Fractions in Disguise Class 8 Notes

Fractions in Disguise Class 8 Notes

Fractions in Disguise Class 8 Notes

Fractions as Percentages

A percentage is simply a'fraction with a denominator of 100. To convert a fraction to a percentage, you multiply it by 100.

Example:

To convert the fraction [latex]\frac{3}{4}[/latex] to a percentage :

Percentage = [latex]\frac{3}{4}[/latex] × 100 = 75%

Why Use Percentages?

Percentages provide a clear understanding of proportions relative to 100, making comparisons easier.

1. Clarity : Percentages simplify comparisons between different quantities.
2. Standardisation : They provide a common basis for understanding data.

Percentage of Some Quantity

To find a percentage of a quantity, you multiply the quantity by the percentage expressed as a fraction.

Example : To find 20% of 50 :

20% of 50 = [latex]\frac{20}{100}[/latex] × 50= 10

The FDPTrio — Fractions, Decimals, and Percentages

1. Interconnectedness : They are different ways to represent the same value.
2. Conversions : Easy to convert between them (e.g., 0.5 = 50 % = [latex]\frac{1}{2}[/latex].
3. Applications : Used in various fields like finance, statistics, and everyday calculations.
4. Understanding : Helps in grasping concepts of ratios and proportions.
5. Mental Math : Knowing their relationships aids in quick calculations.

Converting Fractions and Decimals to Percentages :

To convert a fraction to a percentage, you multiply it by 100.

Example : Convert [latex]\frac{3}{4}[/latex] to a percentage.

[latex]\frac{3}{4}[/latex] × 100 = 75%

To convert a decimal to a percentage, you also multiply by 100.

Example : Convert 0.85 to a percentage.

0.85 × 100 = 85%

Percentage Increase or Decrease

Formula for Percentage Increase : To find the percentage increase from an original value to a new value, you can use the formula :

Percentage Increase

[latex]\frac{\text { New Value }- \text { Original Value }}{\text { Original Value }}[/latex] × 100

Formula for Percentage Decrease : To find the percentage decrease, the formula is :

Percentage Decrease

= [latex]\frac{\text { Original Value - New Value }}{\text { Original Value }}[/latex] × 100

Definitions:

1. Marked Price : The original price set by the seller.
2. Selling Price : The price at which the item is sold.
3. Cost Price : The price at which the item was purchased.

Gross Profit and Net Profit

1. Gross Profit: Revenue from sales minus the cost of goods sold.
2. Net Profit: Gross profit minus all other expenses (like taxes, operating costs).

Taxes

Taxes are mandatory contributions to state revenue, levied on income, property, sales, etc.

Growth and Compounding

1. Interest : The cost of borrowing money or the return on investment.
2. Rate of Interest : The percentage charged or earned on the principal.
3. Principal: The initial amount of money invested or borrowed.

No Compounding vs. Compounding .

No Compounding : Interest is calculated only on the principal.

With Compounding : Interest is calculated on the principal plus any previously earned interest.

Decline and Depreciation

Depreciation : The reduction in the value of an asset over time, often due to wear and tear.

Fractions as Percentages

A percentage is a way of expressing a number as a fraction of 100. The symbol for percentage is

which means ‘per hundred’. For example, when we say 25%, it means 25 out of every 100.

How to Convert Fractions to Percentages:

1. Identify the Fraction : Let’s take the fraction [latex]\frac{3}{4}[/latex].
2. Convert to a Fraction with a Denomi¬nator of 100 : We can find an equivalent fraction that has 100 as the denominator. To do this, we can multiply both the numerator and the denominator by the same number.
3. For [latex]\frac{3}{4}[/latex], [latex]\frac{3}{4}[/latex] = [latex]\frac{3 \times 25}{4 \times 25}[/latex] = [latex]\frac{75}{100}[/latex]
4. This means [latex]\frac{3}{4}[/latex] = 75%.

