Understanding the Alcohol Ratio Problem
The question asks for the ratio of alcohol to water in a mixture. We are given the cost price (CP) of pure alcohol, the selling price (SP) of the final mixture, and the profit percentage made on the sale. Water is added, which has no cost.
Key Information Used
- Cost Price (CP) of Alcohol = Rs. 75 per centiliter (cL)
- Selling Price (SP) of Mixture = Rs. 75 per centiliter (cL)
- Profit Percentage = 50%
- Cost Price (CP) of Water = Rs. 0 per centiliter (cL)
Calculation Steps
Step 1: Calculate the Cost Price of the Mixture
First, we determine the actual cost price (CP) per unit volume of the mixture using the selling price (SP) and the profit percentage.
The formula relating SP, CP, and Profit Percentage is: $ \text{SP} = \text{CP} \times (1 + \frac{\text{Profit Percentage}}{100}) $
Rearranging to find CP:
$ \text{CP}_{\text{mixture}} = \frac{\text{SP}_{\text{mixture}}}{1 + (\text{Profit Percentage}/100)} $
Plugging in the values:
$ \text{CP}_{\text{mixture}} = \frac{75}{1 + (50/100)} = \frac{75}{1 + 0.50} = \frac{75}{1.50} = 50 $
Therefore, the cost price of the mixture is Rs. 50 per cL.
Step 2: Determine the Ratio using Alligation
Alligation is a method used to find the ratio of two ingredients when mixed to produce a mixture of a certain desired value (in this case, cost price).
We list the cost prices:
| Ingredient | Cost Price (Rs./cL) |
|---|---|
| Alcohol | 75 |
| Water | 0 |
The mean cost price of the mixture is Rs. 50/cL.
According to the rule of alligation, the ratio of the quantities of the ingredients is inversely proportional to the difference of their costs from the mean cost.
$ \frac{\text{Quantity of Alcohol}}{\text{Quantity of Water}} = \frac{\text{Mean Cost} - \text{Cost of Water}}{\text{Cost of Alcohol} - \text{Mean Cost}} $
Let $A$ be the quantity of alcohol and $W$ be the quantity of water.
$ \frac{A}{W} = \frac{50 - 0}{75 - 50} = \frac{50}{25} = \frac{2}{1} $
Step 3: State the Ratio
The ratio of Alcohol to Water ($A:W$) is 2:1.
Final Answer Derivation
The calculation shows that the ratio of alcohol to water required to achieve a cost price of Rs. 50/cL (given the cost of alcohol is Rs. 75/cL and water is free) is 2:1. This matches Option A.
Important Questions from Mixture Problems (Notes)
- In 80 litres mixture of milk and water, the ratio of amount of milk to that of amount of water is 7 : 3. In order to make this ratio 2 : 1, how many litres of water should be added ?
- 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 $3:2$?
- 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?
Two solutions $X$ and $Y$ containing ingredients $A, B$ and $C$, in proportions $a:b:c$ and $c:b:a$, respectively, are mixed. For the resultant mixture to have $A, B$ and $C$ in $1:1:1$ proportion, it is necessary that $a:b:c$ is
- A 360 ml aqueous solution contains 40% alcohol. How much will be the approximate percentage of alcohol if 3600 ml of water is added to the solution?
