Ghee and Vegetable Oil Mixing Ratio Problem
This problem involves calculating the ratio in which a shopkeeper mixes pure ghee with a cheaper vegetable oil to sell the mixture at a specific price and make a certain profit. We need to find the proportion of ghee and oil used.
Calculating the Cost Price of the Mixture
First, let's determine the actual cost price (CP) of the mixture. We know the selling price (SP) and the profit percentage.
- Selling Price (SP) of the mixture = ₹91 per kg.
- Profit percentage = 30%.
- The formula relating SP, CP, and Profit is: $SP = CP \times (1 + \frac{Profit\%}{100})$
- Substituting the given values: $91 = CP \times (1 + \frac{30}{100})$
- $91 = CP \times (1 + 0.30)$
- $91 = CP \times 1.30$
- To find the CP, we rearrange the formula: $CP = \frac{91}{1.30}$
- $CP = \frac{910}{13}$
- $CP = 70$
- Therefore, the cost price of the mixture is ₹70 per kg.
Applying the Rule of Alligation
The Rule of Alligation is a method used to solve problems where different quantities of ingredients are mixed together to produce a mixture of a desired characteristic (in this case, a specific cost price).
We compare the cost price of the pure ingredients with the cost price of the mixture.
- Cost Price of Pure Ghee (Dearer Ingredient) = ₹112/kg
- Cost Price of Vegetable Oil (Cheaper Ingredient) = ₹56/kg
- Mean Price (Cost Price of the Mixture) = ₹70/kg
We set up the alligation as follows:
| Cost Price of Dearer | Cost Price of Cheaper | Mean Price |
| $112$ | $56$ | $70$ |
| Difference (Mean - Cheaper) $70 - 56 = 14$ |
Difference (Dearer - Mean) $112 - 70 = 42$ |
The ratio of the quantities of the cheaper ingredient (vegetable oil) to the dearer ingredient (pure ghee) is given by the difference calculated on the opposite side.
- Quantity of Vegetable Oil : Quantity of Pure Ghee = (Difference on Dearer side) : (Difference on Cheaper side)
- Ratio = $42 : 14$
Simplifying the Ratio
Now, we simplify the ratio obtained:
- Ratio = $42 : 14$
- Divide both sides by their greatest common divisor, which is 14:
- Ratio = $\frac{42}{14} : \frac{14}{14}$
- Ratio = $3 : 1$
This ratio represents (Quantity of Vegetable Oil : Quantity of Pure Ghee).
The question asks for the ratio in which he mixes the two, typically meaning (Pure Ghee : Vegetable Oil).
Therefore, the ratio of Pure Ghee to Vegetable Oil is the inverse of the ratio we found:
Ratio (Pure Ghee : Vegetable Oil) = $1 : 3$.
Conclusion
The dishonest shopkeeper mixes pure ghee and vegetable oil in the ratio of 1:3.
Important Questions from Mixture and Alligation
A bottle contains 20 litres of liquid A. 4 litres of liquid A is taken out of it d replaced by same quantity of liquid B. Again 4 litre of the mixture is taken out and replaced by same quantity of liquid B. What is the ratio of quantity of liquid A to that of liquid B in the final mixture?
The average score of a batsman after his 50th innings was 46.4. After 60th innings, his average Score increases by 2.6. What was his average score in the last ten innings?
If 1 litre of water weighs 1 kg, then how many cubic millimeters of water will weigh 0.1 gm?
A vessel full of water weighs 4o kg. If it is one-third filled, its weight becomes 20 kg. What is the weight of the empty vessel?
Two equal glasses of same type are respectively 1/3 and 1/4 full of milk. They are then filled up with water and the contents are mixed in a pot. What is the ratio of milk and water in the pot?
