GANITA PRAKASH · GRADE 8 · PART II · CHAPTER 1

Chapter Summary

Think first. Use AI purposefully. Verify always.

Concept Map

Key Ideas

A percentage expresses a share relative to a whole of 100 equal parts. Always identify what counts as the whole.
Fractions, decimals and percentages are different representations of a number. A recurring decimal may need an approximation; a fraction can preserve the exact value.
A percentage is a multiplier applied to a specified whole. A larger percentage of a smaller whole need not produce a larger amount.
Known percentages can be combined when they refer to the same whole. Reverse percentage recovers the whole from a known part and its percentage.
100% is one whole. A percentage above 100 describes more than one copy of the reference whole; it is not automatically an error.
Normalising to a common whole of 100 compares proportions with different denominators. It does not make the actual totals equal.
Percentage change compares the difference with the original value. A reverse change has a different base.
CP is cost price, MP is marked price, and SP is selling price. Discount uses MP as base; trading profit or loss percentage uses CP. A stated tax rate applies to its specified taxable amount.
Simple interest adds interest calculated on the original principal. Compounding applies each new rate to the updated amount. Repeated decline also changes the base.

Rule Bank

75 % = \frac{75}{100} = \frac{3}{4}

24 % = \frac{24}{100} = \frac{6}{25} = 0.24

x % of Q = \frac{x}{100} \times Q

Whole = part \div \frac{percentage}{100}

150 % = 1.5; 250 % = 2.5

Percentage = \frac{part}{whole} \times 100

Percentage change = \frac{change}{original} \times 100

Profit = SP - CP; discount = MP - SP

p = original principal; t = whole years; r = annual rate as a decimal (10% → 0.10).

A = p (1 + rt)

Annual compounding:

A = p {(1 + r)}^{t}

For a decline at decimal rate d, replace the factor by (1 − d). These models assume a constant stated rate and no additional deposits or withdrawals.

Visual Recall

A hundred equal parts

Predict first. Change the controls, then compare your reasoning.

Shaded cells: 25

Explore all 15 models →

Important Connections

A percentage is a fraction and a decimal multiplier. The same multiplier works with lengths, masses and prices, but the whole and unit must remain clear.

Reverse percentage divides by the multiplier. Successive changes multiply their factors. Net-profit-on-revenue and profit-on-cost use different denominators.

1.05 \times 0.98 \times 0.97 = 0.99813

The mouse-population model is 0.187% below the initial population.

Common Mistakes

Equal parts matter: counting unequal pieces does not measure their share of the whole.
0.5 means 50%, not 0.5%. One third is exactly 100/3%, approximately 33.33%.
Do not compare percentage amounts without checking both wholes.
Reverse percentage is not the same as finding that percentage of the part. A zero percentage cannot determine the whole.
250% of an original amount means a 150% increase, not a 250% increase.
Percentage-only graphs cannot reveal absolute population counts. Different groups have different bases.
Using the new value as the denominator answers a different question. A rise of 20% followed by a fall of 20% does not cancel.
Discount is not automatically a loss. Gross profit excludes other expenses; net profit includes them. Always name the margin denominator.
Use r = 0.10 for 10%, not r = 10. Successive percentage changes multiply; they are not normally added.
Adding successive discounts ignores the changing base: 30% then 20% gives 44%, not 50%.
A 20% rise followed by a 20% fall gives 96% of the original, a 4% decrease.
CP, SP and MP describe different prices; identify each before calculating.
Repeated decline acts on the current amount, not always on the original.
A percentage bar gives a proportion, not the number of people in the group.

Mini Examples

UNMASK · Three quarters of a paint mixture is red.

Given

Three quarters of a paint mixture is red.

To Find

Percentage of the mixture that is red

Reasoning

Equivalent fractions describe the same share.

Solution

$\frac{3}{4}$ = $\frac{75}{100}$ .
75 parts per hundred means 75%.

Result

75% of the mixture is red.

CONVERT · A share of 24%.

Given

A share of 24%.

To Find

The equivalent simplified fraction and decimal

Reasoning

Percent means divided by 100.

