Understanding the Milk Mixture Problem
The question asks us to compare the average cost per litre of a milk mixture with the average cost per litre of the milk initially in Container A. We need to determine how much more or less the mixture's average cost is.
Here's the information given:
- Container A: Contains $30$ litres of pure milk costing $₹40$ per litre.
- Container B: Contains $50$ litres of pure milk costing $₹50$ per litre.
Calculating the Cost of Milk in Each Container
First, let's find the total cost of milk in each container:
- Cost in Container A: The total cost is the quantity multiplied by the cost per litre.
Cost$_A = 30 \text{ litres} \times ₹40/\text{litre} = ₹1200 $. - Cost in Container B: Similarly, we calculate the total cost for Container B.
Cost$_B = 50 \text{ litres} \times ₹50/\text{litre} = ₹2500 $.
Calculating the Mixture's Total Quantity and Cost
When the contents of both containers are mixed, we need to find the total quantity and the total cost of the combined milk.
- Total Quantity: This is the sum of the quantities from both containers.
Total Quantity $= 30 \text{ litres} + 50 \text{ litres} = 80 \text{ litres} $. - Total Cost: This is the sum of the costs from both containers.
Total Cost $= ₹1200 + ₹2500 = ₹3700 $.
Determining the Average Cost Per Litre of the Mixture
The average cost per litre of the mixture is calculated by dividing the total cost by the total quantity.
The formula is:
$ \text{Average Cost per Litre} = \frac{\text{Total Cost}}{\text{Total Quantity}} $
Now, let's plug in the values:
$ \text{Average Cost per Litre (Mixture)} = \frac{₹3700}{80 \text{ litres}} $
Calculating the value:
$ \frac{3700}{80} = \frac{370}{8} = \frac{185}{4} = ₹46.25 \text{ per litre} $
So, the average cost per litre of the mixture is $₹46.25$.
Comparing Mixture Cost to Container A's Cost
The question asks how much the average cost per litre of the mixture is more or less than the average cost per litre of Container A's milk.
- Average Cost of Container A's milk: $₹40$ per litre (given).
- Average Cost of the Mixture: $₹46.25$ per litre (calculated).
To find the difference, we subtract the cost of Container A's milk from the cost of the mixture's milk:
$ \text{Difference} = \text{Average Cost (Mixture)} - \text{Average Cost (Container A)} $
$ \text{Difference} = ₹46.25 - ₹40 $
$ \text{Difference} = ₹6.25 $
Since the result ($₹46.25$) is greater than the cost of Container A's milk ($₹40$), the average cost per litre of the mixture is more than that of Container A's milk.
Therefore, the average cost per litre of the mixture is $₹6.25$ more than the average cost per litre of Container A's milk.
Important Questions from Mixture and Alligation
- I bought two shirts for ₹1,200. I sold the first one at a loss of 8% and the second at a gain of 24%. If, on the whole I made neither a loss nor a gain, find the cost price (in ₹) of the first shirt.
- There are two containers A and B. In container A, the ratio of milk and water is 1:3 and in container B, the ratio of milk and water is \(m:n\). If the mixture in the containers A and B are mixed in the ratio 2:3 to get 20 litres of a mixture having milk and water in the ratio 3:7, then what is the value of \(\frac{m}{n}\)?
- Pure ghee costs ₹112/kg. After adulterating it with vegetable oil costing ₹56/kg, a dishonest shopkeeper sells the mixture at the rate of ₹91/kg, thereby making a profit of 30%. In what ratio does he mix the two?
- Pure ghee costs $₹86/kg$. After adulterating it with vegetable oil costing $₹43/kg$, a dishonest shopkeeper sells the mixture at the rate of $₹80/kg$, thereby making a profit of $60\%$. In what ratio does he mix the two?
- A sample of milk from a vessel contains 4% water. What quantity of pure milk should be added to 8 L of milk in the vessel to reduce the water content to 2%?
