CLASS 8 MATHS · PART 2 · CHAPTER 1 · STUDY NOTES

Fractions in Disguise

Understand fractions in disguise through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.

UNDERSTAND · APPLY · CHECK

Percent means out of one hundred

A percentage is a fraction with denominator 100.

Three quarters = 75%

Orange length: The selected amount.
Full length: The reference whole.

The orange part takes three of four equal quarter-lengths, which is also 75 of 100 equal hundredth-lengths.

Percentage: A way to express a fraction per hundred: p% = p/100.

3/4 = 75/100 = 75%. The whole is the same; only the way we describe its share changes.

Multiply a fraction by 100 to write its percentage. For example 2/5 × 100 = 40, so 2/5 = 40%.

WORKED IDEA

Convert back

32% = 32/100 = 8/25

Write the percentage over 100 and simplify.

Take a moment. What do you think?

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Question

Write 7/20 as a percentage.

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NCERT reference: chapter PDF pages 1, 2, 3, 4, 5.

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Always identify the whole first

A larger percentage does not always mean a larger amount when the wholes differ.

Compare amounts using one common scale

Bar lengths share one amount scale. The second percentage is larger, but its whole is smaller and its selected amount is also smaller.

Percentage amount: p% of W is (p/100) × W.

25% of 120 g is 30 g. But 40% of 60 g is only 24 g. Both the percentage and the reference whole matter.

For mental calculations, 10% is one tenth and 5% is half of that. 15% can be found as 10% + 5% of the same whole.

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Question

Find 15% of 240.

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NCERT reference: chapter PDF pages 6, 7, 13, 14.

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Fraction, decimal and percent name the same value

Move between the three forms without changing the amount.

Three names, one value

Each arrow changes notation, not quantity. The denominator 100 connects decimals and percentages.

Decimal equivalent: A base-ten representation of the same number as a fraction.

Fraction	Decimal	Percent
1/2	0.5	50%
1/4	0.25	25%
1/5	0.2	20%
1/20	0.05	5%

0.35 = 35/100 = 35%. To turn a decimal into a percentage, multiply its value by 100 and add the percent sign.

For a ratio 2:3, the first ingredient is 2 of 5 total parts. Its share is 2/5 = 0.4 = 40%, not 2/3.

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Question

A student earns 27 of 36 marks. Give the fraction, decimal and percentage.

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NCERT reference: chapter PDF pages 8, 9.

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If you know a part, reverse the multiplication

Write an equation for the given percentage, then solve for the whole.

45 km is 30% of 150 km

Orange length: The selected amount.
Full length: The reference whole.

The full bar is the entire 150 km journey. The selected part is 45 km; the rest is 105 km.

Reference whole: The amount treated as 100% in a percentage comparison.

If 30% of a journey is 45 km, then 0.30W = 45. Dividing gives W = 150 km.

The remaining 70% is 150 − 45 = 105 km. It is 70% of the whole, not 70% of the 45 km part.

1
Name the whole W.
2
Write (percentage/100) × W = known part.
3
Divide by the percentage expressed as a decimal.
4
Multiply back to check.

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Question

40% of a number is 28. Find the number.

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NCERT reference: chapter PDF pages 10, 12, 13.

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More than 100% means more than the reference whole

100% is one copy of the reference amount; 150% is one and a half copies.

Copies of the reference amount

Value (number)

Number-line marks · Only horizontal position represents the number.

10 · (0, 0)
21 · (1, 0)
31.5 · (1.5, 0)
42 · (2, 0)

0, 1, 1.5 and 2 copies correspond to 0%, 100%, 150% and 200%.

Read the plotted values

Number-line marks

Value	Height
0	0
1	0
1.5	0
2	0

Percentage of a target: Actual amount ÷ target × 100%.

A target of 200 items and an output of 300 gives 300/200 × 100 = 150% of the target.

Being 150% of the original is the same as increasing by 50%. These two percentage statements have different wording but the same new amount.

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Question

A quantity doubles. What percent of its old value is it, and what is the percentage increase?

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NCERT reference: chapter PDF pages 10, 11, 12.

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Measure the change against the original amount

Percentage change uses the starting amount as its base.

Same change, different bases

The first change is +25% of 80. The second change is −20% of 100. Equal rupee changes need not have equal percentages.

Percentage increase: (New − original)/original × 100%, for a positive original value.
Percentage decrease: (Original − new)/original × 100%, for a positive original value.

A price rising from ₹80 to ₹100 increases by ₹20, which is 20/80 × 100 = 25%.

Returning from ₹100 to ₹80 is a decrease of 20/100 × 100 = 20%. The change is ₹20 both times, but the bases differ.

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Question

A count falls from 160 to 120. Find the percentage decrease.

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NCERT reference: chapter PDF pages 14, 15, 16.

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Cost, marked price and selling price have different roles

Choose the correct base before calculating any percentage.

Track the transaction

Icons show the roles in one seller’s transaction. The same amount can be another trader’s selling price and this trader’s cost price.

