Question

A mixture contains apple juice and water in the ratio 10 : x. When 36 litres of the mixture and 9 litres of water are mixed, the ratio of apple juice and water becomes 5 : 4. The value of x is:

• A. 4
• B. 4.4
• C. 5
• D. 8

Hint:

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.

Answer 1

The Correct Option is B

Let the amount of apple juice in the mixture be 10 parts and water be \(x\) parts. The total quantity of the mixture is \(10 + x\) parts.

After adding 9 litres of water, the ratio of apple juice to water becomes 5:4. This gives:
\(\frac{36 \cdot \frac{10}{10+x}}{36 \cdot \frac{x}{10+x} + 9} = \frac{5}{4}.\)

Simplify:
\(\frac{360}{10+x} = \frac{5}{4} \left( \frac{36x}{10+x} + 9 \right).\)

Clear the fraction and simplify:
\(1440 = 180x + 45(10 + x),\)
\(1440 = 180x + 450 + 45x \quad \Rightarrow \quad 1440 - 450 = 225x.\)
\(990 = 225x \quad \Rightarrow \quad x = \frac{990}{225} = 4.4.\)

Thus, \(x = 4.4\)

Answer 2

Let the amount of apple juice in the mixture be 10 parts and water be \( x \) parts. The total quantity of the mixture is \( 10 + x \) parts.

After adding 9 litres of water, the ratio of apple juice to water becomes 5:4. This gives:

\[ \frac{36 \cdot \frac{10}{10 + x}}{36 \cdot \frac{x}{10 + x} + 9} = \frac{5}{4}. \]

Step 1: Simplify the equation:

Simplifying both sides: \[ \frac{360}{10 + x} = \frac{5}{4} \left( \frac{36x}{10 + x} + 9 \right). \]

Step 2: Clear the fraction and simplify:

Multiply both sides by 4 to eliminate the denominator on the right-hand side: \[ 1440 = 180x + 45(10 + x). \] Expanding the right-hand side: \[ 1440 = 180x + 450 + 45x. \]

Step 3: Combine like terms:

Combine the \( x \)-terms: \[ 1440 = 225x + 450. \] Subtract 450 from both sides: \[ 1440 - 450 = 225x. \] Simplifying: \[ 990 = 225x. \]

Step 4: Solve for \( x \):

Divide both sides by 225: \[ x = \frac{990}{225} = 4.4. \]

Conclusion: The value of \( x \) is \( 4.4 \).