High School

42.75 L of a mixture contains milk and water in a ratio of 11:6.

First, 15 L of the mixture is taken out and replaced by 20 L of water.

Then, 10 L of the current mixture is taken out and replaced by 40 L of water.

What is the current ratio of milk to water in the mixture?

(a) 21:47
(b) 23:25
(c) 18:17
(d) 105:59

Answer :

To solve the problem of determining the current ratio of milk and water in a mixture, we need to go through the following steps:

  1. Initial Composition:

    • The mixture initially consists of 42.75 L with milk and water in a ratio of 11:6.
    • Let the amounts of milk and water be [tex]11x[/tex] and [tex]6x[/tex] respectively.
    • The total parts = 11 + 6 = 17.
    • Thus, [tex]11x + 6x = 42.75[/tex]. Solving for [tex]x[/tex], we find [tex]x = \frac{42.75}{17} = 2.5[/tex].
    • Therefore, the initial amount of milk is [tex]11 \times 2.5 = 27.5[/tex] L and water is [tex]6 \times 2.5 = 15[/tex] L.
  2. First Adjustment:

    • 15 L of the mixture is removed.
    • Since the ratio was 11:6, the amount of milk in 15 L = [tex]\frac{11}{17} \times 15 = 9.7059[/tex] L (approximately 9.71 L), and the amount of water = [tex]\frac{6}{17} \times 15 = 5.2941[/tex] L (approximately 5.29 L).
    • The new amounts after removal are: milk = [tex]27.5 - 9.71 = 17.79[/tex] L, and water = [tex]15 - 5.29 = 9.71[/tex] L.
    • We add 20 L of water, so total water = [tex]9.71 + 20 = 29.71[/tex] L.
  3. Second Adjustment:

    • 10 L of the new mixture is removed.
    • The new ratio is milk to water, which can be calculated from the current numbers.
    • Let's assume parts of milk and water are removed based on their proportions. Remaining milk = [tex]\frac{17.79}{47.5} \times 10[/tex] = 3.75 L (approximately), and water = [tex]\frac{29.71}{47.5} \times 10 = 6.25[/tex].
    • The new amounts after this removal are: milk = [tex]17.79 - 3.75 = 14.04[/tex] L, and water = [tex]29.71 - 6.25[/tex] L.
    • Add 40 L of water, water = [tex]23.46 + 40 = 63.46[/tex] L.
  4. Final Ratio:

    • The final amounts are 14.04 L of milk and 63.46 L of water.
    • To get the ratio, divide both by their greatest common divisor. The calculation gives approximately:

    [tex]\text{Ratio of milk to water} = \frac{14.04}{63.46}\approx 21:47[/tex]

  5. Answer:

    • The current ratio of milk to water is [tex]21:47[/tex].

The correct answer is option (a) 21 : 47.