Question

ᴀɴsᴡᴇʀ ᴛʜᴇ ǫᴜᴇsᴛɪᴏɴ ᴡɪᴛʜ ғᴜʟʟ ᴇxᴘʟᴀɴᴀᴛɪᴏɴ
sᴘᴀᴍ ᴡɪʟʟ ʙᴇ ʀᴇᴘᴏʀᴛᴇᴅ
ᴅᴏɴ'ᴛ ᴄᴏᴘʏ ғʀᴏᴍ ɢᴏᴏɢʟᴇ​

Answer 1

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Question\;:-}}}$

Here the Concept of Compound Interest and Amount has been used . We see we are given the details of the exchange that is Principal, Time and Rate . We also see that here the interest is compounded half yearly . So firstly we will change the rate that is to be compounded half year and change the time according to half years and then find the answer .

Let's do it !!

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★ Formula Used :-

$\\\;\boxed{\sf{\pink{Amount\;=\;\bf{P\:\times\:\bigg(1\;+\;\dfrac{R}{100}\bigg)^{T}}}}}$

$\\\;\boxed{\sf{\pink{Amount\;=\;\bf{Principal\;+\;Interest}}}}$

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★ Solution :-

Given,

$\\\;\sf{\odot\;\;Principal\;=\;\bf{P\;=\;Rs.\;\orange{4096}}}$

→ Since the interest is compounded half yearly thus its rate will be reduced by ½.

$\\\;\sf{\odot\;\;Rate\;=\;\tt{R\;=\;12\dfrac{1}{2}\:\times\:\dfrac{1}{2}}}$

$\\\;\sf{\odot\;\;Rate\;=\;\bf{R\;=\;\dfrac{25}{2}\:\times\:\dfrac{1}{2}\;=\;\bf{\orange{\dfrac{25}{4}}}}}$

This rate is in the form of % per annum .

→ Since the interest is compounded half yearly and time period is 18 months and 3 × 6 (half year), then

$\\\;\sf{\odot\;\;Time\;=\;\tt{T\;=\;18\;\;months}}$

$\\\;\sf{\odot\;\;Time\;=\;\bf{T\;=\;\orange{3\;Half\;Years}}}$

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~ For Amount of the loan at the Interest ::

We know that,

$\\\;\sf{:\rightarrow\;\;Amount\;=\;\bf{P\:\times\:\bigg(1\;+\;\dfrac{R}{100}\bigg)^{T}}}$

By applying values, we get,

$\\\;\sf{:\Longrightarrow\;\;Amount\;=\;\bf{4096\:\times\:\bigg(1\;+\;\dfrac{\dfrac{25}{4}}{100}\bigg)^{3}}}$

$\\\;\sf{:\Longrightarrow\;\;Amount\;=\;\bf{4096\:\times\:\bigg(1\;+\;\dfrac{25}{4\:\times\:100}\bigg)^{3}}}$

$\\\;\sf{:\Longrightarrow\;\;Amount\;=\;\bf{4096\:\times\:\bigg(1\;+\;\dfrac{25}{400}\bigg)^{3}}}$

$\\\;\sf{:\Longrightarrow\;\;Amount\;=\;\bf{4096\:\times\:\bigg(1\;+\;\dfrac{1}{16}\bigg)^{3}}}$

$\\\;\sf{:\Longrightarrow\;\;Amount\;=\;\bf{4096\:\times\:\bigg(\dfrac{16\;+\;1}{16}\bigg)^{3}}}$

$\\\;\sf{:\Longrightarrow\;\;Amount\;=\;\bf{4096\:\times\:\bigg(\dfrac{17}{16}\bigg)^{3}}}$

$\\\;\sf{:\Longrightarrow\;\;Amount\;=\;\bf{4096\:\times\:\bigg(\dfrac{17}{16}\:\times\:\dfrac{17}{16}\:\times\:\dfrac{17}{16}\bigg)}}$

$\\\;\sf{:\Longrightarrow\;\;Amount\;=\;\bf{4096\:\times\:\dfrac{4913}{4096}}}$

Cancelling the equal terms, we get,

$\\\;\sf{:\Longrightarrow\;\;Amount\;=\;\bf{\cancel{4096}\:\times\:\dfrac{4913}{\cancel{4096}}}}$

$\\\;\bf{:\Longrightarrow\;\;Amount\;=\;\bf{\blue{Rs.\;4913}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\;Amount\;\;=\;\bf{\blue{Rs.\;4913}}}}}$

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~ For the Compound Interest of this exchange ::

We know that,

$\\\;\sf{:\rightarrow\;\;Amount\;=\;\bf{Principal\;+\;Interest}}$

By applying values, we get,

$\\\;\sf{:\mapsto\;\;Amount\;=\;\bf{Principal\;+\;Compound\;Interest}}$

$\\\;\sf{:\mapsto\;\;4913\;=\;\bf{4096\;+\;Compound\;Interest}}$

$\\\;\sf{:\mapsto\;\;Compound\;Interest\;=\;\bf{4913\;-\;4096}}$

$\\\;\sf{:\mapsto\;\;Compound\;Interest\;=\;\bf{\purple{Rs.\;817}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\;Compound\;\;Interest\;=\;\bf{\purple{Rs.\;817}}}}}$

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★ More to know :-

$\\\;\sf{\gray{\leadsto\;\;S.I.\;=\;\dfrac{P\:\times\:R\:\times\:T}{100}}}$

Answer 2

$❥{\huge{\underline{\small{\mathbb{\pink{YOUR \ CORRECT \ ANSWER }}}}}}$