Discount — Extra Practice

Competitive Mathematics · Arithmetic

6 Part13 worked examples114 practice questions

NoteThis is the Extra Practice set: lots of questions. To learn the methods first, read the main chapter of the same topic (the one without “Extra Practice” in its name), then practise here.

A shop board says “Jacket ₹2,000 — 30% off”. ₹2,000 is the jacket’s marked price (MP), the price on the label. The customer actually pays less; the amount knocked off is the discount and the price paid is the selling price (SP). One rule runs through the whole chapter in three forms: a discount is always on the marked price, never on the cost price.

∑ Formula

छूट = MP - SP, SP = MP \times \frac{100 - r}{100}, r % = \frac{छूट}{MP} \times 100

The three formulas

Marked price, discount and selling price

MP · discount · SP

Finding the third when two of these are given, and the rate of discount.

Three things, one formula

MP, SP, rate

Finding the SP, the MP and the rate of discount.

The multiplier method: an $r %$ discount means the buyer pays $(100 - r) %$ of the marked price. For 15% off the multiplier is 0.85; for 40% off 0.60. SP = MP × multiplier; MP = SP ÷ multiplier. (This mirrors profit and loss — only the base is the MP, not the CP.)

Example 1 · Basic

A chair marked ₹2,400 is given at a 25% discount. What is the selling price?

Solution, step by step

1Discount $= 25%$ of $2400 = 600$ .
2SP $= 2400 - 600 = 1800$

Answer ₹1,800

Example 2 · Standard

After a 15% discount a suit sold for ₹3,740. What was its marked price?

Solution, step by step

1SP $= 85%$ of MP ⟹ MP $= \frac{3740 \times 100}{85}$ .
2 $= 4400$

Answer ₹4,400

Example 3 · Standard

An article marked ₹5,600 sold for ₹4,760. What was the rate of discount?

Solution, step by step

1Discount $= 5600 - 4760 = 840$ .
2Rate $= \frac{840}{5600} \times 100 = 15%$

Answer 15%

BasicTry it yourself first

An article is marked at ₹1,500 and a 20% discount is allowed. What is the selling price?

BasicTry it yourself first

An article is marked at ₹640 and a 35% discount is allowed. What is the selling price?

StandardTry it yourself first

An article is marked at ₹4,800 and a 12.5% discount is allowed. What is the selling price?

AdvancedTry it yourself first

An article is marked at ₹7,250 and a 18% discount is allowed. What is the selling price?

BasicTry it yourself first

After a 10% discount an article sold for ₹720. What was its marked price?

StandardTry it yourself first

After a 30% discount an article sold for ₹3,150. What was its marked price?

StandardTry it yourself first

After a 15% discount an article sold for ₹2,142. What was its marked price?

AdvancedTry it yourself first

After a 12.5% discount an article sold for ₹5,145. What was its marked price?

BasicTry it yourself first

An article marked ₹500 sold for ₹400. What was the rate of discount?

StandardTry it yourself first

An article marked ₹2,400 sold for ₹2,040. What was the rate of discount?

AdvancedTry it yourself first

An article marked ₹4,750 sold for ₹3,990. What was the rate of discount?

StandardTry it yourself first

Paying an electricity bill before the due date gives a 4% reduction. A man got a reduction of ₹260. What was the amount of the bill?

AdvancedTry it yourself first

Paying an electricity bill before the due date gives a 6% reduction. A man got a reduction of ₹525. What was the amount of the bill?

Successive discounts

A discount on a discount

When several discounts are given one after another, and their single equivalent.

Working out successive discounts

Each discount on what is left

Finding the SP and the single equivalent discount: $a + b - \frac{ab}{100}$ .

