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ICSE • कक्षा 8 • Mathematics • अध्याय 8

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प्रगति: अध्ययनरत

प्रतिशतता (Percentage)

In ICSE Class 8 Mathematics, "Percentage" provides an authoritative, mathematically rigorous master study guide investigating the fractional and decimal arithmetic of per-hundred calculations, percentage increase and decrease, and multi-step real-world commercial applications. This comprehensive chapter explores Meaning of Percentage (Per hundred / Parts per hundred, symbol $\%$; Fraction with denominator 100: $x\% = \frac{x}{100}$), Inter-Conversions (Percentage to fraction/decimal, fraction/decimal to percentage), Finding a Percentage of a Given Quantity ($x\% \text{ of } Q = \frac{x}{100} \times Q$), Expressing One Quantity as a Percentage of Another ($\frac{A}{B} \times 100\%$ with identical units of measurement), Percentage Change: Percentage Increase and Percentage Decrease ($\text{Percentage Increase} = \frac{\text{Increase}}{\text{Original Value}} \times 100\%$; $\text{Percentage Decrease} = \frac{\text{Decrease}}{\text{Original Value}} \times 100\%$; Finding original quantity from increased/decreased value), Successive Percentage Changes (Two successive changes of $x\%$ and $y\%$: $\text{Net Change} = \left( x + y + \frac{xy}{100} \right)\%$), Commodity Price-Consumption Elasticity (If price of a commodity increases by $r\%$, reduction in consumption to keep expenditure constant is $\frac{r}{100 + r} \times 100\%$; If price decreases, increase in consumption is $\frac{r}{100 - r} \times 100\%$), Population Growth and Machine Depreciation ($A = P\left(1 \pm \frac{R}{100}\right)^n$), and Examination Marks/Salary Word Problems aligned with the 2026–27 CISCE ICSE curriculum.

अध्याय CBT ऑनलाइन टेस्ट दें सभी अध्याय देखें

प्रस्तावना अध्याय अवधारणाएँ सूत्र एवं सर्वसमिकाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

Why Does a Store Advertising "Prices Slashed by 20% and Then an Extra 20% Off" Cost You Significantly More Than a 40% Discount?

Walk through a shopping mall during the festival sales season. A neon banner in a clothing store window shouts: "MEGA DOUBLE SAVINGS! Take 20% off all clothes, PLUS an EXTRA 20% off at the checkout register!" Shoppers celebrate, thinking they are getting a $40\%$ discount ($20 + 20 = 40$). But at the billing counter, a shirt priced at $\text{Rs. } 1,000$ does NOT cost $\text{Rs. } 600$! The first $20\%$ cut drops the price to $\text{Rs. } 800$. The second $20\%$ cut is calculated NOT on the original Rs. 1,000, but on the reduced Rs. 800, saving only $\text{Rs. } 160$ and making the final bill Rs. 640! The real net discount was only 36%, not 40%! The store legally kept an extra $4\%$ of your money! Why does a $10\%$ pay raise followed by a $10\%$ pay cut leave you with LESS money than you started with? What is the secret formula to calculate consumption cuts when fuel prices soar? Let's master percentage.

यह अध्याय क्यों महत्वपूर्ण है

Percentage is the universal language of business, banking interest rates, government taxation (GST, income tax), financial inflation metrics, laboratory chemical purity analysis, corporate profit margins, and academic performance grading. Mastering percentage formulas and successive change arithmetic is essential for ICSE Class 8 mathematics.

अध्ययन से पूर्व (आवश्यक ज्ञान)

Operations on fractions and decimals from Class 7.
Solving simple linear equations in one variable.
Units conversion (rupees/paise, meters/centimeters, kilograms/grams).

इस अध्याय के लक्ष्य

Convert seamlessly between percentages, fractions, and decimals.
Calculate percentage increase and decrease on physical and monetary quantities.
Determine the original quantity when given its increased or decreased percentage value.
Apply the successive percentage change formula: $\text{Net} = \left( x + y + \frac{xy}{100} \right)\%$.
Calculate required reductions in household consumption when commodity prices rise to keep expenditure constant.
Solve population growth and machine depreciation problems over multiple years.

अध्याय रूपरेखा एवं प्रगति

1 1. Fundamentals & Inter-Conversions

2 2. Percentage Increase & Decrease

3 3. Successive Percentage Changes

4 4. Price-Consumption Elasticity & P...

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. Fundamentals & Inter-Conversions

2. Percentage Increase & Decrease

3. Successive Percentage Changes

4. Price-Consumption Elasticity & Population Growth

महत्वपूर्ण सूत्र, सर्वसमिकाएँ एवं प्रमेय

Net Successive Percentage Change

$$\text{Net Change (\%)} = x + y + \frac{xy}{100}$$

Use positive for increase and negative for decrease.

Price-Expenditure Consumption Rule

$$\text{Consumption Cut (\%)} = \frac{r}{100 + r} \times 100$$

Maintains constant expenditure when price rises by r%.

