Please answer the questions please and explain it also! if you want br

1.	Please answer the questions please and explain it also! if you want brainlist so answer questions please!
Answer» ${\underline{{\text{Question}}}}$ The simple interest and compound interest on a certain sum of money is Rs 400 and Rs 410 respectively for 2 yrs. find principal and rate of interest? Step-by-step explanation: ${\underline{{\text{Given}}}}$ Simple Interest (I) = Rs 400 Compound Interest (C.I) = Rs 410 Time (t) = 2 yr ${\underline{{\text{To Find:}}}}$ Principal (P) Rate of interest (R) ${\underline{{\text{Solution:}}}}$ LET, the principal be Rs x and the rate of interest be y % per annum. $\text{So, Simple Interest} = \frac{Prt}{100} \\ \\ = \frac{x \times y \times 2}{100} \\ \\ = \frac{2xy}{100} .........(i)$ Again, $\text{Compound Interest}= P \{(1+\dfrac{r}{100})^{t} - 1 \} \\ \\ = x \{(1+\dfrac{y}{100})^{2} - 1 \} \\ \\ = x (1+\dfrac{y}{100} + 1 ) (1 + \frac{y}{100} - 1) \\ \\ \text{[Using} \: \: {a}^{2} - { b}^{2} = (a + b)(a - b)] \\ \\ = x(2 + \frac{y}{100} )( \frac{y}{100} ) \\ \\ = \frac{xy}{100} (2 + \frac{y}{100} ).......(<klux>II</klux>)$ Now, DIVIDING (ii) by (i) I get, $\frac{\frac{xy}{100} (2 + \frac{y}{100} )}{ \frac{2xy}{100} } = \frac{410}{400} \\ \\ \frac{ \cancel\frac{xy}{100} (2 + \frac{y}{100} )}{ \cancel\frac{2xy}{100} } = \frac{41 \cancel0}{40 \cancel0} \\ \\ \frac{(2 + \frac{y}{100} )}{2} = \frac{41}{40} \\ \\ 2 + \frac{y}{100} = 2 \times \frac{41}{40} \\ \\ \frac{y}{100} = \frac{82}{40} - 2 \\ \\ \frac{y}{100} = \frac{2}{40} \\ \\ \frac{y}{ \cancel{100}} = \frac{1}{ \cancel{20}} \\ \\ y = 5\%$ HENCE, the required rate of interest is 5% . So, $\frac{2xy}{100} = 400 \\ \\ \frac{ 2 \times 5 \times x}{100} = 400 \\ \\ \frac{10x}{100} = 400 \\ \\ x = 4000 \times 10 \\ \\ x = 4000$ Hence, the required Principal is Rs 4000. ${\underline{{\text{ Some Important Formula Of Interest }}}}$ $i)I = \dfrac{ P rt}{100}$ Where, I = Simple Interest P = Principal t = time r = rate of interest $ii)C.I = P \{(1+\dfrac{r}{100})^{t} - 1 \}$ Where, C.I = Compound Interest

Please answer the questions please and explain it also! if you want brainlist so answer questions please!

Answer»

${\underline{{\text{Question}}}}$

The simple interest and compound interest on a certain sum of money is Rs 400 and Rs 410 respectively for 2 yrs. find principal and rate of interest?

Step-by-step explanation:

${\underline{{\text{Given}}}}$

Simple Interest (I) = Rs 400

Compound Interest (C.I) = Rs 410

Time (t) = 2 yr

${\underline{{\text{To Find:}}}}$

Principal (P)

Rate of interest (R)

${\underline{{\text{Solution:}}}}$

LET, the principal be Rs x and the rate of interest be y % per annum.

$\text{So, Simple Interest} = \frac{Prt}{100} \\ \\ = \frac{x \times y \times 2}{100} \\ \\ = \frac{2xy}{100} .........(i)$

Again,

$\text{Compound Interest}= P \{(1+\dfrac{r}{100})^{t} - 1 \} \\ \\ = x \{(1+\dfrac{y}{100})^{2} - 1 \} \\ \\ = x (1+\dfrac{y}{100} + 1 ) (1 + \frac{y}{100} - 1) \\ \\ \text{[Using} \: \: {a}^{2} - { b}^{2} = (a + b)(a - b)] \\ \\ = x(2 + \frac{y}{100} )( \frac{y}{100} ) \\ \\ = \frac{xy}{100} (2 + \frac{y}{100} ).......(<klux>II</klux>)$

Now, DIVIDING (ii) by (i) I get,

$\frac{\frac{xy}{100} (2 + \frac{y}{100} )}{ \frac{2xy}{100} } = \frac{410}{400} \\ \\ \frac{ \cancel\frac{xy}{100} (2 + \frac{y}{100} )}{ \cancel\frac{2xy}{100} } = \frac{41 \cancel0}{40 \cancel0} \\ \\ \frac{(2 + \frac{y}{100} )}{2} = \frac{41}{40} \\ \\ 2 + \frac{y}{100} = 2 \times \frac{41}{40} \\ \\ \frac{y}{100} = \frac{82}{40} - 2 \\ \\ \frac{y}{100} = \frac{2}{40} \\ \\ \frac{y}{ \cancel{100}} = \frac{1}{ \cancel{20}} \\ \\ y = 5\%$