When solving ratio problems involving mixtures, the key step is to set up an equation that represents the relationship between the parts. In this case, you used the given ratio of apple juice to water after adding water and converted it into an equation. Always ensure that you carefully simplify and clear fractions by multiplying through to avoid complex fractions. Once you have a linear equation, solving for the unknown becomes straightforward.
Let the quantity of apple juice be 10 parts and the quantity of water be \(x\) parts. The total mixture quantity is \(10 + x\) parts.
Upon adding 9 litres of water, the ratio of apple juice to water becomes 5:4, which can be represented as:
\(\frac{36 \cdot \frac{10}{10+x}}{36 \cdot \frac{x}{10+x} + 9} = \frac{5}{4}.\)
Simplifying the equation yields:
\(\frac{360}{10+x} = \frac{5}{4} \left( \frac{36x}{10+x} + 9 \right).\)
Clearing the fractions and simplifying further:
\(1440 = 180x + 45(10 + x),\)
\(1440 = 180x + 450 + 45x \quad \)
\(\Rightarrow \quad 1440 - 450 = 225x.\)
\(990 = 225x \quad \)
\(\Rightarrow \quad x = \frac{990}{225} = 4.4.\)
Therefore, \(x = 4.4\).