Initial Mixture Analysis
The total volume of the milk and water mixture is 80 litres.
The initial ratio of milk to water is given as 7 : 3.
To find the initial quantities, calculate the total number of parts in the ratio:
Total parts = $ 7 + 3 = 10 $
Calculate the initial amount of milk:
Amount of Milk = $ \frac{7}{10} \times 80 \text{ litres} = 56 \text{ litres} $
Calculate the initial amount of water:
Amount of Water = $ \frac{3}{10} \times 80 \text{ litres} = 24 \text{ litres} $
Target Ratio Calculation
Water is added to the mixture to change the ratio to 2 : 1.
The amount of milk remains constant at 56 litres because only water is added.
Let $x$ represent the litres of water that need to be added.
The new amount of water in the mixture will be $ (24 + x) $ litres.
The desired new ratio of milk to water is 2 : 1.
Solving for Added Water
Set up an equation representing the new ratio:
$ \frac{\text{Amount of Milk}}{\text{New Amount of Water}} = \frac{2}{1} $
Substitute the known values:
$ \frac{56}{24 + x} = \frac{2}{1} $
Solve for $x$ by cross-multiplying:
$ 56 \times 1 = 2 \times (24 + x) $
$ 56 = 48 + 2x $
Subtract 48 from both sides:
$ 56 - 48 = 2x $
$ 8 = 2x $
Divide by 2:
$ x = \frac{8}{2} $
$ x = 4 $
Thus, 4 litres of water should be added.
Important Questions from Mixture Problems (Notes)
- A has a container containing 60 litres of pure milk. He takes out 4 litres of milk and replaces it with the same quantity of water. He sells this mixture to B. B sells 30 litres of the mixture and added 5 litres of water in the remaining mixture. The ratio of milk to water in the remaining mixture is:
- How many liters of water should be added to a 30-liter mixture containing milk and water in the ratio 7:3 such that the resultant mixture has 40% water in it?
- A $595$ litre of mixture contains milk and water in the ratio $17:18$. How much milk must be added to the mixture so that it contains milk and water in the proportion of $3:2$?
- Alloy A is formed by mixing iron (Fe) and nickel (Ni) in the ratio 3:4, while alloy B is formed by mixing Fe and Ni in the ratio 9:5. If equal quantities of alloys A and B are melted together to form a new alloy C, what will be the ratio of Fe to Ni in the alloy C?
Suppose a tap mixes hot water and cold water in a ratio that depends linearly on the proportion of opening. Water out of the tap has temperature 40°C when the tap is half-open, and 30°C when it is three-fourths open. To get water at 50°C, the tap should be_______________.
