Comparing Quantities

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, Ratios and Percentages

In this chapter "Comparing quantities", students of Class 8 will learn to find the Discount when discount percentage is given. Also, we will learn here cost price, sales tax, GST (Goods and Services Tax), along with compound interest.

Fractions, Ratios and Percentages

A fraction represents a part of a whole which consists of numerators and denominators and it is the division of two same quantities.
14, 15, 23, etc.

Ratio is the comparison of one value to the other or the comparison of two different quantities.
Eg: 3:5

Percentage is a value that expresses the part of a whole or a fraction of 100. For example, 10% of 100 is equal to 110 of 100, i.e., 10.

Finding Increase or Decrease Percentage in Situations

Finding new number, when there is increase in percentage. New number = original number + (increase in percentage × number)

Example : The Cost of a mobile phone is Rs 15,000. Find the new price if there is a increaseof 5%
New price = original price + 5% of original price

New price = 15,000 + (5100 × 15,000)

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

Here Rs 750 is increase in the price.

The new number can be found using:
New number = original number × percentage increase

Example:
New price = 15,000 × 105100 = 15,000 × 1.05 = 15,750

Finding new number when there is decrease in percentage:
New number = original number - (decrease in percentage × number)
Also, New number = original number × percentage decrease

Example:
The cost of a mobile phone is Rs 15,000. Find the new price if there is a decrease of 5%
New price = 15,000 × 95100 = 15,000 × 0.95 = 14,250

Finding Discounts

Finding SP without Finding Discount Percentage

A reduction (decrease) 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 selling price is Rs 495.
Solution: Discount = Marked price - Sale price = 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%.
Solution: Discount = 5% of 500 = 5100 × 500 = Rs 25

Estimation of Amounts (In Percentages)

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

Example: Anil bought a pair of shoes priced Rs 650 at a discount of 10%.
Solution: Billing amount = Marked price - discount = Rs 650 - 10100 × 650 = Rs 650 - Rs 65 = Rs 585

Example: Shilpa bought a new mobile for Rs 15,000. She has to pay 2% as delivery charges.
Find the billing amount.
Solution: Billing amount = Marked price + Hike
Billing amount = Rs 15,000 + 2100 × 15,000
Billing amount = Rs 15,000 + Rs 300 = Rs 15,300

Prices Related to Buying and Selling

Prices / Charges Related to Buying and Selling

Profit = Selling price - Cost price
Profit % = ProfitCost Price × 100
Loss = Cost price - Selling price
Loss % = LossCost Price × 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 = Rs 13,500 - Rs 12,000 = Rs 1,500
Profit % = ProfitCost price × 100
Profit % = 150012000 × 100 = 12.5%

Example: Amit sold his laptop, priced Rs 20,000 at Rs 18,000. Find his loss percentage.
Loss = Cost price - Selling price
Loss = Rs 20,000 - Rs 18,000 = Rs 2,000
Loss % = LossCost Price × 100
Loss% = 200020000 × 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 = Tax % of Selling price
Billing Amount = Selling price + VAT

Example: Megha bought a wrist watch for Rs 1,200 and VAT is charged at 8%. Calculate the VAT and billing amount.
VAT = Tax % of selling price
VAT = 8% of 1,200 = 8100 × 1,200 = Rs 96
Billing amount = S.P + VAT = Rs 1,200 + Rs 96 = Rs 1,296

Simple and Compound Interest

Simple Interest

Simple interest is the extra money charged on a loan where the principal amount will be fixed for a particular time period.
Interest is the extra money that a bank gives for saving or depositing money with them.
Similarly, when anybody borrow money, they pay interest.
Simple interest = P × T × R100
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 increases with every time period, as the interest payable is added to the principal.

Example: Find CI on Rs 10,000 for 2 years at 5%.
Solution: 1_st year: P = 10,000, T = 1, R = 5%
I₁ = 10000 × 1 × 5100 = Rs 500
A = 10,000 + 500 = 10,500
2^nd> year: P = 10,500, T = 1, R = 5%
I₂ = 10500 × 1 × 5100 = Rs 525
C.I. = 500 + 525 = Rs 1,025

Deducing a Formula for Compound Interest

Formula for Cost Price

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 interest for the 1^st year, I₁ = P₁ × R100
A₁ = P₁ + I₁ = P₁ + P₁ × R100
A₁ = P₁ × ( 1 + R100 ) = P₂
For 2^nd year, P₂ = P₁ × ( 1 + R100 ), T = 1 year and R = R %
I₂ = P₂ × 1 × R100 = P₂ × R100
I₂ = P₁ × ( 1 + R100 ) × R100
I₂ = P₁ × R100 × ( 1 + R100 )
A₂ = P₂ + I₂
A₂ = P₁ × ( 1 + R100 ) + P₁ × R100 × ( 1 + R100 )
Taking P₁ × ( 1 + R100 ) as common, we get:
A₂ = P₁ × ( 1 + R100 )²
Continuing this way, the amount at the end of n years will be:
A_n = P × ( 1 + R100 )ⁿ
Where, P is the principal amount, R is the rate of interest and n is the number of years.
Compound Interest can be calculated using the formula:
CI = A − P

Rate Compounded Annually and Half Yearly

Rate Compounded Annually or Half-Yearly

If interest is compounded annually, time span, n = 1 year, here the principal amount varies yearly.
Principal amount (A = P + I₁) for first year will serve as the principal for the second year.