Cost price (CP): The amount paid to obtain the item in the stated transaction.
Selling price (SP): The amount received when it is sold.
Marked price (MP): The displayed price before a stated discount.

For CP ₹400 and SP ₹460, profit is ₹60. Profit as a percentage of cost is 60/400 × 100 = 15%.

A 20% discount on MP ₹600 gives SP = 0.8 × 600 = ₹480. Discount uses marked price; this chapter’s profit-on-cost calculation uses cost price.

Let’s look a little closer.

Go deeper: Profit and margin need a named base

Profit divided by selling revenue is a different percentage from profit divided by cost. Gross profit deducts the cost of goods; net profit also deducts the relevant other expenses. Always state which measure and denominator you mean.

A classroom tax example may add a stated hypothetical rate to a tax-exclusive price. It is not a statement of current GST law or rates.

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Question

An item costs ₹250 and sells for ₹225. Find the loss percentage on cost.

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NCERT reference: chapter PDF pages 16, 17, 18, 19, 20.

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Compounding changes the base after each period

When a gain is added back, the next percentage is calculated on a larger amount.

Fixed-base and changing-base growth

Principal plus gains (rupees)

Time (years)

Highlight a line or construction

No compounding · Gain ₹100 each year.
Annual compounding · Multiply the previous amount by 1.1.

Hypothetical mathematics model: constant rate, no fees or tax. Points show year ends; connecting lines guide comparison.

Read the plotted values

No compounding

Time	Principal plus gains
0	1000
1	1100
2	1200
3	1300

Annual compounding

Time	Principal plus gains
0	1000
1	1100
2	1210
3	1331

Principal: The starting amount in the model.
Compounding: Adding each period’s gain to the amount used for the next period.

Start with ₹1000 at a hypothetical 10% each year. With annual compounding: ₹1100 after year 1, ₹1210 after year 2 and ₹1331 after year 3.

Without compounding, interest is ₹100 each year on the original ₹1000, giving total principal plus interest ₹1300 after three years.

WORKED IDEA

Use a decimal rate

A = P(1 + r)^t

For 10%, use r = 0.10, not 10. Here t counts equal compounding periods.

WORKED IDEA

No compounding

A = P(1 + rt)

Interest is Prt; the total also includes P. Rates and time units must match.

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Question

Find the amount from ₹2000 after two years at a hypothetical 5%, compounded annually.

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NCERT reference: chapter PDF pages 20, 21, 22, 23, 24.

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Successive percentages multiply

A decrease leaves a multiplier below 1. Apply each multiplier to the new amount.

Apply one multiplier at a time

Both decreases are 10%, but the second removes ₹450 rather than ₹500. This assumes the rate applies to the current value each period.

Depreciation: A reduction in value in the stated model; actual resale values need not follow a fixed percentage.

A ₹5000 model value falling 10% twice becomes 5000 × 0.9 × 0.9 = ₹4050. The total fall is 19%, not 20%.

A 20% increase followed by a 20% decrease gives 1.2 × 0.8 = 0.96 of the start: a 4% fall. The second percentage uses a different base.

Go deeper: A percentage of a subgroup

If 40% of a group are in Grade 8 and 50% of those Grade 8 students choose art, the art subgroup is 0.4 × 0.5 = 0.2, or 20% of the whole group. Do not add the two percentages.

Keep full precision while calculating; round a final estimate when appropriate. People and discrete objects require a sensible whole-number interpretation.

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Question

Two successive discounts are 20% and 10%. What is their combined discount?

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NCERT reference: chapter PDF pages 24, 25, 26, 27, 28.

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Percentages describe shares, not group sizes

A graph of percentages does not tell you how many people are in each group unless its counts are also known.

Percentage and count answer different questions

Both groups are fictional. The icons identify groups rather than representing one person per drawn figure. Use the stated totals for calculation.

Comparable base: A clearly identified reference quantity used to interpret a share.

If 60% of 20 students and 40% of 50 students use the library, the counts are 12 and 20. The larger percentage has the smaller count.

For a fictional buy-two-get-one-free offer on identical items, you pay for 2 of 3: an effective discount of 1/3, or 33⅓%, if you receive and use all three.

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Question

Why can’t we compare the number of users from percentages alone?

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NCERT reference: chapter PDF pages 29, 30, 31.

Fractions in Disguise Class 8 Notes

Chapter notes

Fractions in Disguise

Percent means out of one hundred

Three quarters = 75%

Convert back

Always identify the whole first

Compare amounts using one common scale

Fraction, decimal and percent name the same value

Three names, one value

If you know a part, reverse the multiplication

45 km is 30% of 150 km

More than 100% means more than the reference whole

Copies of the reference amount

Measure the change against the original amount

Same change, different bases

Cost, marked price and selling price have different roles

Track the transaction

Compounding changes the base after each period

Fixed-base and changing-base growth

Use a decimal rate

No compounding

Successive percentages multiply

Apply one multiplier at a time

Percentages describe shares, not group sizes

Percentage and count answer different questions

Now practise Fractions in Disguise, free