In what ratio, should rice at ₹60 per kg be mixed with rice at ₹42 per kg such that by selling the mixture at ₹56 per kg there is a gain of 12%?
4 ∶ 5
Understanding the Rice Mixing Problem
This problem involves mixing two types of rice with different costs and then selling the mixture at a profit. To find the ratio in which the two types of rice should be mixed, we first need to determine the cost price (CP) of the mixture per kg. We are given the selling price (SP) of the mixture and the gain percentage.
Calculating the Cost Price of the Mixture
The mixture is sold at ₹56 per kg, and this results in a gain of 12%. The formula relating selling price, cost price, and gain percentage is:
$\text{SP} = \text{CP} \times \left(1 + \frac{\text{Gain}%}{100}\right)$
We can plug in the given values:
$56 = \text{CP} \times \left(1 + \frac{12}{100}\right)$
$56 = \text{CP} \times \left(1 + 0.12\right)$
$56 = \text{CP} \times 1.12$
Now, we solve for CP:
$\text{CP} = \frac{56}{1.12}$
To simplify the division, we can multiply the numerator and denominator by 100:
$\text{CP} = \frac{56 \times 100}{1.12 \times 100} = \frac{5600}{112}$
Performing the division:
$\frac{5600}{112} = 50$
So, the cost price of the mixture is ₹50 per kg.
Applying the Rule of Alligation
The rule of alligation is a method used to find the ratio in which two or more ingredients are mixed to produce a mixture of a desired price.
- Cost of the first type of rice (dearer) = ₹60 per kg
- Cost of the second type of rice (cheaper) = ₹42 per kg
- Mean price (cost price) of the mixture = ₹50 per kg
According to the rule of alligation, the ratio of the quantities of the two ingredients is given by the difference between the mean price and the price of each ingredient. We represent this diagrammatically:
| Dearer Rice (₹60) | Cheaper Rice (₹42) | |
| Mean Price (₹50) | ||
| Difference (Dearer - Mean) | ||
| Difference (Mean - Cheaper) | $60 - 50 = 10$ | |
| $50 - 42 = 8$ |
The ratio of the quantity of cheaper rice to the quantity of dearer rice is $(60 - 50) : (50 - 42) = 10 : 8$.
We are asked for the ratio of rice at ₹60 per kg (dearer) to rice at ₹42 per kg (cheaper). This ratio is the inverse of the above, i.e., $(50 - 42) : (60 - 50)$.
Ratio = $8 : 10$
This ratio can be simplified by dividing both parts by their greatest common divisor, which is 2.
Ratio = $\frac{8}{2} : \frac{10}{2} = 4 : 5$
Thus, rice at ₹60 per kg should be mixed with rice at ₹42 per kg in the ratio 4 : 5.
Summary of Steps
- Calculate the cost price (CP) of the mixture using the given selling price (SP) and gain percentage.
- Identify the cost prices of the two types of rice and the calculated mean price (CP of the mixture).
- Apply the rule of alligation to find the ratio of the quantities of the two types of rice needed to achieve the mean price.
- Ensure the final ratio matches the order specified in the question (rice at ₹60 per kg : rice at ₹42 per kg).
Revision Table: Key Concepts
| Concept | Description | Formula/Method |
|---|---|---|
| Profit Percentage | Gain expressed as a percentage of the Cost Price. | $\text{Gain}% = \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100$ |
| Selling Price (SP) | Price at which an item is sold. | $\text{SP} = \text{CP} + \text{Gain}$ or $\text{SP} = \text{CP} \times \left(1 + \frac{\text{Gain}%}{100}\right)$ |
| Cost Price (CP) | Price at which an item is bought. | $\text{CP} = \frac{\text{SP}}{1 + \frac{\text{Gain}%}{100}}$ |
| Rule of Alligation | Method to find ratio for mixing ingredients with different costs to get a desired mean cost. | Ratio of quantity 1 to quantity 2 = (Cost of 2 - Mean Cost) : (Mean Cost - Cost of 1) |
Additional Information: Alligation and Mixtures
The rule of alligation is a specific application of the weighted average concept. When two quantities are mixed in a certain ratio, the average cost (or price, or whatever attribute is being measured) of the mixture is the weighted average of the individual costs.
If quantity Q1 of ingredient 1 with cost C1 is mixed with quantity Q2 of ingredient 2 with cost C2, the mean cost (M) of the mixture is:
$M = \frac{(Q1 \times C1) + (Q2 \times C2)}{Q1 + Q2}$
Rearranging this formula leads to the alligation rule:
$\frac{Q1}{Q2} = \frac{C2 - M}{M - C1}$
So, the ratio $Q1 : Q2$ is $(C2 - M) : (M - C1)$. This matches the differences found in the alligation diagram, where the differences are cross-subtracted. The difference between the dearer price and the mean price gives the quantity proportion of the cheaper ingredient, and the difference between the mean price and the cheaper price gives the quantity proportion of the dearer ingredient.
Important Questions from Mixture Problems
In a mixture of liquid ,1/5 part is acid 2/5 part is alcohol and the remaining part is water. If the total quantity of the mixture is 20 litres, then how much water (in litre) does the mixture contain?
A vessel contains 20 litres containing milk and water in the ratio 3 : 2. Ten litres of this milk is removed and replaced with equal amount of pure milk. If this process is repeated once again, find the final ratio of milk and water.
From a container of 50 liters pure milk, 10 liters is taken out and replaced by 10 liters of water. If this process is repeated thrice, what is the ratio of water and milk finally?
Consider the following statements about a mixture and determine which of the statements is/are correct.
1. A mixture has a variable composition.
2. In compounds, the composition of each new substance is always fixed.
3. A mixture shows the properties of the constituent substances.
At what rate (in percentage) per annum will a sum of money double itself in 16 years on simple interest?
