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CBSE · Class 8 · Mathematics Part 2

Part 2, Chapter 1: Fractions in Disguise

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Fractions, decimals and percentages

From daily life

A shop board says "35% off". It means 35 out of every 100 rupees of the price are taken off, which is the fraction 35/100 or the decimal 0.35 in disguise.

Simple definition

Per cent means "out of 100". A percentage is a fraction with denominator 100: 35% = 35/100 = 0.35. To change a fraction to a percentage, multiply by 100.

In the figure: 35 of the 100 squares are shaded: 35% = 35/100 = 0.35.

Rule 1: Percentage to fraction or decimal — Divide by 100, then simplify.

  • Easy: 25% = 25/100 = 1/4 = 0.25.
  • Medium: 12.5% = 12.5/100 = 1/8 = 0.125.
  • Exam-type: 150% = 150/100 = 3/2 = 1.5 (more than a whole).

Rule 2: Fraction to percentage — Multiply by 100, or make the denominator 100.

  • Easy: 3/5 = 60/100 = 60%.
  • Medium: 7/20 = 35/100 = 35%.
  • Exam-type: 2/3 × 100 = 66.67% (rounded); 3/8 = 0.375 = 37.5%.

Rule 3: Decimal to percentage — Move the decimal point two places to the right.

  • Easy: 0.35 = 35%.
  • Medium: 0.07 = 7%.
  • Exam-type: 0.125 = 12.5%; 0.5% = 0.005.
Remember

Common values worth knowing: 1/2 = 50%, 1/4 = 25%, 3/4 = 75%, 1/5 = 20%, 1/10 = 10%, 1/8 = 12.5%.

Common mistake

Writing 0.35 as 3.5% or 35 as 0.35. Moving the point by one place is wrong; it is always two places.

Shortcut

Per cent means 100. Memory clue: "Per cent, per century - both are hundreds".

Spot the difference

PointFractionPercentage
Example3/560%
Decimal form0.60.6
How to convert× 100 gives the percentage÷ 100 gives the fraction

Practice questions

Easy
  1. 1Write 25% as a fraction.
  2. 2Write 3/5 as a percentage.
  3. 3Write 45% as a decimal.
Medium
  1. 4Write 12.5% as a fraction.
  2. 5Write 9/25 as a percentage.
  3. 6Write 0.07 as a percentage.
Challenging
  1. 7Which is greatest: 3/4, 70% or 0.8?
  2. 8Write 0.5% as a decimal.
Show answers (try them yourself first)
  1. 11/4
  2. 260%
  3. 30.45
  4. 41/8
  5. 536%
  6. 67%
  7. 70.8 (80%)
  8. 80.005

Percentage of a quantity

From daily life

A school has 450 students and 20% walk to school. We need 20% of 450, which is 450 × 20/100 = 90 students.

Simple definition

To find p% of a quantity, multiply the quantity by p/100 (or by the decimal p ÷ 100). "Of" means multiply.

Live figure

Percentage of a quantity

In the app you can move this figure with your finger.

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In the figure: Move the sliders to see the shaded part of the quantity change with the percentage.

Rule 1: Direct multiplication — Quantity × p ÷ 100.

  • Easy: 20% of 450 = 450 × 20 ÷ 100 = 90.
  • Medium: 15% of 240 = 240 × 15 ÷ 100 = 36.
  • Exam-type: 2.5% of 400 = 400 × 2.5 ÷ 100 = 10.

Rule 2: Use easy fractions — 50% is half, 25% is a quarter, 10% is one tenth, 75% is three quarters.

  • Easy: 25% of 18 = 18 ÷ 4 = 4.5.
  • Medium: 75% of 80 = 80 × 3/4 = 60.
  • Exam-type: 90% of 70 = 70 − 10% of 70 = 70 − 7 = 63.

Rule 3: Build from 1% and 10% — Find 1% by dividing by 100 or 10% by dividing by 10, then scale up.

  • Easy: 1% of 3500 = 35.
  • Medium: 5% of 1200: 1% = 12, so 5% = 60.
  • Exam-type: 35% of 200 = 3 × 10% + 5% = 60 + 10 = 70.
Remember

A percentage can be more than 100%: 120% of 50 = 60. And 10% of a number is just the number divided by 10.

Common mistake

Dividing by the percentage instead of by 100: 20% of 450 is not 450 ÷ 20. It is 450 × 20 ÷ 100.

Shortcut

Times the percent, then divide by 100. Memory clue: "Of means times, per cent means over hundred".

