# How many liters of a 60% sugar solution should be added to 40 liters of a 30% sugar solution to produce a 50% sugar solution?

Jun 12, 2018

$80 \text{ litres}$

#### Explanation:

Let the amount of 60% solution be ${S}_{60}$
Let the amount of 30% solution be ${S}_{30} = 40 \text{ litres}$
Let the amount of 50% solution be ${S}_{50} = 40 + {S}_{60} \text{ litres}$

So for this to balance we have:

$\frac{60}{100} {S}_{60} + \left(\frac{30}{100} \times {S}_{30}\right) = \frac{50}{100} \textcolor{w h i t e}{\text{d}} {S}_{50}$

$\frac{60}{100} {S}_{60} + \left(\frac{30}{100} \times 40\right) = \frac{50}{100} \left({S}_{60} + 40\right)$

$\frac{3}{5} {S}_{60} + 12 = \frac{1}{2} {S}_{60} + 20$

$\frac{3}{5} {S}_{60} - \frac{1}{2} {S}_{60} = 20 - 12$

$\frac{1}{10} {S}_{60} = 8$

${S}_{60} = 80 \text{ litres}$
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Check:

$\left(\frac{60}{100} \times 80\right) + \left(\frac{30}{100} \times 40\right) = 48 + 12 = 60$ of solute

60/(40+80)xx100%=50%color(brown)(larr"Answer confimed"

Note that % is a unit of measurement that is worth $\frac{1}{100}$
Consequently you are doing the equivalent of multiplying by 1 when you write: xx100% -> 100xx1/100=100/100 = 1