CBSE Class 8 Maths Notes Chapter 7 Comparing Quantities

Simple and Compound Interest

Simple Interest

Simple Interest is a straightforward method of calculating the extra money charged on a loan or earned on a deposit. The principal amount remains constant throughout the specified time period.

Key Concepts:

• Principal (P): The initial amount of money either borrowed or deposited.
• Time (T): The duration, usually in years, for which the money is borrowed or deposited.
• Rate of Interest (R): The percentage of the principal charged as interest annually.

Formula for Simple Interest: Simple interest = (P.T.R)/100

Compound Interest

Compound Interest differs from Simple Interest in that the interest is calculated on the initial principal, which includes all the accumulated interest from previous periods. This method allows for the interest to "compound" over time, leading to a higher amount compared to Simple Interest. Eg : Find CI on Rs 10,000 for 2 years at an interest rate of $5\%$ . Ans : Interest for the 1st year For 1st year, P = 10,000, T = 1 year, R = 5% I ₁ = PTR/100 = (10000.1.5)/100 = Rs.500 A=P+I ₁ =10,000+500=10,500 Interest for the 2nd year For 2nd year, P = 10,500, T = 1 year, R = 5% I ₂ = PTR/100 = (10500.1.5)/100 = Rs 525 Cost Price, C.I.= I ₁ +I ₂ =Rs 500+Rs 525=Rs 1025

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 1st year, I ₁ = PTR/100 = P ₁ .R/100 A ₁ =P ₁ +I ₁ =P ₁ +P ₁ .R/100 A ₁ =P ₁ (1+R/100) = P2 For 2nd year, P2=P ₁ (1+R/100) T=1 year and R=R% I ₂ = (P ₂ .1.R)/100 = P ₂ .R/100 I ₂ =P ₁ (1+R/100) × R/100 I ₂ =P ₁ R/100(1+R/100) A ₂ =P ₂ +I ₂ A ₂ =P ₁ (1+R/100)+P1R/100(1+R/100) Taking P ₁ (1+R/100) as common, we get; A ₂ =P ₁ (1+R/100)(1+R/100) A ₂ =P ₁ (1+R/100) ² Continuing this way, the amount at the end of n years will be,

A _n =P(1+R/100) ⁿ

Where, P is the principal amount, R is the rate of interest and 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=A−P

Rate Compounded Annually and 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+R/100) ⁿ CI = A – P

Where, P is the principal amount, R is the rate of interest and 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+R/200) ²ⁿ 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 compound interest are :

Calculating the Growth Rate of Population:

• Compound interest is used to determine how a population grows or decreases over time. By applying the compound interest formula, demographers can predict future population sizes, considering factors such as birth rates, death rates, and migration.

Calculating the Change in the Price of an Item:

• Compound interest is also applied to track changes in the price of goods or services over time. This can include inflation, where the price of items increases, or deflation, where prices decrease. The compound interest formula helps in estimating the future cost based on current prices and the rate of change.

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+R/100) ⁿ A=32640000(1+2/100) ² A=32640000×(51/50) ² A=32640000×(51/50)×(51/50) 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−R/100) ⁿ A=160000×(1−10/100) ² A=160000×9/10×91/10 A=129600 ∴ The value of the motorcycle after 2 years is Rs 1,29,600.

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