Question

Hi friends, please help me with this question.
If the compound interest on a sum of money compounded semi-annually in 1 year at 10% per annum is rs 40 more than that of compound interest on the same sum compounded annually at the same time and at the same rate, find the sum.

Answer 1

Given :-

• CI on Semi - annually - CI annually = Rs.40
• Rate is same in Both case = 10% .
• Time is same in Both Case = 1 Year.

To Find :-

• Principal ?

Formula & Concept used :-

• when Interest is compounded semi-annually , Rate Becomes Half , and Time becomes Two Times. (Reason 6 Months * 2 = 1 Year).
• CI = P[ { 1 + (R/100)}^T - 1 ]

Solution :-

Let us Assume That, Given Principal is P.

Than,

Case (1) :- when interest is compounded semi-annually .

→ Rate = 10/2 = 5%

→ Time = 1 * 2 = 2 Years.

→ Principal = P

So,

→ CI = P[ { 1 + (R/100)}^T - 1 ]

→ CI = P[{ 1 + (5/100)}² - 1 ]

→ CI = P [{ 1 + (1/20)}² - 1 ]

→ CI = P [ (21/20)² - 1 ]

→ CI = P [ (441/400) - 1 ]

→ CI = P[ 41/400 ] ----------- Equation (1).

____________________

Case (2) :- when interest is compounded annually .

→ Rate = 10%

→ Time = 1 Years.

→ Principal = P

So,

→ CI = P[ { 1 + (R/100)}^T - 1 ]

→ CI = P[{ 1 + (10/100)} - 1 ]

→ CI = P [{ 1 + (1/10)} - 1 ]

→ CI = P[ (11/10) - 1 ]

→ CI = P ( 1/10) ------------- Equation (2).

___________________

Now, we Have given That :-

➼ CI on Semi - annually - CI annually = Rs.40

➼ Equation (1) - Equation (2) = 40

Putting values of Both Equation we get :-

➻ P[ 41/400] - P[1/10] = 40

➻ P [ (41/400) - (1/10) ] = 40

➻ P [(41 - 40) /400] = 40

➻ P ( 1/400) = 40

➻ P = 40 * 400

➻ P = Rs.16000 (Ans.)

Hence, Required Sum is Rs.16000.

Answer 2

Given :-

The compound interest on a sum of money compounded semi-annually in 1 year at 10% per annum is rs 40 more than that of compound interest on the same sum compounded annually at the same time and at the same rate.

• C.I semi annually - C.I annually = ₹ 40.

• Rate, annually and semi annually = 10 %.

• Time, annually and semi annually = 1 year.

To find :-

The sum (Principal, P).

Solution :-

We know,

$\sf{\displaystyle{CI = P[ { 1 + (\frac{R}{100} )}]^T - 1 }}$

Now,

When interest is compounded semi-annually,

Rate = 5 %    

Time = 2 years

(As we know, when Interest is compounded semi-annually ,rate becomes half , and time becomes two times, because 1 year = 2*6 months.)

Principal = P = ?

Now, putting the known values in the above formula :-

$\displaystyle{\sf{C.I=P[1+(\frac{5}{100})] ^t-1}}\\\\\displaystyle{\sf{\implies C.I=P[1+(\frac{1}{20})]^2-1}}\\\\\\\displaystyle{\sf{\implies C.I=P(\frac{21}{20})^2 -1}}$

$\displaystyle{\sf{\implies C.I=P(\frac{441}{400}-1) }}\\\\\\\displaystyle{\sf{\implies C.I=P(\frac{441-400}{400}) }}\\\\\\\displaystyle{\sf{\implies C.I=P(\frac{41}{400}) }}$                                   ....(i)

When interest is compounded annually,

Rate = 10 %

Time = 1 year

Principal = P = ?

Now, putting the known values in the above formula :-

$\sf{\displaystyle{CI = P[ { 1 + (\frac{10}{100} )}]^1 - 1 }}\\\\\\\displaystyle{\sf{\implies C.I=P[1+\frac{1}{10}]-1 }}\\\\\\\displaystyle{\sf{\implies C.I=P(\frac{11}{10})-1 }}\\\\\\\displaystyle{\sf{\implies C.I=P(\frac{1}{10}) }}$                                  ....(ii)

A/q,

C.I semi annually - C.I annually = ₹ 40

Or, we can say that, eq.(i) - eq.(ii) = ₹ 40

$\displaystyle{\sf{\leadsto P(\frac{41}{400} )-P(\frac{1}{10})=40 }}\\\\\\\sf{\displaystyle{\implies P(\frac{41}{400}-\frac{1}{10})=40 }}\\\\\\\sf{\displaystyle{\implies P(\frac{41-40}{400} )=40}}\\\\\\\sf{\displaystyle{\implies (\frac{P}{400})=40 }}\\\\\\\sf{\displaystyle{\implies P=40*400}}\\\\\\\boxed{\sf{\displaystyle{\implies P=Rs.16000 }}}$

∴ Thus, the required sum is ₹ 16,000.