Question

A sum becomes 450 in certain a
years at rate of 7% and it
becomes 350 in same time if
the rate is 5%. Find the
principle and the time:​

Answer 1

Given :-

• Amount after n time = ₹450 at 7% interest rate.
• Amount after n time = ₹350 at 5% interest rate.

Aim :-

• To find the Principal and time

Answer :-

Let the principal be x and time be y.

$\boxed {\sf Interest = Amount - Principal}$

Let Interest be a and b respectively.

Equation 1 :

450 = x + a

⇒ 450 - a = x

Equation 2 :

350 = x + b

⇒ 350 - b = x

From the above two equations, we can conclude that,

⇒ 350 - b = 450 - a

→ - b + a = 450 - 350

→ a - b = 100

→ a = 100 + a

Formula to use :

$\boxed {\sf Simple\:Interest = \dfrac{Principal \times time \times rate}{100}}$

Simple interest 1 :

• Simple interest = a ⇒ 100 + b
• Principal = 450 - a ⇒ 350 - b
• Time = n
• Rate = 7%

Substituting,

$\implies \sf 100 + b = \dfrac{(350 - b) \times 7 \times n}{100}$

Transposing 100,

$\implies \sf 100(100 + b) = 7(350 - b) \times n$

Transposing 7(350 - b),

$\implies \sf \dfrac{100(100 +b) }{7(350 - b)} = n$

Simple interest 2 :

Let us substitute n for 100(100 + b)/7(350 -b),

$\implies \sf b = \dfrac{(350 - b)\times 5 \times \dfrac{100(100+b)}{7(350-b)} }{100}$

Transposing 100,

$\implies \sf 100b = 5(350 - b) \times \dfrac{100(100 +b)}{7(350-b)}$

Cancelling (350 - b) in the RHS (Right hand side of the equation),

$\implies \sf 100b = 5 \times \dfrac{100(100+b)}{7}$

$\implies \sf 700b = 500(100 +b)$

Transposing 500,

$\implies \sf \dfrac{700b}{500} = 100 + b$

$\implies \sf \dfrac{7\not0\not0b}{5\not0\not0} = 100 + b$

Transposing 5,

$\implies \sf 7b = 5(100 + b)$

$\implies \sf 7b = 500 + 5b$

Transposing 5b to the LHS (Left hand side),

$\implies \sf 7b - 5b = 500$

$\implies \sf 2b = 500$

Transposing 2,

$\implies \sf b = \dfrac{500}{2}$

$\implies \sf b = 250$

Therefore,

⇒ 350 - b = x

→ 350 - 250 = x

→ ₹100 = x

Now that we know the principal,

Time :

$\implies \sf n = \dfrac{100(100+b)}{7(350-b)}$

Substituting b from 250,

$\implies \sf n = \dfrac{100(100+250)}{7(350-250)}$

$\implies \sf n = \dfrac{100 \times 350}{7 \times 100}$

⇒ n = 50 years.

More formulas :

$\boxed {\sf Time = \dfrac{Simple\:interest \times 100}{Principal \times rate} }$

$\boxed{\sf Rate = \dfrac{Simple\:interest\times 100}{Principal\times time} }$

$\boxed {\sf Principal = \dfrac{Simple\:interest\times 100}{Rate\times time} }$