Question

$\huge\fbox\red{Question:-}$


Find the compound interest on 4500 at 10% per annum for 5 years, compounded half-yearly, Find the compound interest of 2000 at the rate of 8% per annum for 18 months when the interest is calculated half yearly.​

Answer 1

$\large\bold{\underline{\underline{Solution:-}}}$

$\underline{\bf{Given\::}}$

Principal, (P) = Rs.4500

Rate, (R) = 10% p.a

Time, (n) = 5 years .

$\underline{\bf{Explanation\::}}$

As we know that formula of the compounded half - yearly;

$\boxed{\bf{Amount = Principal\bigg(1+\frac{R/2}{100} \bigg)^{2n}}}$

A/q

$\mapsto\tt{A =P\bigg(1+ \dfrac{R/2}{100} \bigg)^{2n}}$

$\mapsto\tt{A =4500\bigg(1+ \dfrac{10/2}{100} \bigg)^{2\times 5}}$

$\mapsto\tt{A =4500\bigg(1+ \dfrac{\cancel{10/2}}{100} \bigg)^{10}}$

$\mapsto\tt{A =4500\bigg(1+ \dfrac{5}{100} \bigg)^{10}}$

$\mapsto\tt{A =4500\bigg(1+ \cancel{\dfrac{5}{100}} \bigg)^{10}}$

$\mapsto\tt{A =4500\bigg(1+ \dfrac{1}{20} \bigg)^{10}}$

$\mapsto\tt{A =4500\bigg( \dfrac{20+1}{20} \bigg)^{10}}$

$\mapsto\tt{A =4500\bigg( \dfrac{21}{20} \bigg)^{10}}$

$\mapsto\tt{A =\cancel{4500} \times \dfrac{21}{\cancel{20}} \times \dfrac{21}{\cancel{20}} \times \dfrac{21}{20} \times \dfrac{21}{20} \times \dfrac{21}{20} \times\dfrac{21}{20} \times \dfrac{21}{20} \times \dfrac{21}{20} \times \dfrac{21}{20} \times \dfrac{21}{20} }$

$\mapsto\tt{A =11.25 \times \dfrac{21}{1} \times \dfrac{21}{1} \times \dfrac{21}{20} \times \dfrac{21}{20} \times \dfrac{21}{20} \times\dfrac{21}{20} \times \dfrac{21}{20} \times \dfrac{21}{20} \times \dfrac{21}{20} \times \dfrac{21}{20} }$

$\mapsto\tt{A =\dfrac{187648661004761.3}{25600000000} }$

$\mapsto\tt{A =\cancel{\dfrac{187648661004761.3}{25600000000} }}$

$\mapsto\bf{A = Rs.7330.02}$

Now, as we know that compound Interest;

→ C.I. = Amount - Principal

→ C.I. = Rs.7330.02 - Rs.4500

→ C.I. = Rs.2830.02

Thus,

The compound Interest will be Rs.2830.02 .

Again,

$\underline{\bf{Given\::}}$

Principal, (P) = Rs.2000

Rate, (R) = 8% p.a

Time, (n) = 18 months  [18/12 = 3/2 years ]

$\underline{\bf{Explanation\::}}$

A/q

$\mapsto\tt{A =P\bigg(1+ \dfrac{R/2}{100} \bigg)^{2n}}$

$\mapsto\tt{A =2000\bigg(1+ \dfrac{8/2}{100} \bigg)^{2\times 3/2}}$

$\mapsto\tt{A =2000\bigg(1+ \dfrac{\cancel{8/2}}{100} \bigg)^{\cancel{2} \times 3/\cancel{2}}}$

$\mapsto\tt{A =2000\bigg(1+ \dfrac{4}{100} \bigg)^{3}}$

$\mapsto\tt{A =2000\bigg(1+ \cancel{\dfrac{4}{100}} \bigg)^{3}}$

$\mapsto\tt{A =2000\bigg(1+ \dfrac{1}{25} \bigg)^{3}}$

$\mapsto\tt{A =2000\bigg(\dfrac{25+1}{25} \bigg)^{3}}$

$\mapsto\tt{A =2000\bigg(\dfrac{26}{25} \bigg)^{3}}$

$\mapsto\tt{A =\cancel{2000} \times \dfrac{26}{\cancel{25}} \times \dfrac{26}{\cancel{25}} \times \dfrac{26}{25} }$

$\mapsto\tt{A =3.2 \times \dfrac{26}{1} \times \dfrac{26}{1} \times \dfrac{26}{25} }$

$\mapsto\tt{A =\cancel{\dfrac{56243.2}{25} }}$

$\mapsto\bf{A =Rs.2249.72}$

Now, as we know that compound Interest;

→ C.I. = Amount - Principal

→ C.I. = Rs.2249.72 - Rs.2000

→ C.I. = Rs.249.72

Thus,

The compound Interest will be Rs.249.72 .

Answer 2

Interest for the 1st year =Rs1004500×10×1=Rs450

Amount after the 1st year =Rs4500+Rs450=Rs4950

Interest for the 2nd year =Rs1004950×10×1=Rs495

Amount after the 2nd year =Rs4950+Rs495=Rs5445

∴ Compound Interest =Rs5445−Rs4500=Rs945