Examples:

Convert [latex]\frac{1}{2}[/latex] to a percentage :

[latex]\frac{1}{2}[/latex] = [latex]\frac{1 \times 50}{2 \times 50}[/latex] = [latex]\frac{50}{100}[/latex] = 50%

Convert [latex]\frac{2}{5}[/latex] to a percentage :

[latex]\frac{2}{5}[/latex] = [latex]\frac{2 \times 50}{5 \times 50}[/latex] = [latex]\frac{40}{100}[/latex] = 40%

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value or proportion.

Method 1 : Finding Equivalent Fractions by Multiplication

To find equivalent fractions, we can multiply both the numerator and the denominator by the same number.

For example, to find an equivalent fraction for [latex]\frac{2}{3}[/latex]

Multiply both the numerator and the denominator by 2 : [latex]\frac{2 \times 2}{3 \times 2}[/latex] = [latex]\frac{4}{6}[/latex].

Method 2 : Finding Equivalent Fractions by Division

We can also find equivalent fractions by dividing both the numerator and the denominator by the same number.

For example, for [latex]\frac{6}{9}[/latex],

divide both by 3 : [latex]\frac{6 \div 3}{9 \div 3}[/latex] = [latex]\frac{2}{3}[/latex].

Why Do We Need Percentages ?

1. Simplicity : Percentages simplify comparisons between different quantities, making it easier to understand proportions.
2. Standardisation : They provide a common basis (out of 100) for expressing ratios, which is useful in various fields like finance and statistics.

Kautilya’s Arthashastra

1. The Arthashastra is an ancient Indian treatise on statecraft, economic policy, and military strategy, written by Kautilya (Chanakya).
2. It emphasises the importance of economic prosperity, and the use of intelligence and diplomacy in governance.

Percentage of Some Quantity

Understanding Percentages:

1. A percentage is a way to express a number as a fraction of 100. For example, 25% means 25 out of every 100.
2. To find a percentage of a quantity, you can use the formula:

Percentage of a quantity = [latex]\frac{y}{100}[/latex] × Quantity,

where (y) is the percentage you want to find.

Example : To find 20% of 50 :

20% of 50 = [latex]\frac{20}{100}[/latex] × 50 = 10

The FDP Trio — Fractions, Decimals, and Percentages

1. Fractions represent parts of a whole (e.g., [latex]\frac{1}{2}[/latex]).
2. Decimals are another way to express fractions (e.g., 0.5 for [latex]\frac{1}{2}[/latex]).
3. Percentages express a fraction out of 100 (e.g., 50% for [latex]\frac{1}{2}[/latex]).
4. All three can represent the same value in different forms.
5. Converting between them helps in under-standing and solving problems easily.

Finding the Remaining Distance When 40% is 92 km

We know that 40% of the total distance is 92 km. To find the remaining 60%, we can use three different methods :

Method 1: Direct Calculation

1. If 40% is 92 km, then to find 100% (the total distance), we can set up the equation :

[latex]\frac{40}{100}[/latex] × d = 92

Solving for (d):

d = 92 × [latex]\frac{100}{40}[/latex] = 230 km

The remaining distance (60%) is :

230 km - 92 km = 138 km.

Method 2 : Using Percentages

1. If 40% is 92 km, then 20% is half of that:

20% = [latex]\frac{92}{2}[/latex] = 46 km.

Therefore, 60% is :

92 km + 46 km = 138 km.

Method 3 : Algebraic Approach

1. Let x be the total distance. Since 40% of x equals 92 km :

0.4x = 92

Solving for (x):

x = [latex]\frac{92}{0.4}[/latex] = 230 km

The remaining distance is :

230 km - 92 km = 138 km.

Can there be percentages greater than 100?

Yes, percentages can be greater than 100%.

What it means:

1. A percentage greater than 100% indicates that the quantity is larger than the whole.
2. For example, if a student scores 120% on a test, it means they earned 20% more than the maximum possible score, perhaps through extra credit.

Example : If a shop aims to sell 100 items in a month and sells 150, the percentage of items sold is :

Percentage = [latex]\frac{150}{100}[/latex] × 100% = 150%

This indicates that the shop exceeded its sales target by 50%.

Using Percentages

Converting Fractions and Decimals to Percentages:

To convert fraction to a percentage, you multiply it by 100.

