Question

8. Find the amount and the compound interest on ₹ 10,000 for 1½ years at 10% per annum, compounded half yearly. Would this interest be more that the interest he would get if it was compounded annually?
Need verified answer with full explanation

Answer 1

Given :

• Sum = 10000
• Time = 1 1/2
• Rate of interest = 10% pa
• Time period of change = half yearly

To Find :

The more interest which is given in rate changing of half yearly than yearly.

Formula To be Applied :

We will use the formula of compound interest.

$\tt{a = p {(1 + \frac{r}{100} )}^{t} }$

Solution :

Given, P = 10000

Rate = 10% pa

Time = 1 and 1/2 years

Change of amount = 1/2 years

Means, the time for which the amount will change = 3

Rate for half years = 5%

$\tt{ \implies amount = 10000 {(1 + \frac{5}{100} )}^{3} }$

$\tt{ \implies amount = 10000 {(1 + \frac{1}{20} )}^{3} }$

$\tt{ \implies amount = 10000 \times \frac{9261}{8000} }$

$\tt{ \implies amount = 1.25 \times 9261}$

$\red{ \underline{ \boxed{ \tt{ \bigstar amount = 11576.25}}}}$

Hence, Amount for half yearly changing rate = 11576.25

_____________________

Now, we will calculate the compound interest on yearly conversion:

Given, Sum = 10000

Rate = 10%

Time = 1 and 1/2

$\tt{ \implies amount = 10000 {(1 + \frac{10}{100} )}^{1.5}}$

Note that the power is 1.5

Hence, it is very difficult to calculate.

So, we will use the SI method to calculate CI.

First we will calculate the amount for first year.

$\implies \tt{amount = p(1 + \frac{tr}{100}) }$

$\tt{ \implies amount = 10000(1 + \frac{10 \times 1}{100}) }$

$\tt{ \implies amount = 10000( \frac{11}{10} )}$

$\tt{amount = 11000}$

Now, we will calculate the final amount on using the previous amount as principal.

$\tt{ \implies amount = 11000(1 + \frac{5}{100} )}$

$\tt{ \implies amount = 11000 \times \frac{21}{20} }$

$\tt{ \implies amount = 550 \times 21}$

$\green{ \underline{ \boxed{ \tt{ \bigstar \: amount = 11550}}}}$

Hence, Difference In amount

= 11576.25-11550

= 26.25

_____________________

Hence, we need to pay ₹26.25 more in half yearly conversion rate

If any doubt please ask me

Answer 2

Step-by-step explanation:

$\large{\blue{\bigstar}} \: \: {\underline{\boxed{\red{\sf{ }a = p(1 + \frac{ r }{200} ) {}^{ {}^{2n} } \times (1 + \frac{br}{200} )}}}}$

$\large{\blue{\bigstar}} \: \: {\underline{\boxed{\red{\sf{ = 10000(1 + \frac{10}{200)} {}^{ {}^{(2)} } \times (1 + \frac{ \frac{1}{2} \times 10}{100} ) }}}}}$

$\large{\blue{\bigstar}} \: \: {\underline{\boxed{\red{\sf{ }=10000(1.1025)(1.05)}}}}$

$\large{\blue{\bigstar}} \: \: {\underline{\boxed{\red{\sf{ }}=11576.25}}}$

$\large{\blue{\bigstar}} \: \: {\underline{\boxed{\red{\sf{ }Amount =Rs.11,576.25}}}}$

★Therefore

$\large{\blue{\bigstar}} \: \: {\underline{\boxed{\red{\sf{ }CI=A−P=Rs.1576.25}}}}$