Solution

24% = $\frac{24}{100}$ .
Divide numerator and denominator by 4: $\frac{6}{25}$ .
24 ÷ 100 = 0.24.

Result

24% = $\frac{6}{25}$ = 0.24.

FIND · 120 g biscuits contain 25% sugar; 95 g biscuits contain 35% sugar.

Given

120 g biscuits contain 25% sugar; 95 g biscuits contain 35% sugar.

To Find

Sugar mass in each packet

Reasoning

Percentage amount = percentage ÷ 100 × whole.

Solution

First packet: 0.25 × 120 = 30 g.
Second packet: 0.35 × 95 = 33.25 g.
33.25 − 30 = 3.25 g.

Result

The second packet contains 3.25 g more sugar.

ESTIMATE · 40% of a journey is 92 km.

Given

40% of a journey is 92 km.

To Find

Total journey length and remaining distance

Reasoning

40% corresponds to 92 km.

Solution

20% corresponds to 46 km.
100% corresponds to 5 × 46 = 230 km.
Remaining distance = 230 − 92 = 138 km.

Result

The whole journey is 230 km; 138 km remain.

BEYOND · A sales target is ₹5000; sales are ₹6000.

Given

A sales target is ₹5000; sales are ₹6000.

To Find

Sales as a percentage of the target

Reasoning

Target is the reference whole.

Solution

$\frac{6000}{5000}$ × 100 = 120%.
The excess is 20% of the target.

Result

Sales reached 120% of the target.

COMPARE · Test scores are 42/50 and 70/80.

Given

Test scores are $\frac{42}{50}$ and $\frac{70}{80}$ .

To Find

Which score is the higher percentage

Reasoning

Compare proportions on a common scale.

Solution

$\frac{42}{50}$ × 100 = 84%.
$\frac{70}{80}$ × 100 = 87.5%.

Result

70 out of 80 is the higher percentage.

CHANGE · Attendance falls from 160 to 100.

Given

Attendance falls from 160 to 100.

To Find

Percentage decrease in attendance

Reasoning

Decrease percentage = decrease ÷ original × 100.

Solution

Decrease = 160 − 100 = 60.
$\frac{60}{160}$ × 100 = 37.5%.

Result

Attendance decreased by 37.5%.

TRADE · CP ₹900, MP ₹1800, discount 35%.

Given

CP ₹900, MP ₹1800, discount 35%.

To Find

Selling price and profit percentage on cost

Reasoning

SP = MP × (1 − discount rate); profit percentage uses CP.

Solution

SP = 1800 × 0.65 = ₹1170.
Profit = 1170 − 900 = ₹270.
$\frac{270}{900}$ × 100 = 30%.

Result

SP is ₹1170; profit on cost is 30%.

GROW · Principal ₹6000, 10% per year, 3 years.

Given

Principal ₹6000, 10% per year, 3 years.

To Find

Simple and annually compounded final amounts

Reasoning

Use decimal rate r = 0.10; keep intermediate values unrounded.

Solution

Simple: 6000 × (1 + 0.10 × 3) = ₹7800.
Compound: 6000 × 1.1 × 1.1 × 1.1 = ₹7986.
Difference: 7986 − 7800 = ₹186.

Result

Compounding gives ₹186 more after 3 years.

5-Question Quick Check

Progress: $\frac{0}{5}$ checked correctly

Q1. Express 9/20 as a percentage. Enter the number before %. Your answer (decimal or fraction such as 200/3)

Attempts: 0

Q2. 40% of a journey is 92 km. Enter the full distance in km.Your answer (decimal or fraction such as 200/3)

Attempts: 0

Q3. Attendance falls from 160 to 100. Enter the percentage decrease.Your answer (decimal or fraction such as 200/3)

Attempts: 0

Q4. A 35% markup is followed by a 30% discount. What is the result on cost?5% profit5% loss5.5% loss

Attempts: 0

Q5. 30% then 20% discount: enter the overall discount percentage.Your answer (decimal or fraction such as 200/3)

Attempts: 0

Concept Map

Key Ideas

Rule Bank

Visual Recall

A hundred equal parts

Important Connections

Common Mistakes

Mini Examples

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5-Question Quick Check

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