What sum invested for 1.5 year amounts to $132651$ in $1\dfrac{1}{2}$ years compounded half-yearly at the rate of $4\% $ p.a.?
Answer
Hint: We have to find the principal amount for which $4\% $ interest is compounded half-yearly for $1\dfrac{1}{2}$ years which sum to a final amount of $Rs.132651$ . So, using the compound interest formula which is Final Amount = $principal \times {\left( {1 + \dfrac{{rate}}{{100}}} \right)^{time}}$ we will find the principal amount.
Given:
Interest rate = $4\% $ per annum, for compounded half years the interest rate becomes half that is $2\% $
Final amount =
$Rs.132651$
Time = $1\dfrac{1}{2}$ = $1.5$ years, for compounded half years the time becomes $3$ half years.
Complete step-by-step solution:
The formula for compound interest is
Total Amount = $\text{principal} \times {\left( {1 + \dfrac{{rate}}{{100}}} \right)^{time}}$.
Substituting all the values given in the question, the formula becomes,
$132651$ = $\text{principal} \times {\left( {1 + \dfrac{2}{{100}}} \right)^3}$
Solving the rate of interest part,
Cross
multiplying and making the denominator equal,
$132651$ = $\text{principal} \times {\left( {\dfrac{{100 + 2}}{{100}}} \right)^3}$
$\Rightarrow 132651$ = $\text{principal} \times {\left( {\dfrac{{102}}{{100}}} \right)^3}$
$\Rightarrow 132651$ = $\text{principal} \times {\left( {1.02} \right)^3}$
$\Rightarrow 132651$ = $\text{principal} \times 1.06120$
Dividing the final amount $132651$ by $1.06120$,
$principal = \dfrac{{132651}}{{1.06120}}$
Finally, we get the principal amount
as,
$principal = 125000$
The principal amount is $Rs.125000$
Therefore, $Rs.125000$ is compounded half-yearly for $1.5$ years at a rate of interest of $4\% $ p.a. to get the final amount as $Rs.132651$.
We can cross-check the principal amount by using the same compound interest formula where we substitute principal amount as $Rs.125000$ rate of interest as $2\% $ half-yearly and time as $3$ half years,
Final Amount = $\text{principal} \times {\left( {1 + \dfrac{{rate}}{{100}}}
\right)^{time}}$
Final Amount = $125000 \times {\left( {1 + \dfrac{2}{{100}}} \right)^3}$
Final Amount = $125000 \times {\left( {1 + 0.02} \right)^3}$
Final Amount = $125000 \times {\left( {1.02} \right)^3}$
Final Amount = $125000 \times 1.06120$
Final Amount = $132650 \cong 132651$
Therefore, $Rs.125000$ is the correct principal amount.
Note: Compound interest is the interest calculated on the predominant and the interest accrued over the preceding period. It is distinct from easy interest, where interest isn't introduced to the principal while calculating the interest at some point of the following duration. Compound interest unearths its usage in the maximum of the transactions in the banking and finance sectors and different regions.
Question 1.
Calculate the amount and compound interest on
(i) ₹15000 for 2 years at 10% per annum compounded annually.
(ii) ₹156250 for \(1 \frac{1}{2}\) years at 8% per annum compounded half-yearly.
(iii) ₹ 100000 for 9 months at 4% per annum compounded quarterly.
Solution:
Question 2.
Find the difference between the simple interest and compound interest on ₹4800 for 2 years at 5% per annum, compound interest being reckoned annually.
Solution:
Question 3.
Find the compound interest on ₹3125 for 3 years if the rates of interest for the first, second and third year are respectively 4%, 5% and 6% per annum.
Solution:
Question 4.
Kamla borrowed ₹26400 from a Bank to buy a scooter at a rate of 15% p.a. compounded yearly. What amount will she pay at the end of 2 years and 4 months to clear the loan?
Solution:
Question 5.
Anil borrowed ₹18000 from Rakesh at 8% per annum simple interest for 2 years. If Anil had borrowed this sum at 8% per annum compound interest, what extra amount would he has to pay?
Solution:
Question 6.
Mukesh borrowed 775000 from a bank. If the rate of interest is 12% per annum, find the amount he would be paying after \(1 \frac{1}{2}\) years if the interest is
(i) compounded annually
(ii) compounded
half-yearly.
Solution:
Question 7.
Aryaman invested ₹10000 in a company, he would be paid interest at 7% per annum compounded annually. Find
(i) the amount received by him at the end of 2 years.
(ii) the interest for the 3rd year.
Solution:
Question 8.
What sum of money will amount to ₹9261 in 3 years at 5% per annum compound interest?
Solution:
Question 9.
What sum invested for \(1 \frac{1}{2}\) years compounded half-yearly at the rate 8% p.a. will amount to ₹ 140608?
Solution:
Question 10.
At what rate percent will ₹2000 amount to ₹2315·25 in 3 years at compound interest?
Solution:
Question 11.
If ₹40000 amounts to ₹46305
in \(1 \frac{1}{2}\) years, compound interest payable half-yearly, find the rate of interest per annum.
Solution:
Question 12.
In what time will ₹15625 amount to ₹ 17576 at 4% per annum compound interest?
Solution:
Question 13.
₹ 16000 invested at 10% p.a. compounded semi-annually, amounts to ₹18522. Find the time period of investment.
Solution: