Question

a man borrows money at 3% per annum interest payable yearly and lends it immediately at 5% per annum interest payable half-yearly and gains thereby rupees 165 at the end of year what sum of money does he borrows​

Answer 1

Given :

• a man borrows money at 3% per annum interest payable yearly.

• lends it immediately at 5% per annum interest payable half-yearly .

• Gains thereby rupees 165 at the end of year

To Find :

• what sum of money does he borrows

Solution :

Let sum is x

Interest for one Year :

$: \implies \sf \: \: \: \: \frac{x \times 3}{100} \\ \\ \\ : \implies \sf \: \: \: \: \frac{3x}{100}$

Interest for 6 month :

$: \implies \sf \: \: \: \: \frac{x \times 5}{2 \times 100} \\ \\ \\ : \implies \sf \: \: \: \: \cancel{\frac{5x}{200}} \\ \\ \\ : \implies \sf \: \: \: \: \frac{x}{40}$

Interest for last 6 month :

$: \implies \sf \: \: \: \: \: \: \frac{x}{40} + \frac{ \frac{x}{8} }{2 \times 100} \\ \\ \\ \\ : \implies \sf \: \: \: \: \: \: \frac{x}{40} + \frac{x }{8} \times \frac{1}{200} \\ \\ \\ : \implies \sf \: \: \: \: \: \: \frac{x}{40} + \frac{x}{1600}$

His net gain :

$: \implies \sf \: \: \: \: \frac{x }{40} + \frac{x}{40} + \frac{x}{1600} - \frac{3x}{100} \\ \\ \\ : \implies \sf \: \: \: \: \frac{33x}{1600}$

Now we have :

$: \implies \sf \: \: \: \: \frac{33x}{1600} = 165\\ \\ \\ \\ : \implies \sf \: \: \: \: x = \frac{ 1600 \times \cancel{165}}{ \cancel{33} } \\ \\ \\ : \implies \sf \: \: \: \: x = 1600 \times 5 \\ \\ \\ : \implies \sf \: \: \: \: x = 8000$

Hence 8000 Rs. he borrows

Answer 2

Answer:

Let principal be x.

Interest payable yearly = PRT/100 = 3x/100

Interest payable half-yearly = p{(1+r/200)^2n - 1 }

= 81x/1600

ATQ

81x/1600 - 3x/100 = 330

.............=330

.............

33x/1600 = 330

x = 330×1600/33 = 16000 (Ans)