If interest is compounded half – yearly, time span, n = 6 months, here the principal amount varies half – yearly. Principal amount (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 + R100)ⁿ
CI = A – P
Where, P is the principal amount, R is the rate of interest and 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 the price of an item (increase or decrease).

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

Solution:
P = 3,26,40,000
n = 2 years, R = 2%
Therefore, Population after 2 years
A = P (1 + R100)ⁿ
A = 32640000 (1 + 2100)²
A = 32640000 × (5150)²
A = 32640000 × (5150) × (5150)
A = 13056 × 51 × 51
⇒ A = 33958656
∴ The population after 2 years is 3,39,58,656.

Example : 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:
P = 1,60,000
n = 2 years, R = 10%
A = P (1 − R100)ⁿ
A = 160000 × (1 − 10100)²
A = 160000 × 910 × 910
A = 129600
∴ The value of the motorcycle after 2 years is Rs 1,29,600.

Short Answer Questions(SAQs)

What is the formula for Profit and loss?

The formula for the profit and loss percentage is: Profit percentage = (Profit/Cost Price) x 100. Loss percentage = (Loss/Cost price) x 100.

What is Sales tax?

Sales tax is an amount of money, calculated as a percentage, that is added to the cost of a product or service when purchased by a consumer at a retail location.

What is Compound interest?

Compound interest is the interest which is earned on interest.

MCQs

1. What is the formula to find percentage?

(a) part whole × 100

(b) whole part × 100

(c) part × whole

(d) whole part

► (a) part whole × 100

2. 25% of 200 is:

(a) 25

(b) 50

(c) 75

(d) 100

► (b) 50

3. The selling price is more than the cost price, the seller has a:

(a) Loss

(b) Profit

(c) Discount

(d) None

► (b) Profit

4. Discount is always calculated on:

(a) Cost Price

(b) Selling Price

(c) Marked Price

(d) Profit

► (c) Marked Price

5. What is the simple interest on ₹1000 at 5% per annum for 2 years?

(a) ₹50

(b) ₹100

(c) ₹75

(d) ₹120

► (b) ₹100

6. The formula for compound interest annually is:

(a) P × R × T

(b) P(1 + R 100 )^T

(c) P(1 − R 100 )^T

(d) P + RT

► (b) P(1 + R 100 )^T

7. If you get a 20% discount on ₹500, the discount amount is:

(a) ₹80

(b) ₹120

(c) ₹100

(d) ₹150

► (c) ₹100

8. What is the principal if S.I. is ₹1200, rate is 6%, time is 4 years?

(a) ₹5000

(b) ₹6000

(c) ₹4500

(d) ₹4000

► (a) ₹5000

9. 120 is what percent of 400?

(a) 20%

(b) 25%

(c) 30%

(d) 40%

► (c) 30%

10. Find the amount on ₹4000 at 5% p.a. compound interest for 2 years.

(a) ₹4200

(b) ₹4410

(c) ₹4400

(d) ₹4300

► (b) ₹4410

11. If 30% of x is 60, then x = ?

(a) 150

(b) 200

(c) 100

(d) 180

► (b) 200

12. Profit % =

(a) Profit × 100 Cost Price

(b) Profit × 100 Selling Price

(c) Cost Price × 100 Profit

(d) Selling Price × 100 Profit

► (a) Profit × 100 Cost Price

13. A watch was bought for ₹800 and sold for ₹1000. Find profit %.

(a) 10%

(b) 15%

(c) 20%

(d) 25%

► (d) 25%

14. Compound interest is more than simple interest when:

(a) Time = 1 year

(b) Time > 1 year

(c) Time = 0

(d) Principal is zero

► (b) Time > 1 year

15. A person sold a bike at a 10% loss for ₹9000. What was the cost price?

(a) ₹9500

(b) ₹9900

(c) ₹10000

(d) ₹8900

► (c) ₹10000

16. What is 15% of 300?

(a) 30

(b) 45

(c) 60

(d) 35

► (b) 45

17. Amount =

(a) Principal + Interest

(b) Principal × Time

(c) Interest × Time

(d) Rate + Time

► (a) Principal + Interest

18. Which of these is a type of comparing quantities?

(a) Ratio

(b) Percentage

(c) Profit and loss

(d) All of the above

► (d) All of the above