A large bowl contains a mixture of Himalayan Pink Salt and common salt. When \(1\) kg of common salt is added to the bowl, the ratio, by mass, of Himalayan Pink Salt to common salt becomes \(1 : 2\). When \(1\) kg of Himalayan Pink Salt is added to the new mixture, the ratio becomes \(2 : 3\). Find the ratio of Himalayan Pink Salt to common salt in the original mixture.
Let \(h\) be the amount of Himalayan
Pink Salt, in kgs, in the original mixture.
Let \(c\) be the amount of common salt,
in kgs, in the original mixture.
When \(1\) kg of common salt is added, the ratio of Himalayan Pink Salt to common salt is \(1:2\). Therefore,
\[\frac{h}{c+1} = \dfrac{1}{2}\]
Simplifying, we obtain \(c+1=2h\) and \(c=2h-1\) follows.
When \(1\) kg of Himalayan Pink Salt is added to the new mixture, the ratio becomes \(2:3\). Therefore,
\[\frac{h+1}{c+1} = \frac{2}{3}\]
Since \(c=2h-1\), we have \[\begin{aligned} \frac{h+1}{(2h-1)+1} &= \frac{2}{3}\\ \frac{h+1}{2h} &= \frac{2}{3}\\ 2(2h)&=3(h+1)\\ 4h&=3h+3\\ h&=3 \end{aligned}\]
Substituting \(h=3\) in \(c= 2h-1\), we obtain \(c=2(3)-1=5\).
Therefore, there was originally \(3\) kgs of Himalayan Pink Salt in the bowl and \(5\) kgs of common salt. Thus, the ratio of Himalayan Pink Salt to common salt in the original mixture was \(3:5\).