Math Problem Statement

To make a certain profit, a milkman mixes 30 liters of water in 120 liters of milk. He adds up water to the mixture, after selling one third of it, in order to make up the loss. What is the water to milk ratio at the moment?

a. 3:7 b. 4:9 c. 6:11 d. 7:12

Solution

To solve this, let’s break down the problem:

  1. Initial Mixture:

    • The milkman starts with 120 liters of milk and 30 liters of water.
    • Total volume of the mixture = 120+30=150120 + 30 = 150 liters.
  2. Mixture Sold:

    • He sells one-third of the total mixture, which is 13×150=50\frac{1}{3} \times 150 = 50 liters.
    • The 50 liters sold will contain both milk and water in the same ratio as the original mixture.

    The initial milk-to-water ratio is: 12030=4:1\frac{120}{30} = 4:1

  3. Amount of Milk and Water in 50 Liters Sold:

    • Water in the 50 liters = 15×50=10\frac{1}{5} \times 50 = 10 liters
    • Milk in the 50 liters = 45×50=40\frac{4}{5} \times 50 = 40 liters
  4. Remaining Mixture:

    • After selling 50 liters, he has:
      • Water: 3010=2030 - 10 = 20 liters
      • Milk: 12040=80120 - 40 = 80 liters
    • Remaining mixture = 80+20=10080 + 20 = 100 liters.
  5. Adding Water to Compensate:

    • He adds water to bring the mixture back to the original total volume of 150 liters.
    • Required amount of water to add = 150100=50150 - 100 = 50 liters.
  6. Final Mixture Composition:

    • Total water = 20+50=7020 + 50 = 70 liters
    • Total milk = 80 liters
  7. Final Ratio of Water to Milk: Water : Milk=70:80=7:8\text{Water : Milk} = 70 : 80 = 7 : 8

Thus, the correct answer is not in the options given.

If any questions or details needed?
Here are 5 questions to expand understanding:

  1. How to calculate the proportion of substances when the ratio is given?
  2. Why did the milk and water ratio remain the same in each fraction of the mixture sold?
  3. What other ways could the milkman adjust the mixture to compensate for the water?
  4. How can we determine the final ratios when scaling quantities up or down?
  5. In what real-life situations are these mixture ratio calculations applicable?

Tip: Always verify the volume of individual components before and after operations in mixture problems for accuracy.

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Math Problem Analysis

Mathematical Concepts

Ratios
Mixture and Alligation
Proportion

Formulas

Ratio of components in mixture = component1 : component2
Amount after proportionate mixture removal

Theorems

Conservation of Ratio in Mixtures

Suitable Grade Level

Grades 8-10