In what ratio sugar of Rs. 30 per kg should be mixed with sugar of Rs. 40 per kg so that the mixture is worth Rs. 37 per kg?
3 ∶ 7
Understanding the Sugar Mixing Problem
This problem requires us to determine the specific proportion in which two different types of sugar, one costing Rs. 30 per kg and another costing Rs. 40 per kg, should be combined to create a mixture that has a value of Rs. 37 per kg.
Problems of this type, involving the mixing of two or more ingredients with different costs or properties to achieve a mixture with a desired average cost or property, can be efficiently solved using the rule of alligation or through algebraic methods based on weighted averages.
Solving with the Alligation Method
The rule of alligation is a handy graphical or tabular technique used to find the ratio of the quantities of two components that need to be mixed to obtain a mixture with a specific mean value. It works by looking at the differences between the individual values and the mean value.
Applying Alligation Steps to Find the Sugar Ratio
Here's how we apply the alligation method to find the required ratio for mixing the sugar:
- Identify the price of the cheaper ingredient: Rs. 30/kg.
- Identify the price of the dearer ingredient: Rs. 40/kg.
- Identify the desired mean price of the mixture: Rs. 37/kg.
According to the rule of alligation, the ratio of the quantities of the two ingredients is inversely proportional to the differences between their respective prices and the mean price. We set up the differences diagonally.
| Item | Price (Rs/kg) | Difference from Mean Price (Quantity Ratio Component) |
|---|---|---|
| Sugar @ Rs. 30/kg (Cheaper) | 30 | \(|40 - 37| = 3\) (This value goes diagonally opposite Cheaper Price, representing quantity of Cheaper) |
| Sugar @ Rs. 40/kg (Dearer) | 40 | \(|30 - 37| = 7\) (This value goes diagonally opposite Dearer Price, representing quantity of Dearer) |
The difference between the dearer price (40) and the mean price (37) is \(40 - 37 = 3\). This difference gives us the proportional part of the quantity of the cheaper sugar (Rs. 30/kg).
The difference between the mean price (37) and the cheaper price (30) is \(37 - 30 = 7\). This difference gives us the proportional part of the quantity of the dearer sugar (Rs. 40/kg).
The ratio of the quantity of sugar at Rs. 30/kg to the quantity of sugar at Rs. 40/kg is therefore the ratio of these differences:
Quantity of Rs. 30/kg Sugar : Quantity of Rs. 40/kg Sugar \(=\) \(3 : 7\).
Solving with the Algebraic Method
Alternatively, we can solve this sugar mixing problem using algebraic principles, specifically the concept of weighted averages. Let \(Q_{30}\) represent the quantity (in kg) of the sugar costing Rs. 30 per kg, and \(Q_{40}\) represent the quantity (in kg) of the sugar costing Rs. 40 per kg that are mixed.
Setting up and Solving the Algebraic Equation
- The total cost of \(Q_{30}\) kg of the first type of sugar is \(30 \times Q_{30}\).
- The total cost of \(Q_{40}\) kg of the second type of sugar is \(40 \times Q_{40}\).
- The total cost of the mixture is the sum of these costs: \(30 Q_{30} + 40 Q_{40}\).
- The total quantity of the mixture is the sum of the quantities: \(Q_{30} + Q_{40}\).
The mean price of the mixture is the total cost divided by the total quantity. We are given that the mean price is Rs. 37 per kg. So, we can write the following equation:
\[ \text{Mean Price} = \frac{\text{Total Cost}}{\text{Total Quantity}} \] \[ 37 = \frac{30 Q_{30} + 40 Q_{40}}{Q_{30} + Q_{40}} \]Now, we solve this equation to find the ratio \(Q_{30} : Q_{40}\).
