SP of y ⇒ profit% = (x − y) × 100/y.

Same SP, x% profit and x% loss ⇒ loss of x²/100 per cent.

False weight: gain% = (T − F) × 100/F; with a declared profit or loss multiply the factors.

Chains of sales: multiply factors, never add percentages.

Concepts and rules

Marked price (MP), also called list price, is the price printed on the article. Discount is a reduction given on the MP, so SP = MP − Discount.

Discount% = Discount × 100/MP, and SP = MP × (100 − d)/100. Discount is on MP; profit is on CP. Never mix the two bases.

Successive discounts a% and b% equal a single discount of (a + b − ab/100)%. The order of the discounts does not matter. For three discounts multiply the factors (100 − a)(100 − b)(100 − c).

If the MP is m% above CP and a discount of d% is given, profit% = m − d − md/100 (negative means a loss).

Key relation: CP × (100 + p)/100 = MP × (100 − d)/100, so MP : CP = (100 + p) : (100 − d). Almost every discount question reduces to this one equation.

'Buy x, get y free' means you pay for x but receive x + y articles, so the effective discount is y × 100/(x + y) per cent.

A single discount of 20% is better for the buyer than successive discounts of 10% and 10% (which equal 19%).

Method and shortcuts

Step 1: write the chain CP → MP → SP. Step 2: put CP = 100 (or MP = 100) and apply the mark-up and the discount factors. Step 3: match the rupee value given to its place in the chain.

Ratio shortcut: MP/CP = (100 + p)/(100 − d). With 12% profit after a 20% discount, MP/CP = 112/80 = 7/5, so MP is 40% above CP.

Successive discounts: combine two at a time with a + b − ab/100. Example: 20% and 10% give 30 − 2 = 28%.

Free articles: treat them as a discount on the total quantity, then use the same MP–CP relation.

When it fails: a + b − ab/100 combines discounts only. A mark-up followed by a discount uses m − d − md/100 (the second term has a minus sign).

Example 1: Simple discount

Question: A shirt is marked ₹1,250 and sold at a discount of 12%. Find the selling price.

Solution: Discount = 12% of 1,250 = ₹150, so SP = 1,250 − 150 = ₹1,100. Directly: 1,250 × 88/100 = 1,100.

Answer: ₹1,100

Example 2: Successive discounts

Question: A jacket marked ₹2,500 is sold after successive discounts of 20% and 10%. Find the selling price and the single equivalent discount.

Solution: SP = 2,500 × 80/100 × 90/100 = ₹1,800. Single discount = 20 + 10 − 200/100 = 28%; check: 28% of 2,500 = 700 = 2,500 − 1,800.

Answer: ₹1,800; 28%

Example 3: Buy 4 get 1 free

Question: A store offers 'buy 4, get 1 free' on soaps. What is the effective discount per cent?

Solution: The buyer pays for 4 soaps and gets 5, so the discount is the price of 1 soap out of 5: 1 × 100/5 = 20%. It is not 25%, because the discount is measured against the MP of all 5 soaps received.

Answer: 20%

Example 4: Mark-up and discount

Question: A shopkeeper marks his goods 40% above cost price and allows a discount of 15%. What is his profit per cent? (a) 25% (b) 21% (c) 19% (d) 17%

Solution: CP = 100, MP = 140, SP = 140 × 85/100 = 119, so profit = 19%. Formula: 40 − 15 − 600/100 = 19%.

Answer: (c) 19%

Example 5: Marking for a target profit

Question: A trader buys a cooler for ₹1,900. He wants a profit of 8% after allowing successive discounts of 10% and 5%. At what price must he mark it? (a) ₹2,400 (b) ₹2,350 (c) ₹2,280 (d) ₹2,500

Solution: Required SP = 1,900 × 108/100 = ₹2,052. The discounts leave 90/100 × 95/100 = 0.855 of MP.

So MP × 0.855 = 2,052 and MP = 2,052 ÷ 0.855 = ₹2,400. Check: 2,400 → 2,160 → 2,052.

Answer: (a) ₹2,400

Example 6: Gap between MP and CP

Question: After giving a 20% discount, a dealer still makes a profit of 12%. If the marked price exceeds the cost price by ₹1,200, find the cost price. (a) ₹2,800 (b) ₹3,200 (c) ₹3,600 (d) ₹3,000

Solution: MP : CP = (100 + 12) : (100 − 20) = 112 : 80 = 7 : 5. The gap of 7 − 5 = 2 parts equals ₹1,200, so 1 part = ₹600.

CP = 5 parts = ₹3,000 and MP = ₹4,200. Check: 4,200 × 80/100 = 3,360 = 3,000 × 112/100.

Answer: (d) ₹3,000

Common mistakes

Adding successive discounts: 20% and 10% is 28%, not 30%.

Taking discount on CP or profit on MP. Discount is on MP, profit is on CP.

Treating 'buy 4 get 1 free' as a 25% discount. The base is the 5 articles received, so it is 20%.

Writing profit = mark-up − discount. A 40% mark-up with a 15% discount gives 19%, not 25%, because the discount is on the larger MP.

Practice questions

Q1. What single discount is equivalent to successive discounts of 25%, 20% and 10%? (a) 55% (b) 45% (c) 50% (d) 46%

Q2. A bag marked ₹800 is sold at a discount of 15%, and the seller still makes a profit of 25%. What is the cost price of the bag? (a) ₹560 (b) ₹520 (c) ₹544 (d) ₹576

Q3. A shopkeeper offers 'buy 3, get 2 free' and still earns a profit of 20%. By what per cent is his marked price above the cost price? (a) 80% (b) 100% (c) 120% (d) 60%

Answers

Q1. (d) Remaining = 75/100 × 80/100 × 90/100 = 0.54 of MP, so discount = 100 − 54 = 46%.

Q2. (c) SP = 800 × 85/100 = ₹680; CP = 680 × 100/125 = ₹544.

Q3. (b) For 5 articles he receives the MP of 3: 3 × MP = 5 × CP × 120/100 = 6 × CP, so MP = 2 × CP, which is 100% above CP.

Quick revision

SP = MP × (100 − d)/100; discount on MP, profit on CP.

Successive discounts: a + b − ab/100; mark-up then discount: m − d − md/100.

MP : CP = (100 + p) : (100 − d).

Buy x get y free: discount = y × 100/(x + y) per cent.