Comparing Quantities Class 8 Notes given here has been carefully put together by experts to help students understand all the concepts given in chapter 8 clearly and at the same time allow them to practice sums effectively. The notes are further designed to help students complete timely revisions and score better marks in the exams.

## Introduction to Fraction and Ratios

### Fractions and Ratios

A **fraction** represents a** part of a whole** which consists of **numerators** and **denominators** and it is the division of two **same quantities**.

Eg: \(\frac{3}{5}\)

**Ratio** is the comparison of one value to the other or the **comparison of two different quantities.**

Eg:3:5

## Finding the Increase or Decrease in Percent

### Finding Increase or Decrease Percentage in Situations

Finding new number, when there is **increase** in percentage.

New number = original number + (increase in percentage Ã— number)

Ex : The Cost of a mobile phoneÂ Â is Rs 15,000. Find the new price if there is a increaseof 5%

New price = original price + 5% ofÂ original price

New price = 15,000+\(\frac{5}{100}\)
Ã—15,000

New price = 15,000+750 = 15,750

Here Rs 750 isÂ **increaseÂ **in the price.

The new number can be found out using,

New number = original number Ã— percentage increase

Ex : New price = 15,000Ã—105Ã·100=15,000Ã—1.05=15,750

Finding new number, when there is **decrease** in percentage.

New number = original number –Â (decrease in percentage Ã— number)

Also,Â New number = original number Ã— percentage decrease

Ex :Â The Cost of a mobile phoneÂ Â is Rs 15,000. Find the new price if there is a decrease of 5%

New price = 15,000Ã—95Ã·100=15,000Ã—0.95=14,250

## Finding Discounts

### Finding SP without Finding Discount Percentage

A **reductionÂ **(decrease)Â on the **marked price **is known as **discount.**

If the discount is given in numbers then it is calculated by

**Discount = Marked price – Sale price**

If the discount is given in percentage then it is calculated by

**Discount = Discount % of Marked price**

### Finding Discounts

If the **discount** is given in numbers.

Example : Marked price of a shirt is Rs 535. Its sellingÂ price is Rs 495. Find the discount.

Solution :Â Discount = Marked price – Sale price

Discount = Rs 535 – Rs 495 = Rs 40

If the **discount** is given in percentage.

Example : A toy priced Rs 500 is available at a discount of 5%. Find the discount.

Solution : Discount = Discount % of Marked price

Discount = 5%Â ofÂ 500=\(\frac{5}{100}\)Â Ã—Â 500

Discount = Rs 25

### Estimation of Amounts (In Percentages)

Estimating **amounts** when there is a **discount** or **hike** on the marked price.

**Example **: Anil bought aÂ pair of shoes priced Rs 650, at a discount of 10%. Find the billing amount.

**Solution** : Billing amount = Marked price – discount

Billing amount = RsÂ 650âˆ’\(\frac{10}{100}\)
Ã—650

Billing amount = RsÂ 650âˆ’RsÂ 65=RsÂ 585

**Example** : Shilpa bought a new mobileâ€‹â€‹â€‹â€‹â€‹â€‹ for Rs 15,000. She has to pay 2% asÂ delivery charges.

Find theÂ billing amount.

**Solution** : Billing amount = Marked price + Hike

Billing amount = RsÂ 15,000+\(\frac{2}{100}\)Â Ã—Â 15000

Billing amount = RsÂ 15,000+RsÂ 300=RsÂ 15,300

## Prices Related to Buying and Selling

### Prices / Charges Related to Buying and Selling

Profit= Selling price – Cost price

Profit%=\(\frac{Profit}{Cost\;price}\)Ã—100

Loss= Cost price – Selling price

Loss%=\(\frac{Loss}{Cost\;price}\)Ã—100

### Finding Prices / Charges Related to Buying and Selling

**Example** : A shopkeeper sold a T.V priced Rs 12,000 at Rs 13,500. Find his profit percentage.

**Profit **= Selling price – Cost price

Profit = RsÂ 13,500âˆ’RsÂ 12,000=RsÂ 1,500

**Profit ****%** =\(\frac{Profit}{Cost\;price}\)Ã—100

Profit % = \(\frac{1500}{12000}\)Ã—100=12.5%

**Example **:Â Amit sold hisÂ laptop, priced Rs 20,000 at Rs 18,000. Find his loss percentage.

**Loss** = Cost price – Selling price

Loss = RsÂ 20,000âˆ’RsÂ 18,000=RsÂ 2000

**Loss ****%** =\(\frac{Loss}{Cost\;price}\)Ã—100

Loss% = \(\frac{2000}{20,000}\)Ã—100=10%

## Sales Tax and Value Added Tax

### Sales Tax / VAT

**Sales tax** or **value added tax(VAT)** is the **tax** that should be **paid to the government** on sale of an item

and it is **added to the bill amount.**

Normally, VAT is included in the price of items like groceries.

### Finding Sales Tax / VAT

Sales tax or VAT =Â Tax % ofÂ Selling price

Billing Amount = Selling price + VAT

**Example **:Â Megha bought a wrist watch for Rs 1,200 and VAT is charged at 8%. Calculate the VAT and billing amount.

**Solution** : VAT =Â Tax % of selling price

VAT = 8%Â ofÂ 1,200=\(\frac{8}{100}\)Ã—1200=RsÂ 96

Billing amount = S.P + VAT = Rs 1,200 + Rs 96 = Rs 1296.

