A mixture contains milk and water in the ratio of 5 ∶ 3, respectively. On adding 7 litres of water, the ratio of milk to water becomes 1 ∶ 2. Find the quantity of milk in the mixture.
5 litres
Solving Milk and Water Mixture Ratio Problems
This problem involves a mixture of milk and water where the ratio changes upon adding more water. We need to find the initial quantity of milk in the mixture.
Setting up the Initial Mixture
The initial ratio of milk to water is given as 5 : 3. This means for every 5 parts of milk, there are 3 parts of water. We can represent the initial quantities using a variable.
- Let the common ratio factor be \(x\).
- Initial quantity of milk = \(5x\) litres.
- Initial quantity of water = \(3x\) litres.
Understanding the Change
7 litres of water are added to the mixture. The quantity of milk remains the same, but the quantity of water increases.
- Quantity of milk after adding water = \(5x\) litres.
- Quantity of water after adding water = \((3x + 7)\) litres.
Forming the Equation from the New Ratio
After adding 7 litres of water, the new ratio of milk to water becomes 1 : 2. We can set up an equation using this new ratio and the quantities after the change.
The new ratio is given by:
\(\frac{\text{Quantity of Milk}}{\text{Quantity of Water}} = \frac{1}{2}\)
Substituting the expressions for the quantities:
\(\frac{5x}{3x + 7} = \frac{1}{2}\)
Solving the Equation to Find x
Now, we need to solve this equation for \(x\). We can do this by cross-multiplication.
\(2 \times (5x) = 1 \times (3x + 7)\)
\(10x = 3x + 7\)
To find the value of \(x\), we need to isolate \(x\) on one side of the equation. Subtract \(3x\) from both sides:
\(10x - 3x = 7\)
\(7x = 7\)
Now, divide both sides by 7:
\(x = \frac{7}{7}\)
\(x = 1\)
Finding the Quantity of Milk
The quantity of milk in the initial mixture was represented by \(5x\).
Substitute the value of \(x = 1\) into the expression for the quantity of milk:
Quantity of milk = \(5x = 5 \times 1 = 5\) litres.
Therefore, the quantity of milk in the mixture is 5 litres.
Let's check this:
- Initial milk = 5 litres, Initial water = 3 litres. Ratio = 5:3.
- Add 7 litres water. Milk = 5 litres, Water = 3 + 7 = 10 litres.
- New ratio = Milk : Water = 5 : 10 = 1 : 2.
This matches the information given in the problem.
Revision Table: Key Concepts
| Concept | Explanation | Application in Problem |
|---|---|---|
| Ratio | A comparison of two quantities by division. Expressed as a:b or a/b. | Used to represent the proportion of milk and water in the mixture. |
| Algebraic Representation | Using variables (like \(x\)) to represent unknown quantities based on the ratio. | Initial quantities were \(5x\) and \(3x\). |
| Setting up Equation | Translating the problem statement into a mathematical equation. | The new ratio (after adding water) was used to form the equation \(\frac{5x}{3x + 7} = \frac{1}{2}\). |
| Solving Linear Equation | Finding the value of the variable that satisfies the equation. | Solved \(10x = 3x + 7\) to find \(x=1\). |
Additional Information: Ratio and Proportion Basics
Ratio and proportion problems are common in quantitative aptitude. Understanding the basics is crucial.
- Ratio: Compares two quantities of the same unit. A ratio a:b is equivalent to a fraction a/b. Ratios can be simplified by dividing both parts by their greatest common divisor.
- