Question

A can contains a mixture of two liquids A and B is the ratio 7:5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7:9. How many litres of liquid A was contained by the can initially?
A.10
B.20
C.21
D.25

Answer 1

Hint: The answer could easily be found out by assuming the litre of liquid as x in the initial ratio. When 9 litres are drawn off from the mixture, the 9 litres contained liquids A and B in 7:5 ratio, and the rest added was only B and not the ratio of A and B.

Complete answer:
The correct answer to this question is option C, 21 litre of liquid A was contained initially.
Let us see how, the ratio in which the solution contains A and B is 7:5 respectively. Thus assuming x litre of solution, we can say that
\[A=7x\] and \[B=5x\]
Now, in 9 litre of mixture the amount of liquid A is
\[A=\dfrac{7x}{12x}\times 9\], here 12x is the total amount of solution with respect to x.
Therefore, \[A=\dfrac{21}{4}l\], where l represents litre.
Similarly for liquid B,
\[B=\dfrac{5x}{12x}\times 9\]
Therefore, \[B=\dfrac{15}{4}l\]
Now, this was for the old solution. For the new solution we have to replace the old amount of liquid with a new amount. For liquid A, the amount of new solution, after drawing off 9 litres of solution, will be
\[A=7x-\dfrac{21}{4}\] litres
And similarly, for liquid B
\[B=5x-\dfrac{15}{4}\] litres
Now, after filling the solution with liquid B, more 9 litres of liquid B will be added to the solution.
Therefore, \[B=5x-\dfrac{15}{4}+9\] litres.
Now, the ratio of new solutions is given in the question, which is 7:9.
Therefore, \[\dfrac{7x-\dfrac{21}{4}}{5x-\dfrac{15}{4}+9}=\dfrac{7}{9}\]
After solving it, we get \[\dfrac{28x-21}{20x+21}=\dfrac{7}{9}\]
Therefore, \[252x-189=140x+147\]
Therefore, \[112x=336\]
Therefore, \[x=3\]
As x is found, the initial amount of liquid A in the first solution can be found out easily.
Therefore as said above, \[A=7x\], putting x equals 3,
\[A=7\times 3=21\] litres.
Therefore, 21 litres of liquid A was there initially in the solution.

Note:
Assumption is necessary in these kinds of questions, without assuming the amount of liquid correct answer cannot be found. Also, when 9 litre of solution is drawn off from the solution, it was taken in the ratio of 7:5 of liquids A and B respectively. And then only liquid B was added in the solution, not any mixture.