Question

30,000 is invested for some period at 8% p.a. to earn 2400 as the simple interest. If th
same sum is invested for the same period at the same rate of interest compounde
half-yearly, find the compound interest.
​

Answer 1

$\sf\large\underline\purple{Given:-}$

$\sf{\implies Principal=Rs.30000}$

$\sf{\implies Rate=8\%}$

$\sf{\implies Simple\:interest=Rs.2400}$

$\sf\large\underline\purple{To\: Find:-}$

$\sf{\implies Time\:_{(for\: simple\: interest)}=?}$

$\sf{\implies CI\:_{(for\: compound\: interest)}=?}$

$\sf\large\underline\purple{Solution:-}$

To calculate compound interest at first we have to find out time for simple interest by applying formula of SI then calculate compound interest. As given in the question that in compound interest the time is given in half yearly so, we have to change time and rate. For changing time we have to multiply by 2 and for rate dividing by 2:-]

$\sf\small\underline\green{Calculation\:for\:SI:-}$

$\sf\small\underline{Here,\:\:P=30000\:\:R=8\%\:\:SI=2400:-}$

$\tt{\implies SI=\dfrac{P\times\:T\times\:R}{100}}$

$\tt{\implies 2400=\dfrac{30000\times\:T\times\:8}{100}}$

$\tt{\implies 2400=300\times\:T\times\:8}$

$\tt{\implies 2400=2400T}$

$\tt\pink{\implies T=1\: year}$

$\sf\small\underline\green{Calculation\:for\:CI:-}$

$\sf\small\underline{Here,\:\:P=30000\:\:R=8/2=4\%\:\:T=1*2=2\: years:-}$

$\tt{\implies CI=P\bigg(1+\dfrac{r}{100}\bigg)^{n}-P}$

$\tt{\implies CI=30000\bigg(1+\dfrac{8}{100}\bigg)^{2}-30000}$

$\tt{\implies CI=30000\bigg(\dfrac{100+8}{100}\bigg)^{2}-30000}$

$\tt{\implies CI=30000\bigg(\dfrac{108}{100}\bigg)^{2}-30000}$

$\tt{\implies CI=30000\times\:(1.08)^{2}-30000}$

$\tt{\implies CI=30000\times\:1.1664-30000}$

$\tt{\implies CI=34992-30000}$

$\tt\pink{\implies CI=Rs.4992}$

$\sf\large{Hence,}$

$\sf\blue{\implies Time\:_{(for\: simple\: interest)}=1\: year}$

$\sf\blue{\implies CI\:_{(for\: compound\: interest)}=Rs.4992}$

Answer 2

Answer :-

★ Time Period = 1 year

★ Compound Interest = ₹ 4992

_____________________

★ Concept :-

Here the concept of Simple Interest (SI) and Compound Interest (CI) has been used. These are given as :-

• SI = (P×R×T) / 100

• CI = P[1 + (R/100)]ⁿ - P

Where P is the Principal, R is the rate and T is the time.

________________________________

★ Solution :-

In order to calculate our final answer, we must first calculate Time Period by applying the values in the formula of Simple Interest. Once we got that, we can then apply the values we got into the formula of Compound Interest.

Given,

» Principal Amount = P = ₹ 30,000

» Rate = R = 8 %

» Let the Time be 'T'.

» Simple Interest = SI = ₹ 2400

Now let us calculate our first step.

✍ SI = (PRT) / 100

✍ SI × 100 = P × R × T

✍ 2400 × 100 = 30000 × 8 × T

Cancelling the zeroes, both sides, we get,

✍ T = 24 / 24 = 1 year

• Hence, the time period for the simple interest is 1 year.

Now let us calculate the Compound Interest. Here the time is given half yearly, so we must we must calculate it for 6 months. Thus,

=> n = T × 2 = 1 × 2 = 2 years

And,

=> Rate = ½ × 8 = 4 %

✒ CI = P[1 + (r/100)]ⁿ - P

Here we are subtracting P, because we have to subtract the principal of previous year, but calculate next from initial year. Even, both half of rate and twice of year is cancelling each other, so we are directly applying its values.

✒ CI = (30000)[1 + (8/100)]² - 30000

✒ CI = (30000)[(100 + 8)/100]² - 30000

✒ CI = (30000)[108/100]² - 30000

✒ CI = 30000 × (1.08)² - 30000

✒ CI = 30000 × 1.1664 - 30000

✒ CI = 34992 - 30000

✒ CI = 4992 Rs.

Hence we get the compound interest

= ₹ 4992.

_____________________

★ More to know :-

• Simple Interest is the form of money that when a Principal Amount is given, the amount of net interest on it will depend on Principal, Time and Rate.

• Compound Interest is the sum total interest for a given period of time.

• Principal is the initial amount that is to be given.

• Rate is the rate at which the interest is calculated.

• Time is the time period for which the interest has been done.