What should I do to get how much juice is to be added?

Question

A certain brand of beverage contains $60\%$ orange and $40\%$ mango juice. How much mango juice must be added to $2.4$ liters of the beverage in order to get a mixture which is $30\%$ orange and $70\%$ mango juice ?

Answer 1

If you have $V_t = 2.4 $ litres of your beverage at first then it means that there is $V_o=0.6\times 2.4 = 1.44$ litre of orange juice and $V_m =0.4 \times 2.4 = 0.96$ litre of mango juice .

You wonder what is the volume $x$ of mango juice to add in order you to have $30\%$ of orange juice and $70 \%$ of mango juice in your beverage.

So you can solve:

$$V_o=30 \% (V_t+x)=0.3(V_t+x)$$

Therefore:

$$ x = \dfrac{V_0}{0.3}-V_t = 2.4 \text{ litres}$$

But this could have done more quickly, because if you only add mango juice then if you want the proportion of orange juice in the beverage to be half what it was, it means you just need to add mango juice for the total volume to be the double as it was originally.