Spot the difference

PointRightWrong
20% of 450450 × 20 ÷ 100 = 90450 ÷ 20 = 22.5
Ideapercent means out of 100percent means out of the percent
Result9022.5

Practice questions

Easy
  1. 1Find 20% of 450.
  2. 2Find 75% of 80.
  3. 3Find 25% of 18.
Medium
  1. 4Find 12% of 250.
  2. 5Find 5% of 1200.
  3. 6Find 8% of ₹2500.
Challenging
  1. 7Find 120% of 50.
  2. 8Find 35% of 200 using 10% and 5%.
Show answers (try them yourself first)
  1. 190
  2. 260
  3. 34.5
  4. 430
  5. 560
  6. 6₹200
  7. 760
  8. 870

Finding the percentage and the whole

From daily life

Asha scores 36 out of 50 and Bela scores 45 out of 60. To see who did better, change both to percentages: 72% and 75%. Bela did slightly better.

Simple definition

Part as a percentage: (part ÷ whole) × 100. If a percentage of the whole is known, the whole = (part ÷ percentage) × 100.

20%100% = ?301% = 1.5, 100% = 150

In the figure: 20% is 30, so 1% is 1.5 and the whole is 150.

Rule 1: What percentage is a part? — Divide the part by the whole, then multiply by 100.

  • Easy: 15 out of 60: 15/60 × 100 = 25%.
  • Medium: ₹30 out of ₹250: 30/250 × 100 = 12%.
  • Exam-type: 250 g of 2 kg: 250/2000 × 100 = 12.5% (same unit first).

Rule 2: Find the whole — Find 1% by dividing the part by its percentage, then multiply by 100.

  • Easy: 25% of a number is 17: 17 × 4 = 68.
  • Medium: 15% of a number is 45: 45 ÷ 15 × 100 = 300.
  • Exam-type: A student passes with 60 marks which is 40%: total = 60 ÷ 40 × 100 = 150.

Rule 3: Compare using percentages — Change every quantity to a percentage of its own whole, then compare.

  • Easy: 36/50 = 72% and 45/60 = 75%: the second is higher.
  • Medium: Class of 40 with 18 girls: boys are 22/40 = 55%.
  • Exam-type: A shop sold 35% and 78 items are left: 65% = 78, total = 120.
Remember

Both the part and the whole must be in the same unit before you compute the percentage.

Common mistake

Dividing the whole by the part: 15 out of 60 is 15 ÷ 60, not 60 ÷ 15. Also comparing 36 and 45 directly when the totals differ.

Shortcut

Part over whole, times 100. Memory clue: "Little over big, then hundred".

Spot the difference

PointFinding the percentageFinding the whole
Formula in words(part ÷ whole) × 100(part ÷ percentage) × 100
Example15/60 × 100 = 25%45 ÷ 15 × 100 = 300
Answer isa percentagea quantity

Practice questions

Easy
  1. 1What percentage of 60 is 15?
  2. 2What percentage of 48 is 36?
  3. 320% of a number is 30. Find the number.
Medium
  1. 4₹30 is what percentage of ₹250?
  2. 515% of a number is 45. Find the number.
  3. 612 minutes is what percentage of 1 hour?
Challenging
  1. 7Asha has 36/50 and Bela has 45/60. Find the higher percentage.
  2. 8A shop sold 35% of its items and 78 are left. Find the original number.
Show answers (try them yourself first)
  1. 125%
  2. 275%
  3. 3150
  4. 412%
  5. 5300
  6. 620%
  7. 7Bela, 75%
  8. 8120

Percentage increase and decrease

From daily life

Tomatoes cost ₹30 a kilogram three years ago and ₹42 now. The price rose by ₹12, which is 12/30 × 100 = 40% of the old price.

Simple definition

Percentage increase = (increase ÷ original) × 100. Percentage decrease = (decrease ÷ original) × 100. The base is always the ORIGINAL amount.

80100oldnew

In the figure: The rise is 20 on a base of 80, so the increase is 20/80 × 100 = 25%.

Rule 1: Find the percentage change — Change ÷ old × 100.

  • Easy: 80 → 100: 20/80 × 100 = 25% increase.
  • Medium: 200 → 150: 50/200 × 100 = 25% decrease.
  • Exam-type: 800 → 500: 300/800 × 100 = 37.5% decrease.

Rule 2: Apply a percentage change — New = old × (100 + p) ÷ 100 for an increase; old × (100 − p) ÷ 100 for a decrease.

  • Easy: 150 up by 20%: 150 × 1.2 = 180.
  • Medium: 400 down by 15%: 400 × 0.85 = 340.
  • Exam-type: ₹60000 bike loses 10%: 60000 × 0.9 = 54000.

Rule 3: Up then down is not back to start — The second change is on a different base.

