Rice worth ₹35 per kilogram and ₹38 per kilogram are mixed with a third variety in the ratio 2 ∶ 1 ∶ 2. If the mixture is worth ₹42 per kilogram, the price of the third variety per kilogram is:
Understanding the Rice Mixture Problem
This question asks us to find the price of a third variety of rice when it's mixed with two other varieties in a specific ratio, and the average price of the resulting mixture is known. This is a classic example of a mixture problem often encountered in quantitative aptitude.
Setting up the Rice Mixture Calculation
We are given three varieties of rice:
- Variety 1: Price ₹35 per kilogram (kg)
- Variety 2: Price ₹38 per kilogram (kg)
- Variety 3: Price unknown, let's call it 'p' per kilogram (kg)
These varieties are mixed in the ratio 2 ∶ 1 ∶ 2. This means for every 2 parts of Variety 1, there is 1 part of Variety 2, and 2 parts of Variety 3.
Let's assume the quantities of the three varieties in the mixture are 2k, 1k, and 2k kilograms respectively, where 'k' is a constant representing the size of one part in the ratio. The total quantity of the mixture will be the sum of these quantities.
Total quantity of mixture $= 2k + 1k + 2k = 5k$ kilograms.
Calculating the Total Value of the Mixture
The total value of the mixture is the sum of the values of each variety used in the mix.
- Value of Variety 1 = Quantity of Variety 1 × Price of Variety 1 $= 2k \times 35 = 70k$
- Value of Variety 2 = Quantity of Variety 2 × Price of Variety 2 $= 1k \times 38 = 38k$
- Value of Variety 3 = Quantity of Variety 3 × Price of Variety 3 $= 2k \times p = 2pk$
Total value of the mixture $= 70k + 38k + 2pk = (108 + 2p)k$.
Using the Average Price to Find the Unknown Price
We are given that the mixture is worth ₹42 per kilogram. The average price of the mixture is calculated by dividing the total value of the mixture by the total quantity of the mixture.
Average price $= \frac{\text{Total Value}}{\text{Total Quantity}}$
We are given the average price is ₹42/kg. So, we can set up the equation:
\(42 = \frac{(108 + 2p)k}{5k}\)
Assuming \(k > 0\) (since we have a mixture), we can cancel 'k' from the numerator and denominator:
\(42 = \frac{108 + 2p}{5}\)
Now, we need to solve this equation for 'p'.
Multiply both sides by 5:
\(42 \times 5 = 108 + 2p\)
\(210 = 108 + 2p\)
Subtract 108 from both sides:
\(210 - 108 = 2p\)
\(102 = 2p\)
Divide both sides by 2:
\(p = \frac{102}{2}\)
\(p = 51\)
Therefore, the price of the third variety of rice is ₹51 per kilogram.
| Variety | Ratio Part | Assumed Quantity (kg) | Price per kg (₹) | Total Value (₹) |
|---|---|---|---|---|
| 1 | 2 | \(2k\) | 35 | \(35 \times 2k = 70k\) |
| 2 | 1 | \(1k\) | 38 | \(38 \times 1k = 38k\) |
| 3 | 2 | \(2k\) | \(p\) | \(p \times 2k = 2pk\) |
| Total | 5 | \(5k\) | Mixture Price = 42 | \((108 + 2p)k\) |
Using the formula: \( \text{Average Price} = \frac{\text{Sum of (Quantity } \times \text{ Price)}}{\text{Sum of Quantities}} \)
\( 42 = \frac{70k + 38k + 2pk}{2k + 1k + 2k} \)
\( 42 = \frac{(108 + 2p)k}{5k} \)
\( 42 \times 5 = 108 + 2p \)
\( 210 = 108 + 2p \)
\( 2p = 210 - 108 \)
\( 2p = 102 \)
\( p = \frac{102}{2} \)
\( p = 51 \)
The price of the third variety per kilogram is ₹51.
Revision Table: Key Concepts
| Concept | Description | Application in Problem |
|---|---|---|
| Ratio | Compares the quantities of different items. | Given as 2:1:2 for the three rice varieties. |
| Mixture Problem | Involves calculating properties (like price) of a mix based on the properties and quantities of components. | Finding the price of one component given the mix ratio and average price. |
| Weighted Average | The average where each component's value is weighted by its quantity. The average price of the mixture is a weighted average of the prices of the components. | \( \text{Average Price} = \frac{\sum (\text{Quantity}_i \times \text{Price}_i)}{\sum \text{Quantity}_i} \) |
Additional Information on Mixture Problems
Mixture problems often involve mixing liquids (like milk and water, or different solutions), solids (like different types of grains or metals), or investments with different interest rates. The core idea is always to balance the total quantity and the total value or property of the mixture.
For instance, if you mix \(Q_1\) kg of ingredient A at price \(P_1\) and \(Q_2\) kg of ingredient B at price \(P_2\), the average price \(P_{avg}\) of the mixture is given by:
\( P_{avg} = \frac{Q_1 P_1 + Q_2 P_2}{Q_1 + Q_2} \)
In our case with three varieties, the formula extended to:
\( P_{avg} = \frac{Q_1 P_1 + Q_2 P_2 + Q_3 P_3}{Q_1 + Q_2 + Q_3} \)
We used \(Q_1 = 2k\), \(P_1 = 35\), \(Q_2 = 1k\), \(P_2 = 38\), \(Q_3 = 2k\), \(P_3 = p\), and \(P_{avg} = 42\). Plugging these values into the formula allows us to solve for the unknown price 'p'.
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