Percentage: Successive Changes & Price-Consumption Dynamics

ⁿ + for annual population growth - for machinery depreciation • Double Discounts: 20% + 20% ≠ 40%! Actual discount = 20 + 20 - 4 = 36% PERCENT = PER HUNDRED • NET CHANGE = x + y + xy/100 • EXPENDITURE CUT = r/(100+r)

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1

Percent means parts per hundred; x% = x / 100.

मुख्य बिंदु 2

To convert fractions/decimals to percentages, multiply by 100%; to remove %, divide by 100.

मुख्य बिंदु 3

Percentage increase or decrease is always calculated over the Original Base value.

मुख्य बिंदु 4

Successive percentage changes x% and y% yield Net Change = (x + y + xy/100)%.

मुख्य बिंदु 5

An increase of 10% followed by a decrease of 10% results in an overall 1% net reduction.

मुख्य बिंदु 6

If commodity price rises by r%, consumption must drop by [r / (100 + r)] * 100% to keep spending constant.

मुख्य बिंदु 7

If commodity price drops by r%, consumption can increase by [r / (100 - r)] * 100%.

मुख्य बिंदु 8

Population growth after n years: A = P(1 + R/100)^n.

मुख्य बिंदु 9

Machine depreciation value after n years: A = P(1 - R/100)^n.

मुख्य बिंदु 10

Two successive discounts of 20% and 20% equal a net single discount of 36% (not 40%).

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

If the price of petrol increases by $25\%$, by what percentage must a motorist reduce his consumption so that his total expenditure on petrol remains unchanged?

उत्तर एवं व्याख्या देखें

उत्तर: Let the original price of petrol be $\text{Rs. } 100$ per liter and original consumption be $100\text{ liters}$.
Original Expenditure $= 100 \times 100 = \text{Rs. } 10,000$.
New price of petrol $= 100 + 25 = \text{Rs. } 125$ per liter.
To keep expenditure unchanged at $\text{Rs. } 10,000$, new consumption is:
$$\text{New Consumption} = \frac{10,000}{125} = 80\text{ liters}$$
Reduction in consumption $= 100 - 80 = 20\text{ liters}$.
$$\text{Percentage Reduction} = \frac{20}{100} \times 100\% = \mathbf{20\%}$$

• *Direct Formula Verification:*
$$\text{Reduction} = \left( \frac{r}{100 + r} \times 100 \right)\% = \left( \frac{25}{100 + 25} \times 100 \right)\% = \frac{25}{125} \times 100\% = \mathbf{20\%}$$.
Apply $[r / (100 + r)] \times 100\% = (25 / 125) \times 100\% = 20\%$.

An employee's salary was increased by $20\%$, but due to poor company performance, the new salary was subsequently reduced by $20\%$. What is the net percentage change in his salary?

उत्तर एवं व्याख्या देखें

उत्तर:

Apply the Successive Percentage Change formula:

$$\text{Net Change} = \left( x + y + \frac{xy}{100} \right)\%$$

Here $x = +20$ (increase) and $y = -20$ (decrease):

$$\text{Net Change} = 20 - 20 + \frac{(20)(-20)}{100}$$

$$= 0 - \frac{400}{100} = \mathbf{-4\%}$$

• The negative sign indicates a reduction.
• The employee suffers a net $4\%$ decrease in his original salary!

Net change $= 20 - 20 + (20 \times -20)/100 = 0 - 4 = -4\%$. (A net 4% decrease).

In an examination, $35\%$ marks are required to pass. A student gets $145$ marks and fails by $30$ marks. Find the maximum total marks of the examination.

उत्तर एवं व्याख्या देखें

उत्तर:

Step 1: Calculate passing marks required:

$$\text{Passing Marks} = 145 + 30 = 175\text{ marks}$$

Step 2: Let the maximum total marks be $M$. According to the question:

$$35\% \text{ of } M = 175$$

$$\frac{35}{100} \times M = 175$$

$$M = \frac{175 \times 100}{35}$$

Divide $175$ by $35$ ($175 \div 35 = 5$):

$$M = 5 \times 100 = \mathbf{500\text{ marks}}$$

.
The maximum total marks of the examination is $500$.

Pass marks $= 145 + 30 = 175$. Since $35\%$ of Total $= 175$, Total $= (175 \times 100) / 35 = 500$.

The population of a town increases at the rate of $5\%$ per annum. If its current population is $185,220$, find its population 3 years ago.

उत्तर एवं व्याख्या देखें

उत्तर:

Let the population 3 years ago be $P$.
Annual growth rate $R = 5\%$, Time $n = 3\text{ years}$, Current Population $A = 185,220$.
Apply the Population Growth formula:

$$A = P \left( 1 + \frac{R}{100} \right)^n$$

$$185,220 = P \left( 1 + \frac{5}{100} \right)^3 = P \left( 1 + \frac{1}{20} \right)^3 = P \left( \frac{21}{20} \right)^3$$

$$185,220 = P \times \frac{9261}{8000}$$

Solve for $P$:

$$P = \frac{185,220 \times 8000}{9261}$$

Notice that $185,220 \div 9261 = 20$:

$$P = 20 \times 8000 = \mathbf{160,000}$$

.
The population 3 years ago was $160,000$.

