I did not understand the approach needed to solve this problem on inverse proportionality

Question

The concentration of an acid solution is inversely proportional to the volume the solution if the amount of the acid is not changed. A $40$% hydrochloric acid solution becomes $30$% solution when $30$ L of water is added to it. Find the original volume of the solution.

I understood this much that ratio of the concentration will be $4/3$ hence,ratio of the volume will be $3/4$ since it has been given in the above question that concentration is inversely proportional to the volume. But, I did not understand how to solve the question ahead when 30L is added to $40%$ of the hydrochloric acid.

Answer 1

Hints:

Volume 2=Volume 1+30L.

Volume 1$\times$Concentration 1=Volume 2$\times$Concentration 2

Answer 2

We can proceed this way. $$\begin{array} &\text{Total volume } & \text{ Water } & \text{ HCl }\\ x & 0.6x & 0.4 x\\ x+30 & 0.6x +30 & 0.4x\\ \end{array}$$

Thus, we have $$\frac{0.4x}{x+30} = \frac{3}{10} \Rightarrow x =90$$ Hope it helps.

Answer 3

Let $a$ denote the volume of the acid.

Let $w$ denote the volume of the water.

Then we need to solve the following system of $2$ equations in $2$ variables:

• $\frac{a}{a+w}=\frac{40}{100}$
• $\frac{a}{a+w+30}=\frac{30}{100}$

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$\frac{a}{a+w}=\frac{40}{100}\implies100a=40a+40w\implies60a=40w\implies3a=2w$

$\frac{a}{a+w+30}=\frac{30}{100}\implies100a=30a+30w+900\implies70a=30w+900\implies7a=3w+90$

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$[3a=2w]\wedge[7a=3w+90]\implies$

$[21a=14w]\wedge[21a=9w+270]\implies$

$[14w=9w+270]\implies$

$[5w=270]\implies$

$[w=54]$

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$[3a=2w]\wedge[w=54]\implies$

$[3a=108]\implies$

$[a=36]$

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Hence the original volume of the solution is $54+36=90$.

Answer 4

Let Initial concentration = a

Final concentration = b

Initial volume = x

Final volume = y

Concentration of an acid solution is inversely proportional to the volume.

a inversely proportional to x.

a = k.$\frac{1}{x}$ (some constant)

a.x = k

Also,

b.y = k

So from both equations we have

a.x = b.y

40% . x = 30% . y

y = $\frac{4}{3}$x

Also,

y - x = 30

$\frac{4}{3}$x - x = 30

x = 90

So intial volume is 90.