∑ Formula

SP = MP (1 - \frac{r _{1}}{100}) (1 - \frac{r _{2}}{100}), r_{eq} = r_{1} + r_{2} - \frac{r _{1} r _{2}}{100}

Two successive discounts

TrapTwo discounts of 20% and 10% make 28%, not 30%:

20 + 10 - \frac{200}{100} = 28

. The second discount acts on an already reduced price; swapping the order does not change the answer, but adding is wrong.

Example 4 · Standard

An article marked ₹8,000 gets successive discounts of 20% and 10%. Find the SP and the total discount.

Solution, step by step

1SP $= 8000 \times 0.8 \times 0.9 = 5760$ .
2Total discount $= 8000 - 5760 = 2240$ , i.e. $28%$ .

Answer ₹5,760 and ₹2,240

Example 5 · Standard

Three successive discounts of 15%, 10% and 20% equal what single discount?

Solution, step by step

1Remaining fraction $= 0.85 \times 0.90 \times 0.80 = 0.612$ .
2Single discount $= 1 - 0.612 = 0.388$ , i.e. $38.8%$ .

Answer 38.8%

BasicTry it yourself first

A refrigerator is marked ₹2,000. Successive discounts of 10% and 8% are given. What is the final selling price?

StandardTry it yourself first

A refrigerator is marked ₹6,400. Successive discounts of 25% and 10% are given. What is the final selling price?

StandardTry it yourself first

A refrigerator is marked ₹4,500. Successive discounts of 20% and 20% and 10% are given. What is the final selling price?

AdvancedTry it yourself first

A refrigerator is marked ₹12,500. Successive discounts of 12% and 10% and 5% are given. What is the final selling price?

BasicTry it yourself first

What single discount is equivalent to successive discounts of 10% and 10%?

StandardTry it yourself first

What single discount is equivalent to successive discounts of 30% and 20%?

StandardTry it yourself first

What single discount is equivalent to successive discounts of 25% and 20% and 10%?

AdvancedTry it yourself first

What single discount is equivalent to successive discounts of 15% and 12% and 20%?

Comparing offers and “free” schemes

Which one is cheaper

Single versus successive discounts, and turning “buy $x$ get $y$ free” into a discount.

To weigh two offers work out the final price of each, or find the single equivalent discount and compare. In “buy $x$ get $y$ free” you receive $x + y$ articles but pay for $x$ , so the discount $= \frac{y}{x + y} \times 100%$ . (“Buy 2 get 1 free” = $33 \frac{1}{3} %$ off, not 50%.)

Example 6 · Standard

Two offers on an article: (A) a single 28% discount, (B) 20% then 10%. Which is better for the buyer?

Solution, step by step

1The single equivalent of B is $20 + 10 - 2 = 28%$ .
2They are equal — either will do.

Answer Both equal

Example 7 · Standard

“Buy 3, get 1 free” equals what per cent discount?

Solution, step by step

1You receive 4 articles and pay for 3.
2Discount $= \frac{1}{4} \times 100 = 25%$

Answer 25%

StandardTry it yourself first

For a ₹5,000 article there are two offers — (A) a single 25% discount, (B) successive 20% and 10% discounts. Which is cheaper for the buyer?

AdvancedTry it yourself first

For a ₹8,000 article there are two offers — (A) a single 30% discount, (B) successive 20% and 15% discounts. Which is cheaper for the buyer?

BasicTry it yourself first

A shop runs a scheme: buy 4, get 1 free. This equals what per cent discount?

StandardTry it yourself first

A shop runs a scheme: buy 3, get 2 free. This equals what per cent discount?

AdvancedTry it yourself first

A shop runs a scheme: buy 7, get 3 free. This equals what per cent discount?

StandardTry it yourself first

On a marked price the selling prices after a single 30% discount and after successive 20% and 10% discounts differ by ₹60. What is the marked price?

AdvancedTry it yourself first

On a marked price the selling prices after a single 40% discount and after successive 20% and 20% discounts differ by ₹100. What is the marked price?