$185,220 = P \times (21/20)^3 = P \times (9261/8000) \implies P = 20 \times 8000 = 160,000$.

A scooter was bought for $\text{Rs. } 42,000$. Its value depreciates at the rate of $8\%$ per annum. Find its value after 1 year.

उत्तर एवं व्याख्या देखें

उत्तर: Given: Initial cost $P = \text{Rs. } 42,000$, Depreciation rate $R = 8\%$, Time $n = 1\text{ year}$.
Apply Depreciation formula:
$$\text{Value after 1 year} = P \left( 1 - \frac{R}{100} \right)^1$$
$$= 42,000 \times \left( 1 - \frac{8}{100} \right) = 42,000 \times \left( \frac{92}{100} \right)$$
$$= 420 \times 92 = \mathbf{\text{Rs. } 38,640}$$.
Value $= 42,000 \times (1 - 0.08) = 42,000 \times 0.92 = \text{Rs. } 38,640$.

What percentage of $2\text{ kilograms}$ is $65\text{ grams}$?

उत्तर एवं व्याख्या देखें

उत्तर:

Step 1: Convert both quantities to the same unit ($1\text{ kg} = 1000\text{ g}$):

$$2\text{ kg} = 2 \times 1000\text{ g} = 2000\text{ grams}$$

Step 2: Calculate percentage:

$$\text{Percentage} = \frac{65}{2000} \times 100\% = \frac{65}{20}\% = \frac{13}{4}\% = \mathbf{3.25\%}$$

Convert 2 kg to 2000 g: $(65 / 2000) \times 100\% = 3.25\%$.

The price of a television set inclusive of $18\%$ GST is $\text{Rs. } 33,040$. Find the original price of the television before GST was added.

उत्तर एवं व्याख्या देखें

उत्तर:

Let the original price of the TV be $\text{Rs. } x$.
GST rate $= 18\%$.
Price including GST:

$$x + 18\% \text{ of } x = 33,040$$

$$x \left( 1 + \frac{18}{100} \right) = 33,040$$

$$x \left( \frac{118}{100} \right) = 33,040$$

$$x = \frac{33,040 \times 100}{118}$$

Divide $33,040$ by $118$ ($33,040 \div 118 = 280$):

$$x = 280 \times 100 = \mathbf{\text{Rs. } 28,000}$$

.
The original price of the TV was $\text{Rs. } 28,000$.

Set $1.18x = 33,040 \implies x = 33,040 / 1.18 = \text{Rs. } 28,000$.

A number is first increased by $10\%$ and then decreased by $10\%$. Another number is decreased by $20\%$. Compare the results.

उत्तर एवं व्याख्या देखें

उत्तर:

• For the first number (let $N = 100$):
Increased by $10\% \to 110$. Then decreased by $10\% \to 110 - 11 = 99$.
Net change is a $1\%$ decrease.
• For the second number: decreased directly by $20\%$ (from 100 to 80).
• The second number suffers a much larger reduction ($20\%$) than the first number ($1\%$).

First number suffers a net 1% decrease ($100 \to 110 \to 99$); second number drops by 20% ($100 \to 80$).

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प्रतिशतता (Percentage)

Why Does a Store Advertising "Prices Slashed by 20% and Then an Extra 20% Off" Cost You Significantly More Than a 40% Discount?

यह अध्याय क्यों महत्वपूर्ण है

अध्ययन से पूर्व (आवश्यक ज्ञान)

इस अध्याय के लक्ष्य

अध्याय रूपरेखा एवं प्रगति

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. Fundamentals & Inter-Conversions

A. Meaning of Percentage:

B. Expressing One Quantity as a Percentage of Another:

2. Percentage Increase & Decrease

A. The Master Change Formulas:

B. Direct Calculation of New Values:

3. Successive Percentage Changes

4. Price-Consumption Elasticity & Population Growth

A. Expenditure Invariance Formula:

B. Population Growth & Machine Depreciation:

महत्वपूर्ण सूत्र, सर्वसमिकाएँ एवं प्रमेय

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प्रतिशतता (Percentage)

Why Does a Store Advertising "Prices Slashed by 20% and Then an Extra 20% Off" Cost You Significantly More Than a 40% Discount?

यह अध्याय क्यों महत्वपूर्ण है

अध्ययन से पूर्व (आवश्यक ज्ञान)

इस अध्याय के लक्ष्य

अध्याय रूपरेखा एवं प्रगति

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. Fundamentals & Inter-Conversions

A. Meaning of Percentage:

B. Expressing One Quantity as a Percentage of Another:

2. Percentage Increase & Decrease

A. The Master Change Formulas:

B. Direct Calculation of New Values:

3. Successive Percentage Changes

4. Price-Consumption Elasticity & Population Growth

A. Expenditure Invariance Formula:

B. Population Growth & Machine Depreciation:

महत्वपूर्ण सूत्र, सर्वसमिकाएँ एवं प्रमेय

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