  • Easy: 100 up 50% is 150; 150 down 50% is 75.
  • Medium: 1000 down 10% is 900; down 10% again is 810.
  • Exam-type: Two cuts of 10% make 19% in all, not 20%.
Remember

A 100% increase doubles the amount; a 100% decrease makes it zero.

Common mistake

Using the new amount as the base. 80 → 100 is 25% up (20/80), but 100 → 80 is 20% down (20/100). The two answers differ.

Shortcut

Divide by where you started. Memory clue: "Change over old".

Spot the difference

Point80 → 100100 → 80
Change2020
Base (old)80100
Result25% increase20% decrease

Practice questions

Easy
  1. 1A price rises from ₹80 to ₹100. Find the percentage increase.
  2. 2150 is increased by 20%. Find the new value.
  3. 3A price falls from ₹200 to ₹150. Find the percentage decrease.
Medium
  1. 4A population grows from 5000 to 5600. Find the percentage increase.
  2. 5400 is decreased by 15%. Find the new value.
  3. 6Rent rises from ₹8000 to ₹9200. Find the percentage increase.
Challenging
  1. 7100 is increased by 50% and then decreased by 50% of the new value. Find the final value.
  2. 8Footfall falls from 800 to 500. Find the percentage decrease.
Show answers (try them yourself first)
  1. 125%
  2. 2180
  3. 325%
  4. 412%
  5. 5340
  6. 615%
  7. 775
  8. 837.5%

Profit, loss and discount

From daily life

A shopkeeper buys a sweater for ₹300 (cost price), writes ₹480 on the tag (marked price) and after bargaining sells it for ₹430 (selling price). He earns ₹130 profit.

Simple definition

Profit = SP − CP and Loss = CP − SP. Profit % and Loss % are taken on the cost price. Discount = MP − SP and Discount % is taken on the marked price.

Live figure

Profit, loss and discount

In the app you can move this figure with your finger.

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In the figure: Change the marked price and the discount to see how the selling price changes.

Rule 1: Profit and loss percentage — Profit or loss ÷ cost price × 100.

  • Easy: CP ₹400, SP ₹500: profit 100, 100/400 × 100 = 25%.
  • Medium: CP ₹500, SP ₹450: loss 50, 50/500 × 100 = 10%.
  • Exam-type: Buy 20 pens at ₹8, sell at ₹10: profit 2/8 × 100 = 25%.

Rule 2: Find SP or CP — SP = CP × (100 ± p) ÷ 100. To find CP from SP, divide by (100 ± p) and multiply by 100.

  • Easy: CP ₹250, profit 20%: SP = 250 × 1.2 = ₹300.
  • Medium: SP ₹1150, profit 15%: CP = 1150 ÷ 115 × 100 = ₹1000.
  • Exam-type: SP ₹720, loss 10%: CP = 720 ÷ 90 × 100 = ₹800.

Rule 3: Discount and tax — Take the discount off the marked price; add tax on the bill.

  • Easy: MP ₹800, discount 15%: SP = 800 − 120 = ₹680.
  • Medium: MP ₹600, SP ₹510: discount 90/600 × 100 = 15%.
  • Exam-type: Bill ₹500 with 18% tax: 500 + 90 = ₹590.
Remember

Two successive discounts of 10% are not 20%: 1000 → 900 → 810, a total discount of 19%.

Common mistake

Taking profit % on the selling price. It is always on the cost price. Also taking discount % on the cost price instead of the marked price.

Shortcut

Profit on Cost, Discount on Mark. Memory clue: "PC and DM: Profit-Cost, Discount-Marked".

Spot the difference

PointProfit / loss %Discount %
Taken oncost pricemarked price
Example100/400 × 100 = 25%90/600 × 100 = 15%
Amount isSP − CPMP − SP

Practice questions

Easy
  1. 1CP ₹400 and SP ₹500. Find the profit percentage.
  2. 2MP ₹800 with a 15% discount. Find the selling price.
  3. 3CP ₹250 and profit 20%. Find the selling price.
Medium
  1. 4CP ₹500 and SP ₹450. Find the loss percentage.
  2. 5MP ₹600 and SP ₹510. Find the discount percentage.
  3. 6A bill of ₹500 has 18% tax. Find the total.
Challenging
  1. 7SP is ₹1150 with a profit of 15%. Find the CP.
  2. 8Two discounts of 10% each are given on ₹1000. Find the final price.
Show answers (try them yourself first)
  1. 125%
  2. 2₹680
  3. 3₹300
  4. 410%
  5. 515%
  6. 6₹590
  7. 7₹1000
  8. 8₹810