Class 8 Maths Comparing Quantities | Compound Interest |

__Compound Interest__

Simple interest is calculated on the principal, or original, amount of a loan.

Let Principle = P, Rate = R% per annum, Time = n years.

Now, Simple Interest (SI) = (P * R * n)/100

Compound interest is calculated on the principal amount and also on the accumulated interest of

previous periods, and can thus be regarded as “interest on interest.”

- When interest is compounded annually

Amount = P(1 + R/100)^{n}

- When interest is compounded Half-yearly

Amount = P{1 + (R/2)/100}^{2n}

- When interest is compounded annually but time is in fraction say 3

Amount = P(1 + R/100)^{3 }* {1 + (2R/5)/100}

Now, Compound Interest (CI) = Amount - Principle (P)

__Problem:__ Fabina borrows ₹ 12,500 per annum for 3 years at simple interest and Radha borrows the same amount for the same time period at 10% per annum, compounded annually. Who pays more interest and by how much?

Solution:

Here,

Principal (P) = ₹ 12,500, Time (T) = 3 years, Rate of interest (R) = 12% p.a.

Simple Interest for Fabina = (P * R * T)/100 = (12500 * 12 * 3)/100

= 125 * 12 * 13

= ₹ 4,500

Amount for Radha, P = ₹ 12,500, R = 10% and n = 3 years

Now, amount (A) = P(1 + R/100)^{n}

= 12500(1 + 10/100)^{3}

= 12500(1 + 1/10)^{3}

= 12500(11/10)^{3}

= 12500 * 11/10 * 11/10 * 11/10

= ₹ 16,637.50

So, C.I. for Radha = A – P = ₹ 16,637.50 – ₹ 12,500 = ₹ 4,137.50

__Problem:__ Arif took a loan of ₹ 80,000 from a bank. If the rate of interest is 10% per annum, find the difference in amounts he would be paying after 1 years if the interest is:

(i) Compounded annually. (ii) Compounded half yearly

Solution:

(i) Here,

Principal (P) = ₹ 80,000, Time (n) = 1 years, Rate of interest (R) = 10%

Amount for 1 year (A) = P(1 + R/100)^{n }

= 80000(1 + 10/100)^{1}

= 80000(1 + 1/10)^{1}

= 80000(11/10)^{1}

= 80000 * 11/10

= ₹ 88,000

Interest for 1/2 year = (88000 * 10 *1)/(100 * 2) = ₹ 4,400

Total amount = ₹ 88,000 + ₹ 4,400 = ₹ 92,400

(ii) Here, Principal (P) = ₹ 80,000, Time (n) = 1 year = 3 half-years (compounded half yearly)

Rate of interest (R) = 10% = 5% (compounded half yearly)

Amount (A) = P(1 + R/100)^{n}

= 80000(1 + 5/100)^{3}

= 80000(1 + 1/20)^{3}

= 80000(21/20)^{3}

= 80000 * 21/20 * 21/20 * 21/20

= ₹ 92,610

Difference in amounts = ₹ 92,610 – ₹ 92,400 = ₹ 210

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