$7:36$
Solving the Ghee and Vegetable Oil Mixture Ratio Problem
This problem involves calculating the ratio in which two ingredients, pure ghee and vegetable oil, are mixed. We are given the cost price of each ingredient, the selling price of the mixture, and the profit percentage made by the shopkeeper. We need to find the ratio of mixing using the concept of alligation.
Step 1: Calculate the Cost Price (CP) of the Mixture
The shopkeeper sells the mixture at $₹80/kg$ making a profit of $60\%$. We first need to find the actual cost price of the mixture per kg.
Let the Cost Price (CP) of the mixture be $₹ C_{mix}/kg$.
The Selling Price (SP) is $₹80/kg$.
The Profit Percentage is $60\%$.
The formula relating SP, CP, and Profit is:
$ SP = CP \times \left(1 + \frac{\text{Profit Percentage}}{100}\right) $
Substituting the given values:
$ ₹80 = C_{mix} \times \left(1 + \frac{60}{100}\right) $
$ ₹80 = C_{mix} \times \left(1 + 0.60\right) $
$ ₹80 = C_{mix} \times 1.60 $
Now, solve for $C_{mix}$:
$ C_{mix} = \frac{₹80}{1.60} $
$ C_{mix} = \frac{₹800}{16} $
$ C_{mix} = ₹50 $
So, the cost price of the mixture is $₹50/kg$.
Step 2: Apply the Rule of Alligation
Alligation is a method used to find the ratio of the quantities when two or more ingredients having different prices are mixed to produce a mixture of a desired mean price.
We have:
- Cost Price of Pure Ghee (Higher Price, $C_1$) = $₹86/kg$
- Cost Price of Vegetable Oil (Lower Price, $C_2$) = $₹43/kg$
- Mean Cost Price of the Mixture ($C_{mix}$) = $₹50/kg$
We set up the alligation as follows:
| Cost Price of Dearer Item ($C_1$) | Cost Price of Cheaper Item ($C_2$) | Mean Price ($C_{mix}$) |
|---|---|---|
| $₹86$ | $₹43$ | $₹50$ |
Difference ($C_1 - C_{mix}$) |
Difference ($C_{mix} - C_2$) |
Calculate the differences:
- Difference 1 (Quantity of Cheaper Item - Oil): $C_{mix} - C_2 = ₹50 - ₹43 = ₹7$
- Difference 2 (Quantity of Dearer Item - Ghee): $C_1 - C_{mix} = ₹86 - ₹50 = ₹36$
The ratio in which the two ingredients are mixed is given by the ratio of these differences:
Ratio = (Quantity of Ghee) : (Quantity of Vegetable Oil)
$ \text{Ratio} = (C_{mix} - C_2) : (C_1 - C_{mix}) $
$ \text{Ratio} = 7 : 36 $
Conclusion
The shopkeeper mixes pure ghee and vegetable oil in the ratio 7:36.
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?
- Container A contains $30$ litres of pure milk costing $₹40$ per litre, and container B contains $50$ litres of pure milk costing $₹50$ per litre. The entire contents are mixed together. By how much is the average cost per litre of the mixture more or less than the average cost per litre of container A's milk?
- 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%?