Example : Convert [latex]\frac{3}{4}[/latex] to a percentage.

[latex]\frac{3}{4}[/latex] × 100 = 75%

To convert a decimal to a percentage, you also multiply by 100.

Example : Convert 0.85 to a percentage.

0.85 × 100 = 85%

Percentage Increase or Decrease

Formula for Percentage Increase : To find the percentage increase from an original value to a new value, you can use the formula :

Percentage Increase

= [latex]\frac{\text { New Value - Original Value }}{\text { Original Value }}[/latex] × 100

Example : If the price of a book increases from ₹ 200 to ₹ 250 :

Original Value = ₹ 200

New Value = ₹ 250 Percentage Increase :

[latex]\frac{250-200}{200}[/latex] × 100 = [latex]\frac{50}{200}[/latex] × 100 = 25%

Formula for Percentage Decrease : To find the percentage decrease, the formula is :

Percentage Decrease

= [latex]\frac{\text { Original Value - New Value }}{\text { Original Value }}[/latex] × 100

Example : If the price of a shirt decreases from ₹ 400 to ₹ 300 :

Original Value = ₹ 400

New Value = ₹ 300

Percentage Decrease :

[latex]\frac{400-300}{400}[/latex] × 100 = [latex]\frac{100}{400}[/latex] × 100 = 25%

Profit and Loss :

Cost Price (CP) : The price at which an item is purchased.

Selling Price (SP) : The price at which an item is sold.

Marked Price (MP) : The originial price set by the seller before any discounts.

Profit: Profit occurs when the Selling Price is greater than the Cost Price.

Profit = Selling Price - Cost Price

Example : If a book is bought for ₹ 150 and sold for ₹ 200 :

Profit = ₹ 200 - ₹ 150 = ₹ 50

Profit Percentage : To find the profit percentage :

Profit Percentage = [latex]\frac{\text { Profit }}{\text { Cost Price }}[/latex] × 100

Example : Profit Percentage = [latex]\frac{50}{150}[/latex] × 100 = 33.33%

Loss : Loss occurs when the Selling Price is less than the Cost Price,

Loss = Cost Price - Selling Price Example : If shirt is bought for ₹ 300 and sold for ₹ 250 :

Loss = ₹ 300 - ₹ 250 = ₹ 50

Loss Percentage : To find the loss percentage :

Loss Percentage = [latex]\frac{\text { Loss }}{\text { Cost Price }}[/latex] × 100

Taxes

Taxes are often expressed as percentages of the total amount. For example, if a product costs ₹ 100 and there is a 10% tax, the tax amount is :

Tax Amount = [latex]\frac{10}{100}[/latex] × 100 = ₹ 10.

Thus, the total amount paid would be

₹ 100 + ₹ 10 = ₹ 110.

Growth and Compounding

Principal (P) : The initial amount of money deposited or borrowed.

Rate of Interest (r) : The percentage at which interest is calculated.

Interest : The money paid for the use of borrowed money or earned on deposited money.

No Compounding : In simple interest, the interest is calculated only on the principal amount. Total Amount = P + (p × r × t).

Example : If ₹ 1000 is deposited at a 5% interest rate for 2 years. Total Amount = 1000 + (1000 × 0.05 × 2) = 1000 + 100 = ₹ 1100.

With Compounding : In compound interest, interest is calculated cn the principal and also on the accumulated interest from previous periods. Total Amount = P × (1 × r)^t.

Example : If ₹ 1000 is deposited at a 5% interest rate compounded annually for 2 years : Total Amount = 1000 × (1 + 0.05)² = 1000 + 1.1025 = ₹ 1102.50.

Decline

Definition : Decline refers to a decrease in value over time, often expressed as a percentage decrease.

Depreciation : Depreciation is the reduction in the value of an asset over time, often due to wear and tear. It can be calculated similarly to percentage decrease.

Example : If a car originally worth ₹ 500,000 depreciates to ₹ 400,000 :

Depreciation = ₹ 500,000 - ₹ 400,000

= ₹ 100,000

Depreciation Percentage :

Depreciation Percentage

= [latex]\frac{100,000}{500,000}[/latex] × 100 = 20%.