Question

A vessel is filled with liquid, 3 parts of which is water and 5 parts syrup. What part of the mixture should be drawn off and replaced with water so that the mixture contains half water and half syrup?

• A. \( \frac{1}{4} \)
• B. \( \frac{1}{5} \)
• C. \( \frac{1}{6} \)
• D. \( \frac{1}{7} \)

Hint:

When solving mixture problems, write equations based on the change in the amount of each component after the replacement, and use the desired proportion to solve for the unknown.

Answer 1

The Correct Option is B

Let the total volume of the liquid in the vessel be \( V \). Initially, the liquid consists of 3 parts water and 5 parts syrup, so the total amount of water is \( \frac{3}{8}V \) and the total amount of syrup is \( \frac{5}{8}V \). Let \( x \) be the fraction of the mixture to be drawn off and replaced with water. After this, the amount of water in the mixture becomes: \[ \text{Water} = \frac{3}{8}V - x \times \frac{3}{8}V + x \times V = \frac{3}{8}V(1 - x) + xV \] This should be half of the total mixture, i.e., \( \frac{1}{2}V \). Thus, we set up the equation: \[ \frac{3}{8}V(1 - x) + xV = \frac{1}{2}V \] Simplifying: \[ \frac{3}{8}(1 - x) + x = \frac{1}{2} \] Multiply through by 8: \[ 3(1 - x) + 8x = 4 \] Expanding: \[ 3 - 3x + 8x = 4 \] Simplifying: \[ 3 + 5x = 4 \] Solving for \( x \): \[ 5x = 1 \quad \Rightarrow \quad x = \frac{1}{5} \] Thus, the fraction of the mixture that should be drawn off and replaced with water is \( \frac{1}{5} \).