Question

One test tube contains some acid and another test tube contains an equal quantity of water. To prepare a solution, 20 g of the acid is poured into the second test tube. Then, two-thirds of the so formed solution is poured from the second test tube into the first. If the fluid in the first test tube is four times that in second, what quantity of water was taken initially.

Answer 1

Hint: Assume unknown quantity denoted by x (variable), form the mathematical equations as stated in the language of question.
The word ‘twice’ implies the multiplication by ‘2’, similarly, ‘two times’ also implies multiplication by ‘2’.

Complete step by step solution:
Step 1
Let the amount of water in second test tube $ = x$ g

Step 2
Amount of acid in first test tube = amount of water in second test tube $(\because {\text{given}})$
                   $\therefore $ amount of acid in first test tube $ = x$ g

Step 3
20 g of acid in poured into second test tube to form solution: $(\because {\text{stated in question)}}$
\[\therefore \] amount of solution in second test tube $ = (x + 20)$ g
$\because $ amount of acid in first test tube $ = (x - 20)$ g

Step 4
Two-thirds of the solution of second test tube is poured into the first test tube:
$(\because {\text{stated in question)}}$
$\therefore $amount of fluid in first test tube $ = [(x - 20) + \dfrac{2}{3}(x + 20)]$ g
$\because $amount of solution left in second test tube $ = \dfrac{1}{3}(x + 20)$ g
Step 5
Amount of fluid in first test tube (in g) = four times the amount of fluid in second test tube (in g)
                                                                                                                                                         $(\because {\text{given}})$
               $[(x - 20) + \dfrac{2}{3}(x + 20)] = 4 \times \dfrac{1}{3}(x + 20)$
\[ \Rightarrow \] $x - 20 + \dfrac{{2x}}{3} + \dfrac{{40}}{3} = \dfrac{{4x}}{3} + \dfrac{{80}}{3}$
$ \Rightarrow $ $x + \dfrac{{2x}}{3} - \dfrac{{4x}}{3} = \dfrac{{80}}{3} + 20 - \dfrac{{40}}{3}$
$ \Rightarrow $ \[\dfrac{{3x + 2x - 4x}}{3} = \dfrac{{80 + 60 - 40}}{3}\]
$ \Rightarrow $ $\dfrac{x}{{{3}}} = \dfrac{{100}}{{{3}}}$
$ \Rightarrow $ x = 100

Therefore, the amount of water in the second test tube is x = 100 g.

Note:
Similar patterns should be followed in the questions of one unknown variable and/or more variables.
In question language ‘more than’ indicates addition to the latter sentence and ‘less than’ indicates subtraction from the latter sentence.
Students are likely to make mistakes while shifting the terms from one side of the equation to another, keep in mind that sign always changes when shifted to another side of the equation.