## Simple and Compound Interest

### SI

**Simple interest** is the **extra money**Â charged on a loan where the **principal amount will be fixed** for aÂ **particular time** period.

Interest is the extra money that a bank gives for saving or depositing money with them.

Similarly,Â when anybody borrow money, they pay interest.

Simple interest =\(\frac{P.T.R}{100}\), where

P is the principal amount

T is the number of years.

R is the interest rate

### Calculating CI

**Compound interest** is the interest, calculated on the **principal** and the **interest for the previous period.**

The **principal** amount Â **increases with every time period**, as the **interest** payable is **added to the principal**.

Eg : Find CI on RsÂ 10,000 for 2Â years at an interest rate of 5%.

Ans : Interest for the 1st year

For 1st year, P = 10,000, T = 1 year, R = 5%

I1=\(\frac{P.T.R}{100}\)Â = \(\frac{10000.1.5}{100}\)Â =RsÂ 500

A=P+I1=10,000+500=10,500

Interest for the 2nd year

For 2ndÂ year, P = 10,500, T = 1 year, R = 5%

I2=\(\frac{P.T.R}{100}\)=\(\frac{10500.1.5}{100}\)Â =RsÂ 525

C.I=I1+I2=RsÂ 500+RsÂ 525=RsÂ 1025

## Deducing a Formula for Compound Interest

### Formula for CI

Calculation of **compound interest** can be generalized.

let P1 be the sum on which the interest is compounded annually at the rate of R

Then the interestÂ for the 1st year,

I1=\(\frac{P_{1}. 1.R} {100}\)
=\(\frac{P_{1}.R} {100}\)

A1=P1+I1=P1+\(\frac{P_{1}.R}{100}\)

A1=P1(1+\(\frac{R}{100}\))=P2

For 2nd year,

P2=P1(1+\(\frac{R}{100}\)),T=1Â yearÂ andÂ R=R%

I2=\(\frac{P_{2}.1.R} {100}\)=\(\frac{P_{2}.R}{100}\)

I2=P1(1+\(\frac{R}{100}\))Ã—\(\frac{R}{100}\)

I2=\(\frac{P_{1} R}{100}\)(1+\(\frac{R}{100}\))

A2=P2+I2

A2=P1(1+\(\frac{R}{100}\))+\(\frac{P_{1}R}{100}\)(1+\(\frac{R}{100}\))

A2=P1(1+\(\frac{R}{100}\))(1+\(\frac{R}{100}\))Â [taking P1(1+\(\frac{R}{100}\)) as common ]
A2=P1(1+\(\frac{R}{100}\))2

Continuing this way, the amount at the end of n years will be,

An=P(1+\(\frac{R}{100}\))n

i.e., A=P(1+\(\frac{R}{100}\))n

Where,Â P is the principal amount,Â R is the rate of interest andÂ n is the number of years.

We get the formula for the amount to be paid at the end of n years.

Compound Interest can be calculated using the formula,

CI=Aâˆ’P

## Rate Compounded Annually and Half Yearly

### Rate Compounded Annually or Half-Yearly

If interestÂ is **compounded annually,**

time span, nÂ = 1 year, here the principal amount **varies yearly**.

Principal amount (A=P+I1)Â Â for first year will serve as the **principal**Â for the second year.

If interest is **compounded half – yearly,**

time span, nÂ = 6 months, here the principal amount **varies half – yearly**.

Principal amountÂ (A=P+I1) for first 6 months will be the **principal** for the next 6 months.

### Finding CI When Rate Compounded Annually or Semi – Annually

When compound interest is compounded annually,

A=P(1+\(\frac{R}{100}\))

^{n}C.I=Aâˆ’P

Where, P is the principal amount,Â R is the rate of interest andÂ n is the number of years.

When compound interest is compounded half yearly,

the **interest rate will be half of the annual interest rate **and the **time period will be doubled**.

A=P(1+\(\frac{R}{200}\))

^{2n}C.I=Aâˆ’P

Where, P is the principal amount,Â R is the rate of interest andÂ n is the number of years.

## Application of Compound Interest

### Application of Formula of CI

**Application of compound interest** are :

1.To calculate the growth rate of population (increase or decrease).

2. To calculate change in theÂ price of an item (increase or decrease).

**Example** :Â If the population of a town increases 2% annually and the present population is 3,26,40,000, find its population after 2Â years.

**Solution.**Â P = 3,26,40,000Â Â nÂ = 2 years, R = 2%

Therefore,Â Â Â Population after 2Â years

A=P(1+\(\frac{R}{100}\))n

A=32640000(1+\(\frac{2}{100}\))2

A=32640000Ã—(\(\frac{51}{50}\))2

A=32640000Ã—\(\frac{51}{50}\)Â Ã— \(\frac{51}{50}\)

A=13056Ã—51Ã—51

â‡’A=33958656

âˆ´The population after 2 years isÂ 3,39,58,656

**Example :Â **A motorcycle is bought at Rs 1,60,000. Its value depreciates at the rate of 10% per annum. Find its value after 2 years.

**Solution.**Â P = 1,60,000 Â nÂ = 2 years, R = 10%

A=P(1âˆ’\(\frac{R}{100}\))n

A=160000Ã—(1âˆ’\(\frac{10}{100}\))2

A=160000Ã—\(\frac{9}{10}\)Â Ã— \(\frac{9}{10}\)

A=129600

âˆ´Â The value of the motorcycle after 2 yearsÂ is RsÂ 1,29,600.

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