A depositor deposited Tk. 4,000 at x% simple interest and Tk. 5,000 at y % simple interest. He received annual interest of Tk. 320 on his deposited amounts at the year end. If he could deposit Tk. 5,000 at x% simple interest and Tk. 4,000 at y% simple interest, he would receive annual interest of Tk. 310. Find the value of x and y.

(Interest)

প্রশ্নটিতে বলা হয়েছে, 4,000 টাকা x% এবং 5,000 টাকা y% সুদে ডিপোজিট করলে বছর শেষে মোট 320 টাকা সুদ পাওয়া যায়। এরপর বলা হয়েছে, 5,000 টাকা x% এবং 4,000 টাকা y% সুদে ডিপোজিট করলে বছর শেষে মোট 310 টাকা সুদ পাওয়া যায়। আপনাকে x এবং y এর মান বের করতে হবে।

We know, $I = \frac{prt}{100}$

Where, I = Interest; p = Principal; t = Time

Here, according to question for 1^st & 2^nd part $\frac{4000 \times x \times 1}{100} + \frac{5000 \times y \times 1}{100} = 320 . . . . . . . . \left(i\right)$

And, $\frac{5000 \times x \times 1}{100} + \frac{4000 \times y \times 1}{100} = 310 . . . . . . . . \left(ii\right)$

Now, from (i) we have,

$$\begin{gathered} \frac{4000x}{100} + \frac{5000y}{100} = 320 \\ \Rightarrow \frac{4000x + 5000y}{100} = 320 \\ \Rightarrow 4000x + 5000y = 320 \times 100 \\ \Rightarrow 1000 \left(4x+ 5y\right) = 32 \times 1000 \\ \Rightarrow 4x + 5y = 32 [Divide both side by 1,000] \\ \therefore 4x + 5y = 32 . . . . . . . . \left(iii\right) \end{gathered}$$

Again, from (ii) we have,

$$\begin{gathered} \frac{5000x}{100}+ \frac{4000y}{100} = 310 \\ \Rightarrow \frac{5000x + 4000y}{100}= 310 \\ \Rightarrow 5000x + 4000y = 310 \times 100 \\ \Rightarrow 1000 \left(5x + 4y\right) = 31 \times 1,000 \\ \Rightarrow 5x + 4y = 31 [Divide both side by 1,000] \\ \therefore 5x + 4y = 31 . . . . . . . \left(iv\right) \end{gathered}$$

$Now, \left(iii\right) \times 4-\left(iv\right) \times 5, we have$

$$\begin{gathered} \frac{16x + 20y = 128 \\ 25x + 20y = 155}{-9x = -27} \\ \Rightarrow x = \frac{-27}{-9} \\ \therefore x = 3 \end{gathered}$$

$$\begin{gathered} Now, putting x = 3 in \left(iii\right) \\ (4 \times 3)+5y=32 \\ \Rightarrow 12+5y=32 \\ \Rightarrow 5y=32-12 \\ \Rightarrow 5y=20 \\ \therefore y = \frac{20}{5} = 4 \end{gathered}$$

ans: The required value, x = 3 & y = 4