Question

Nihal took a loan at 5% simple interest for
three years and lent it at 5% compound interest,
compounded annually, for the same period. As
a result he made a profit of Rs.76.25. Find the
principal.​

Answer 1

Given :-

• Rate on simple interest = 5%
• Rate on compound interest = 5%
• Total profit earn by them = 76.25

To Find :-

• Loan taken by Nihal = ?

Solution :-

To calculate the sum of money which is taken by Nihal at first we have find simple interest and compound interest by substituting the value.

Calculation for SI :-

⟹ Let , Sum of money be "P"

• P = P T = 3 years. R = 5%

⟹ SI = P × T × R / 100

⟹ SI = P × 3 × 5 / 100

⟹ SI = 3P / 20

⟹ A = P + SI

⟹ A = P + 3P / 20

⟹ A = 23P / 20 -------(i)

Calculation for CI :-

⟹ Let , Sum of money be "P"

• P = P. T = 3 years R = 5%

⟹ CI = P ( 1 + r/100 )^n - P

⟹ CI = P ( 1 + 5 / 100 ) ³ - P

⟹ CI = P ( 1 + 1 /20 )³ - P

⟹ CI = P ( 21/20 )³ - P

⟹ CI = 9261P / 8000 - P

⟹ CI = 9261P - 8000P / 8000

⟹ CI = 1261P / 8000

⟹ A = P + 1261P / 8000

⟹ A = 9261P / 8000 -------(ii)

Now calculate loan taken by Nihal :-

⟹ Substracting equation from (ii) and (i)

⟹ Amount for CI - Amount for SI = Profit

⟹ 9261P / 8000 - 23P / 20 = 76.25

⟹ 9261P - 9200P / 8000 = 76.25

⟹ 61P / 8000 = 76.25

⟹ 61P = 76.25 × 8000

⟹ 61P = 610000

⟹ P = 10000

Hence,

• The loan taken by Nihal = Rs. 10000

Answer 2

Given :-

Nihal took a loan at 5% simple interest for  three years and lent it at 5% compound interest,  compounded annually, for the same period. As  a result he made a profit of Rs.76.25.

To Find :-

Principal

Solution :-

According to the question

$\sf SI <CI=Profit$

So,

Profit = CI - SI

Let the principal be P

$\sf 76.25 = \bigg\lgroup P \times \bigg(1 + \dfrac{R}{100}\bigg)^n - P \bigg \rgroup -\bigg\lgroup \dfrac{PRT}{100} + P\bigg\rgroup$

$\sf 76.25 = \bigg\lgroup P \times \bigg(1 + \dfrac{5}{100}\bigg)^3 - P\bigg\rgroup - \bigg\lgroup \dfrac{P \times 5 \times 3}{100}+P\bigg\rgroup$

$\sf 76.25 = \bigg\lgroup P \times \bigg(\dfrac{100 +5}{100}\bigg)^3 - P\bigg\rgroup-\bigg\lgroup \dfrac{15P+100P}{100}\bigg\rgroup$

$\sf 76.25 =\bigg\lgroup P\times\dfrac{21}{20}\times \dfrac{21}{20} \times \dfrac{21}{20} - P\bigg\rgroup- \dfrac{115P}{100}$

$\sf 76.25 = \bigg(\dfrac{9261P - 8000P}{8000}\bigg) - \dfrac{115P}{100}$

$\sf 76.25= \dfrac{1261P + 8000P}{8000} - \dfrac{115P}{100}$

$\sf 76.25=\dfrac{9261P - 9200P}{8000}$

$\sf 76.25\times 8000 = 61P$

$\sf 61000=61P$

$\sf\dfrac{61000}{61} = P$

$\sf 10000 = P$