Question

Find the amount and the compound interest on ₹ 10,000 for 1 ½ years at 10% per annum, compounded half yearly. Would this interest be more than the interest he would get if it was compounded annually?​

Answer 1

Answer:

What is known: Principal, Time Period, and Rate of Interest

What is unknown: Amount and Compound Interest (C.I.)

Reasoning:

A = P[1 + (r/100)]n

P = ₹ 10,000

n = 1 1/2

years

R = 10% p.a. compounded annually and half-yearly

where , A = Amount, P = Principal, n = Time period and R = Rate percent

For calculation of C.I. compounded half-yearly, we will take the Interest rate as 5% and n = 3

A = P[1 + (r/100)]n

A = 10000[1 + (5/100)]3

A = 10000[1 + (1/20)]3

A = 10000 × (21/20)3

A = 10000 × (21/20) × (21/20) × (21/20)

A = 10000 × (9261/8000)

A = 5 × (9261/4)

A = 11576.25

Interest earned at 10% p.a. compounded half-yearly = A - P

= ₹ 11576.25 - ₹ 10000 = ₹ 1576.25

Now, let's find the interest when compounded annually at the same rate of interest.

Hence, for 1 year R = 10% and n = 1

A = P[1 + (r/100)]n

A = 10000[1 + (10/100)]1

A = 10000[1 + (1/10)]

A = 10000 × (11/10)

A = 11000

Now, for the remaining 1/2 year P = 11000, R = 5%

A = P[1 + (r/100)]n

A = 11000[1 + (5/100)]

A = 11000[(105/100)]

A = 11000 × 1.05

A = 11550

Thus, amount at the end of 1 and a half years

when compounded annually = ₹ 11550

Thus, compound interest = ₹ 11550 - ₹ 10000 = ₹ 1550

Therefore, the interest will be less when compounded annually at the same rate.

Answer 2

Given,

$\sf \: P\:=\:₹\;10,000 \\ \\ \sf Time\:=\: 1 \frac{1}{2} \: \:or \: \:1.5\:years \\ \\ \sf \: Rate\:of\:Intrest\:=\: 10\:\%$

Amount received on a certain sum of money of P invested at the rate of r % per annum compounded semi-annually for n years is given by :-

$\sf { \color{lime} \boxed{Amount\:=\:P \left[\:1\:+\:\frac{r}{200} \right] ^{2n}}}$

Substituting the values,

$\sf \: Amount\:=\:10000\: \left[\:1+\: \frac{10}{200} \right]^{2 \: \times \: 1.5}$

$\sf \: Amount\:=\:10000\: \left[\:1+\: \frac{1}{20} \right]^{3}$

LCM of 1 and 20 = 20

$\sf \: Amount\:=\:10000\: \left[ \: \frac{1 \times 20 + 1}{20} \right]^{3}$

$\sf \: Amount\:=\:10000\: \left[ \: \frac{21}{20} \right]^{3}$

$\color{aqua} \boxed{\therefore \: Amount\:=\:₹\:10500}$

Compound Interest = Amount - Principal

$\sf \: Compound\:Intrest \:=\:10,500\:-\:10,000$

$\sf \: Compound\:Intrest \:=\:₹\:500$

➜ Amount = ₹ 10,500

➜ Compound Interest = ₹ 500

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$\color{orange} \boxed{ \begin {array} { |c|c|c|} Statement&Formulas \\ \\ \\❑ \: Amount \: on \: a \: certain \: sum \: of \: money \: of \: P \: invested \: at \: the \: rate \: of \: r \% \: per \: \\ annum \: compounded \: annually \: for \: n \: years \: is \: given \: by &❑ \: Amount\:=\:P[1\:+\: \frac{r}{100}]^n \\ \\ ❑ \: Amount \: on \: a \: certain \: sum \: of \: money \: of \: P \: invested \: at \: the \: rate \: of \: r \% \: per \\ \: annum \: compounded \: quarterly \: for n \: years \: is \: given \: by &❑ \:Amount\:=\:P[1\:+\: \frac{r}{200}]^{4n} \\ \\ ❑ \:Amount \: on \: a \: certain \: sum \: of \: money \: of \: P \: invested \: at \: the \: rat e \: of \: r \% \: per \: \\ annum \: compounded \: monthly \: for \: n \: years \: is \: given \: b y&❑\:Amount\:=\:P\:[1\:+\: \frac{r}{1200}^{12n}\end {array}}$