What sum invested for amount to 1326517 1 1/2 years compounded half-yearly at the rate of 4% pa will

The given equation of line passes through the point A(0, 1, 2) and the direction ratios of the line are 1, 2, 3.

Let `bar"a"` be the position vector of point A.

Let `bar"b"` be the vector parallel to this line.

∴ `bar"a" = hat"j" + 2hat"k"` and `bar"b" = hat"i" + 2hat"j" + 3hat"k"`

The vector equation of a line passing through a point with position vector `bar"a"` and parallel to `bar"b"` is `bar"r" = bar"a" + lambdabar"b"`.

The vector equation of the given line is `bar"r" = (hat"j" + 2hat"k") - lambda(hat"i" + 2hat"j" + 3hat"k")`

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

Verified

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.

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