Discount together with profit and loss

The CP–MP–SP relation

A shopkeeper first marks the price above cost and then gives a discount — the two per cents have different bases.

Three prices on one ladder

CP → MP → SP

CP to MP by the mark-up per cent, MP to SP by the discount; the gap between SP and CP is the profit or loss.

∑ Formula

MP = CP \times \frac{100 + m}{100}, SP = MP \times \frac{100 - d}{100}, लाभ % = m - d - \frac{m d}{100}

For a mark-up

m %

and discount

d %

To find	Method
CP, given MP, discount and profit	SP = MP × (100−d)/100, then CP = SP × 100/(100+g)
Mark-up needed (profit $g$ , discount $d$ )	MP = CP × (100+g)/(100−d)
CP : MP	(100−d) : (100+g)
No profit no loss	discount $= \frac{m}{100 + m} \times 100$

Example 8 · Standard

A shopkeeper marks goods 40% above cost and allows a 20% discount. What is his profit per cent?

Solution, step by step

1CP = 100 ⟹ MP = 140; SP = $140 \times 0.8 = 112$ .
2Profit $= 12%$ (formula: $40 - 20 - \frac{800}{100} = 12$ )

Answer 12%

Example 9 · Standard

A shopkeeper allows a 10% discount on an article marked ₹4,500 and still gains 8%. What is the cost price?

Solution, step by step

1SP $= 4500 \times 0.90 = 4050$ .
2CP $= \frac{4050 \times 100}{108} = 3750$

Answer ₹3,750

Example 10 · Advanced

By what per cent above cost should an article be marked so that after a 15% discount there is still a 36% profit?

Solution, step by step

1CP = 100 ⟹ required SP 136. SP = $85%$ of MP ⟹ MP $= \frac{136 \times 100}{85} = 160$ .
2Mark-up $= 60%$

Answer 60%

BasicTry it yourself first

A bicycle is marked ₹2,400. The shopkeeper allows a 10% discount and still earns a 8% profit. What is the cost price?

StandardTry it yourself first

A bicycle is marked ₹3,600. The shopkeeper allows a 20% discount and still earns a 20% profit. What is the cost price?

StandardTry it yourself first

A bicycle is marked ₹5,250. The shopkeeper allows a 12% discount and still earns a 10% profit. What is the cost price?

AdvancedTry it yourself first

A bicycle is marked ₹7,800. The shopkeeper allows a 15% discount and still earns a 36% profit. What is the cost price?

BasicTry it yourself first

A shopkeeper wants a 8% profit even after allowing a 10% discount. By what per cent above the cost price should he mark the article?

StandardTry it yourself first

A shopkeeper wants a 20% profit even after allowing a 20% discount. By what per cent above the cost price should he mark the article?

StandardTry it yourself first

A shopkeeper wants a 50% profit even after allowing a 25% discount. By what per cent above the cost price should he mark the article?

AdvancedTry it yourself first

A shopkeeper wants a 40% profit even after allowing a 12.5% discount. By what per cent above the cost price should he mark the article?

BasicTry it yourself first

A shopkeeper marks goods 25% above cost and allows a 20% discount. What is his profit per cent?

StandardTry it yourself first

A shopkeeper marks goods 50% above cost and allows a 30% discount. What is his profit per cent?

StandardTry it yourself first

A shopkeeper marks goods 20% above cost and allows a 30% discount. What is his loss per cent?

AdvancedTry it yourself first

A shopkeeper marks goods 35% above cost and allows a 40% discount. What is his loss per cent?

StandardTry it yourself first

A shopkeeper sells at a 10% discount and earns a 8% profit. What is the ratio of the cost price to the marked price?

AdvancedTry it yourself first

A shopkeeper sells at a 20% discount and earns a 25% profit. What is the ratio of the cost price to the marked price?

StandardTry it yourself first

A shopkeeper marks goods 25% above cost. What discount should he give to neither gain nor lose?

AdvancedTry it yourself first

A shopkeeper marks goods 60% above cost. What discount should he give to neither gain nor lose?

Mixed questions

The SP–MP–profit loop

When both prices are given for an article and one unknown must be found.

Example 11 · Standard

A seller marked an article costing ₹6,000 up by 25% and gave a 4% discount. Find the SP and the profit.

Solution, step by step

1MP $= 6000 \times 1.25 = 7500$ ; SP $= 7500 \times 0.96 = 7200$ .
2Profit $= 7200 - 6000 = 1200$ , i.e. $20%$ .

Answer ₹7,200 and ₹1,200 (20%)

Example 12 · Advanced

Selling an article at a 20% discount makes the shopkeeper lose 10%. What would his profit per cent be with no discount?

Solution, step by step

1Let MP $= 100$ ⟹ SP $= 80$ , which is $90%$ of CP ⟹ CP $= \frac{80}{0.9} = 88.89$ .
2With no discount SP $= 100$ ; profit $= \frac{100 - 88.89}{88.89} \times 100 = 12.5%$

Answer 12.5%

StandardTry it yourself first

Without any discount a shopkeeper would gain 25%. What will his gain be after allowing a 10% discount?

Traps and fast tricks

Slips · tricks

Where most marks are lost in discount questions.

The commonest mistakes

Six traps

The slips that cost marks.

Trap 1Taking the discount on the CP. A discount is on the marked price; profit and loss are on the cost price.

Trap 2Adding successive discounts. 20% + 10% = 28%, not 30%.

Trap 3Using $\frac{y}{x}$ for “buy $x$ get $y$ free”. The right ratio is

\frac{y}{x + y}

Trap 4Multiplying when finding MP. MP = SP ÷ (1 − rate), not SP × (1 − rate).

Trap 5Subtracting mark-up and discount to get profit. 40% mark-up − 20% discount ≠ 20% profit; the right answer is

40 - 20 - 8 = 12%

Trap 6Confusing the profit without discount with the profit after it. The no-discount SP is the MP.

Fast tricks

Multipliers and assuming 100

Tricks that shorten the work.

Assume CP = 100 when no amount is given; then MP and SP are simple numbers.
Single equivalent discount: $a + b - \frac{ab}{100}$ ; for three, pair them up.
Mark-up $m$ , discount $d$ ⟹ profit $= m - d - \frac{m d}{100}$ (a negative value means a loss).
Discount for no profit no loss $= \frac{m}{100 + m} \times 100$ .
“Free” schemes: free ÷ total.

Example 13 · Standard

If the mark-up is 60% and the discount 25%, what is the profit per cent?

Solution, step by step

1Formula: $60 - 25 - \frac{60 \times 25}{100}$ .
2 $= 35 - 15 = 20%$

Answer 20%

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

BasicTry it yourself first

An article is marked at ₹1,200 and a 15% discount is allowed. What is the selling price?

BasicTry it yourself first

An article is marked at ₹850 and a 20% discount is allowed. What is the selling price?

BasicTry it yourself first

After a 10% discount an article sold for ₹540. What was its marked price?

BasicTry it yourself first

After a 5% discount an article sold for ₹1,900. What was its marked price?

BasicTry it yourself first

An article marked ₹800 sold for ₹640. What was the rate of discount?

BasicTry it yourself first

An article marked ₹1,500 sold for ₹1,200. What was the rate of discount?

BasicTry it yourself first

A refrigerator is marked ₹1,000. Successive discounts of 10% and 10% are given. What is the final selling price?

BasicTry it yourself first

A refrigerator is marked ₹5,000. Successive discounts of 20% and 10% are given. What is the final selling price?

BasicTry it yourself first

What single discount is equivalent to successive discounts of 20% and 10%?

BasicTry it yourself first

What single discount is equivalent to successive discounts of 50% and 20%?

BasicTry it yourself first

A shop runs a scheme: buy 1, get 1 free. This equals what per cent discount?

