Question

8.
fü) If(10x+3y):(5x+2y)=9:5, let us show that, (2x+y): (x+2y)=11:13
Let us calculate what term should be added to both terms of the ratio 2:5 to make the
ratio 6:11
​

Answer 1

Answer:

11/13   and   8/5

Step-by-step explanation:

$\implies \sf{\dfrac{10x+3y}{5x+2y} = \dfrac{9}{5} }\\\\\\\implies \sf{\dfrac{10x\frac{y}{y} +3y}{5x\frac{y}{y}+2y}=\dfrac{9}{5} }\\\\\\\implies \sf{\dfrac{y\big(10\frac{x}{y}+3\big)}{y\big(5\frac{x}{y}+2\big)}=\dfrac{9}{5} }$

Let $\dfrac{x}{y}=k$,

$\implies\sf{\dfrac{10k+3}{5k+2}=\dfrac{9}{5}} \\\\\implies\sf{5(10k+3)=9(5k+2) }\\\\\implies\sf{k=\dfrac{3}{5}}$

   Hence,

$\implies \sf{\dfrac{2x+y}{x+2y} }\\\\\\\implies \sf{\dfrac{2x\frac{y}{y} +y}{x\frac{y}{y}+2y} }\\\\\\\implies \sf{\dfrac{y\big(2\frac{x}{y}+1\big)}{y\big(\frac{x}{y}+2\big)} }$

$\implies\sf{\dfrac{2k+1}{k+2} =\dfrac{2\big(\frac{3}{5}\big)+1}{\frac{3}{5}+2} }\\\\\implies\sf{ \dfrac{11}{13} }$

Question 2:

Let the x should be added,

⇒ (2 + x)/(5 + x) = 6/11

⇒ 11(2 + x) = 6(5 + x)

⇒ 22 + 11x = 30 + 6x

⇒ 11x - 6x = 30 - 22

⇒ 5x = 8

⇒ x = 8/5

      8/5 should be subtracted

Answer 2

Answer:

Question No 1 :-

$\longrightarrow$ If (10x + 3y) : (5x + 2y) = 9 : 5, let us show that (2x + y) : (x + 2y) = 11 : 13.

Given :-

$\mapsto$ $\sf (10x + 3y) : (5x + 2y) = 9 : 5.$

Show That :

$\mapsto$ $\sf (2x + y) : (x + 2y) = 11 : 13.$

Solution :-

$\implies$ $\sf 10x + 3y : 5x + 2y = 9 : 5$

Then, we can write as :

$\implies$ $\sf \dfrac{10x + 3y}{5x + 2y} =\: \dfrac{9}{5}$

By doing cross multiplication we get,

$\implies$ $\sf 5(10x + 3y) =\: 9(5x + 2y)$

$\implies$ $\sf 50x + 15y =\: 45x + 18y$

$\implies$ $\sf 50x - 45x =\: 18y - 15y$

$\implies$ $\sf 5x =\: 3y$

$\implies$ $\sf \dfrac{x}{y} =\: \dfrac{3}{5}$

Let,

$\longmapsto$ $\sf \dfrac{x}{y} =\: \dfrac{3}{5} =\: k\: [where\: k \neq 0]$

Then,

➲ $\sf x =\: 3k$

➲ $\sf y =\: 5k$

Now by taking L.H.S = (2x + y) : (x + 2y)

↦ $\sf (2 \times 3k + 5k) : (3k + 2 \times 5k)$

↦ $\sf (6k + 5k) : (3k + 10k)$

↦ $\sf 11{\cancel{k}} : 13{\cancel{k}}$

➠ $\bold{\red{11 : 13}}$ = R.H.S

$\leadsto$ $\sf\boxed{\bold{\pink{(PROVED)}}}$

_______________________________________

Question No 2 :-

$\longrightarrow$ Let us calculate what term should be added to both terms of the ratio 2 : 5 to make the ratio 6 : 11.

Solution :-

Let, the number should be added be x

Now, according to the question,

⇒ $\sf \dfrac{2 + x}{5 + x} =\: \dfrac{6}{11}$

By doing cross multiplication we get,

⇒ $\sf 11(2 + x) =\: 6(5 + x)$

⇒ $\sf 22 + 11x =\: 30 + 6x$

⇒ $\sf 11x - 6x =\: 30 - 22$

⇒ $\sf 5x =\: 8$

➠ $\sf\bold{\red{x =\: \dfrac{8}{5}}}$

$\therefore$ $\sf\bold{\dfrac{8}{5}}$ is should be added to the number.