Question

$\huge\underline\bold\green{Question \: : }$

A sum of Rs. 10,500 amounts to Rs. 13,650 in 2 years at a certain rate percent per annum simple interest. The same sum will amount to what in 1 year at the same rate, if the interest is compounded half yearly (nearest to Rs.1)?

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Answer 1

Given :-

A sum of Rs. 10,500 amounts to Rs. 13,650 in 2 years at a certain rate percent per annum simple interest compounded half yearly.

Concept :-

➻ Simple interest = (Principal × rate × time)/100

➻ Simple interest = Amount - Principal

➻ Compound interest = Amount - Principal

➻ Amount = Principal(1 + (r/100))t

If interest is compounded half yearly then time become double and rate become half.

Calculation :-

⇒ Simple interest = 13,650 - 10,500

⇒ Simple interest = Rs. 3150

⇒ 3150 = (10,500 × r × 2)/100

⇒ 3150 = 105 × r × 2

⇒ r = 3150/(105 × 2)

⇒ r = 15%

Now,

For compounded half yearly

⇒ Rate = (15/2)%

⇒ Time = 2 years

⇒ Amount = 10,500(1 + (15/200))2

⇒ Amount = 10,500(1 + (3/40))2

⇒ Amount = 10,500 × (43/40) × (43/40)

⇒ Amount = 12,134

∴ The sum will amount is 12,134.

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Answer 2

$\sf\underline{\underline{{ Question : \: }{ \: }}}$

A sum of Rs. 10,500 amounts to Rs. 13,650 in 2 years at a certain rate percent per annum simple interest compounded half yearly.

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$\sf\underline{\underline{{ Given : \: }{ \: }}}$

ㅤㅤㅤㅤ• Principal = ₹10,500

ㅤㅤㅤㅤ• Amount = ₹13,650

ㅤㅤㅤㅤ• Time = 2 years

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$\sf\underline{\underline{{ To \: Find : \: }{ \: }}}$

ㅤㅤㅤㅤ• If it was compounded half yearly, Amount = ?

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$\sf\underline{\underline{{ Solution : \: }{ \: }}}$

~ Formula Used :

ㅤㅤ• Simple interest

$\bf\huge\pink{\textsf{⇢}} \: \small{\color{purple}{\fbox{\textsf{\textbf{ Simple interest = Amount - Principal\: }}}}}$

ㅤㅤ• Rate

$\bf\huge \pink{\textsf{⇢}} \: \small{\color{purple}{\fbox{\textsf{\textbf{Simple interest = (Principal × rate × time)/100 \: }}}}}$

ㅤㅤ• Amount

$\bf\huge\pink{\textsf{⇢}} \: \small{\color{purple}{\fbox{\textsf{\textbf{ Amount = Principal(1 + (r/100))t}}}}}$

ㅤㅤㅤㅤㅤ________________________________

~ Concept Used :

If interest is compounded half yearly then time become double and rate become half.

ㅤㅤㅤㅤㅤ________________________________

~ Calculating the Simple Interest :

$\sf{{ \: \: \: \: \: \: \: \: ⟼ \: \: \: \: \: Simple \: interest = ₹13,650 - ₹10,500}{ \: }}$

$\sf{{ \: \: \: \: \: \: \: \: \: \: \: \: \: ⟼ \: \: \: \: \: \: Simple \: interest = \pink{₹3150}\: }{ \: }}$

ㅤㅤㅤㅤㅤ________________________________

~ Finding the Rate :

$\sf{{ \: \: \: \: \: \: \: \: \: ⟼ \: \: \: \: Amount = Principal(1 + (r/100))\: }{ \: }}$

$\sf{{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: ⟼ \: \: \: \: \: \: 3150 = (10,500 × r × 2)/100\: }{ \: }}$

$\sf{{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ⟼ \: \: \: \: \: \: 3150 = 105 × r × 2 \: }{ \: }}$

$\sf{{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ⟼ \: \: \: \: \: \: \: r = 3150/(105 × 2)\: }{ \: }}$

$\sf{{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ⟼ \: \: \: \: \: \: \: \: r = \green{15\%} }{ \: }}$

ㅤㅤㅤㅤㅤ________________________________

$\sf\underline{\underline{{ Now : \: }{ \: }}}$

$\sf \underline{{ For \: compounding \: half \: yearly \: }{ : }}$

ㅤㅤㅤㅤ• Rate = (15/2)%

ㅤㅤㅤㅤ• Time = 2 years

ㅤㅤㅤㅤㅤ________________________________

~ Calculating the Sum of money :

$\sf{{ \: \: \: \: \: \: \: \: \: \: \: \: \: ⟼ \: \: \: \: \: \: \: Amount = 10,500(1 + (15/200))²\: }{ \: }}$

$\sf{{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ⟼ \: \: \: \: \: \: Amount = 10,500(1 + (3/40))² \: }{ \: }}$

$\sf{{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ⟼ \: \: \: \: \: \: Amount = 10,500 × (43/40) × (43/40)\: }{ \: }}$

$\sf{{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ⟼ \: \: \: \: \: \: Amount = \blue{₹ 12,134}\: }{ \: }}$

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Therefore :

The sum will amount is ₹12,134.

$\small{\color{green}{ \fbox{\textsf{\textbf{ ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬\: }}}}}$