Multiply both sides of the equation by \((Q_{30} + Q_{40})\) to remove the denominator:
\[ 37 (Q_{30} + Q_{40}) = 30 Q_{30} + 40 Q_{40} \]Distribute 37 on the left side:
\[ 37 Q_{30} + 37 Q_{40} = 30 Q_{30} + 40 Q_{40} \]Now, rearrange the terms to group \(Q_{30}\) terms on one side and \(Q_{40}\) terms on the other side:
\[ 37 Q_{30} - 30 Q_{30} = 40 Q_{40} - 37 Q_{40} \]Simplify both sides:
\[ 7 Q_{30} = 3 Q_{40} \]To find the ratio \(Q_{30} : Q_{40}\), divide both sides by \(7 Q_{40}\):
\[ \frac{7 Q_{30}}{7 Q_{40}} = \frac{3 Q_{40}}{7 Q_{40}} \] \[ \frac{Q_{30}}{Q_{40}} = \frac{3}{7} \]This means the ratio of the quantity of sugar at Rs. 30/kg to the quantity of sugar at Rs. 40/kg is \(3 : 7\).
Conclusion: Determining the Sugar Mixing Ratio
Both the alligation method and the algebraic method consistently show that to obtain a mixture worth Rs. 37 per kg, sugar costing Rs. 30 per kg must be mixed with sugar costing Rs. 40 per kg in the ratio of \(3 : 7\).
The required mixing ratio is \(3 : 7\).
Revision Table: Key Concepts for Mixture Problems
| Concept | Relevance in this Problem |
|---|---|
| Mixture | Combining two types of sugar. |
| Ingredient Prices | The costs of the individual sugar types (Rs. 30/kg and Rs. 40/kg). |
| Mean Price | The desired price of the final mixture (Rs. 37/kg). |
| Alligation Rule | A direct method to find the quantity ratio from ingredient prices and mean price. |
| Weighted Average | The mathematical basis for the mean price, where quantity is the weight for each price. |
| Ratio | The required proportion of quantities of the two sugar types. |
Additional Information: Mixtures and Alligation Rule Details
The rule of alligation is a powerful tool particularly useful in quantitative aptitude questions involving mixtures of two items. It helps determine the ratio in which two ingredients with known per-unit costs or properties should be mixed to get a blend with a desired mean per-unit cost or property. The core principle is that the ratio of the quantities is inversely related to the deviations of the individual prices/properties from the mean price/property.
If you have two items with per-unit values \(V_1\) and \(V_2\), and you mix them to get a mean value \(V_m\), where \(V_1 < V_m < V_2\), the ratio of the quantity of item 1 (\(Q_1\)) to the quantity of item 2 (\(Q_2\)) is given by the formula: \(Q_1 : Q_2 = (V_2 - V_m) : (V_m - V_1)\). This formula is derived directly from the weighted average equation \(V_m = \frac{Q_1 V_1 + Q_2 V_2}{Q_1 + Q_2}\), which we demonstrated in the algebraic solution.
This method is not limited to mixing commodities like sugar. It can be applied to various scenarios, including finding the ratio of liquids with different concentrations to get a mixture of a specific concentration, determining the ratio of investments at different interest rates to achieve a target average interest rate, or calculating the ratio of speeds over different parts of a journey to find an average speed.
Mastering both the alligation technique for quick calculations and the algebraic method for understanding the underlying principle is key to confidently solving mixture and alligation problems in exams.
Important Questions from To Make a Mixture from Two Mixtures
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Two vessels P and Q contain liquid A and liquid B in the ratio $4 : 3$ and $5 : 4$ respectively. In what ratio must the mixtures from vessel P and vessel Q be combined to obtain a new mixture in vessel R containing liquid A and liquid B in the ratio $11 : 8$?
In a vessel, a mixture of milk and water is in ratio $9 : 5$, while in another vessel mixture of milk and water is in ratio $3 : 8$. In what ratio mixture of both the vessels should be mixed together so that in the resultant mixture ratio of milk and water becomes $13 : 19$?