BasicTry it yourself first

A shop runs a scheme: buy 2, get 1 free. This equals what per cent discount?

BasicTry it yourself first

A bicycle is marked ₹1,500. The shopkeeper allows a 20% discount and still earns a 20% profit. What is the cost price?

BasicTry it yourself first

A shopkeeper wants a 20% profit even after allowing a 20% discount. By what per cent above the cost price should he mark the article?

BasicTry it yourself first

A shopkeeper marks goods 20% above cost and allows a 10% discount. What is his profit per cent?

BasicTry it yourself first

A shopkeeper marks goods 10% above cost. What discount should he give to neither gain nor lose?

BasicTry it yourself first

A shopkeeper sells at a 20% discount and earns a 25% profit. What is the ratio of the cost price to the marked price?

BasicTry it yourself first

Paying an electricity bill before the due date gives a 4% reduction. A man got a reduction of ₹100. What was the amount of the bill?

BasicTry it yourself first

For a ₹2,000 article there are two offers — (A) a single 15% discount, (B) successive 10% and 5% discounts. Which is cheaper for the buyer?

BasicTry it yourself first

On a marked price the selling prices after a single 20% discount and after successive 10% and 10% discounts differ by ₹20. What is the marked price?

Level 2 — Standard

Two ideas combined

Exam-level questions.

StandardTry it yourself first

An article is marked at ₹3,640 and a 12.5% discount is allowed. What is the selling price?

StandardTry it yourself first

An article is marked at ₹9,500 and a 16% discount is allowed. What is the selling price?

StandardTry it yourself first

After a 15% discount an article sold for ₹2,380. What was its marked price?

StandardTry it yourself first

After a 25% discount an article sold for ₹3,375. What was its marked price?

StandardTry it yourself first

An article marked ₹3,600 sold for ₹2,970. What was the rate of discount?

StandardTry it yourself first

An article marked ₹7,200 sold for ₹6,120. What was the rate of discount?

StandardTry it yourself first

A refrigerator is marked ₹7,500. Successive discounts of 20% and 15% are given. What is the final selling price?

StandardTry it yourself first

A refrigerator is marked ₹10,000. Successive discounts of 10% and 20% and 25% are given. What is the final selling price?

StandardTry it yourself first

A refrigerator is marked ₹3,200. Successive discounts of 25% and 10% are given. What is the final selling price?

StandardTry it yourself first

What single discount is equivalent to successive discounts of 40% and 25%?

StandardTry it yourself first

What single discount is equivalent to successive discounts of 10% and 20% and 30%?

StandardTry it yourself first

A shop runs a scheme: buy 5, get 2 free. This equals what per cent discount?

StandardTry it yourself first

A shop runs a scheme: buy 4, get 3 free. This equals what per cent discount?

StandardTry it yourself first

A bicycle is marked ₹4,200. The shopkeeper allows a 5% discount and still earns a 14% profit. What is the cost price?

StandardTry it yourself first

A bicycle is marked ₹6,600. The shopkeeper allows a 10% discount and still earns a 8% profit. What is the cost price?

StandardTry it yourself first

A shopkeeper wants a 19% profit even after allowing a 15% discount. By what per cent above the cost price should he mark the article?

StandardTry it yourself first

A shopkeeper wants a 40% profit even after allowing a 30% discount. By what per cent above the cost price should he mark the article?

StandardTry it yourself first

A shopkeeper marks goods 60% above cost and allows a 25% discount. What is his profit per cent?

StandardTry it yourself first

A shopkeeper marks goods 40% above cost and allows a 35% discount. What is his loss per cent?

StandardTry it yourself first

A shopkeeper marks goods 40% above cost. What discount should he give to neither gain nor lose?

StandardTry it yourself first

A shopkeeper sells at a 25% discount and earns a 20% profit. What is the ratio of the cost price to the marked price?

