Question

How many grams of water must be evaporated from $100$ kilograms of a $40\%$ salt solution to produce a $50\%$ salt solution?

Answer 1

Hint: In this question we have been provided with the data that we have a $40\%$ salt solution which constitutes a total of $100$ kilograms. We need to evaporate a certain amount of water to make the solution a $50\%$ salt solution without increasing or decreasing the quantity of the salt present in the solution. We will consider the quantity of water evaporated to be $x$ and solve for its value to get the required solution.

Complete step by step solution:
We know that the solution is a $40\%$ salt solution. This implies:
$\Rightarrow \text{40 }\!\!\%\!\!\text{ =}\dfrac{\text{40 kg salt}}{\text{100 kg solution}}$
Now consider the quantity of water to be removed to get a $50\%$ solution to be $x$.
This means that we have to subtract $x$ from the $\text{100 kg}$ solution.
And since there is no other alteration, we can write it as:
$\Rightarrow \dfrac{40}{100-x}=50\%$
Now we know that $50\%=\dfrac{50}{100}$ therefore, on substituting, we get:
$\Rightarrow \dfrac{40}{100-x}=\dfrac{50}{100}$
On cross multiplying, we get:
$\Rightarrow 40\times 100=50\left( 100-x \right)$
On simplifying both the sides, we get:
$\Rightarrow 4000=5000-50x$
On transferring the term $-50x$ from the right-hand side to the left-hand side, we get:
$\Rightarrow 50x+4000=5000$
On transferring the term $4000$ from the left-hand side to the right-hand side, we get:
$\Rightarrow 50x=5000-4000$
On simplifying, we get:
$\Rightarrow 50x=1000$
On transferring the term $50$ from the left-hand side to the right-hand side, we get:
$\Rightarrow x=\dfrac{1000}{50}$
On simplifying, we get:
$\Rightarrow x=20$, which implies that we have to evaporate $20$ kilograms of water to make the $40\%$ salt solution to a $50\%$ salt solution.

Note: It is to be remembered that in real life the maximum solubility of salt in water is $26\%$, upon adding more salt in the solution, it will never completely dissolve therefore, this question is a hypothetical question. It is to be remembered that water is generally measured in liters and $\text{1 liter water = 1 kilogram water}$.