StandardTry it yourself first

A shopkeeper sells at a 12% discount and earns a 10% profit. What is the ratio of the cost price to the marked price?

StandardTry it yourself first

Paying an electricity bill before the due date gives a 5% reduction. A man got a reduction of ₹240. What was the amount of the bill?

StandardTry it yourself first

For a ₹6,000 article there are two offers — (A) a single 38% discount, (B) successive 25% and 20% discounts. Which is cheaper for the buyer?

StandardTry it yourself first

For a ₹9,000 article there are two offers — (A) a single 35% discount, (B) successive 20% and 20% discounts. Which is cheaper for the buyer?

StandardTry it yourself first

On a marked price the selling prices after a single 30% discount and after successive 20% and 10% discounts differ by ₹80. What is the marked price?

Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

AdvancedTry it yourself first

An article is marked at ₹18,750 and a 22% discount is allowed. What is the selling price?

AdvancedTry it yourself first

After a 15% discount an article sold for ₹7,616. What was its marked price?

AdvancedTry it yourself first

After a 32% discount an article sold for ₹12,240. What was its marked price?

AdvancedTry it yourself first

An article marked ₹8,400 sold for ₹6,720. What was the rate of discount?

AdvancedTry it yourself first

A refrigerator is marked ₹16,000. Successive discounts of 12.5% and 10% and 20% are given. What is the final selling price?

AdvancedTry it yourself first

A refrigerator is marked ₹24,000. Successive discounts of 25% and 20% and 10% are given. What is the final selling price?

AdvancedTry it yourself first

What single discount is equivalent to successive discounts of 12% and 15% and 25%?

AdvancedTry it yourself first

What single discount is equivalent to successive discounts of 45% and 20% and 10%?

AdvancedTry it yourself first

A shop runs a scheme: buy 9, get 4 free. This equals what per cent discount?

AdvancedTry it yourself first

A shop runs a scheme: buy 11, get 5 free. This equals what per cent discount?

AdvancedTry it yourself first

A bicycle is marked ₹9,100. The shopkeeper allows a 12% discount and still earns a 30% profit. What is the cost price?

AdvancedTry it yourself first

A bicycle is marked ₹12,600. The shopkeeper allows a 20% discount and still earns a 25% profit. What is the cost price?

AdvancedTry it yourself first

A shopkeeper wants a 31% profit even after allowing a 18% discount. By what per cent above the cost price should he mark the article?

AdvancedTry it yourself first

A shopkeeper wants a 52% profit even after allowing a 24% discount. By what per cent above the cost price should he mark the article?

AdvancedTry it yourself first

A shopkeeper marks goods 45% above cost and allows a 30% discount. What is his profit per cent?

AdvancedTry it yourself first

A shopkeeper marks goods 75% above cost and allows a 40% discount. What is his profit per cent?

AdvancedTry it yourself first

A shopkeeper marks goods 150% above cost. What discount should he give to neither gain nor lose?

AdvancedTry it yourself first

A shopkeeper sells at a 16% discount and earns a 25% profit. What is the ratio of the cost price to the marked price?

AdvancedTry it yourself first

A shopkeeper sells at a 24% discount and earns a 12.5% profit. What is the ratio of the cost price to the marked price?

AdvancedTry it yourself first

Paying an electricity bill before the due date gives a 7% reduction. A man got a reduction of ₹868. What was the amount of the bill?

AdvancedTry it yourself first

For a ₹15,000 article there are two offers — (A) a single 45% discount, (B) successive 30% and 20% discounts. Which is cheaper for the buyer?

AdvancedTry it yourself first

On a marked price the selling prices after a single 50% discount and after successive 30% and 25% discounts differ by ₹125. What is the marked price?

AdvancedTry it yourself first

On a marked price the selling prices after a single 45% discount and after successive 25% and 25% discounts differ by ₹80. What is the marked price?

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.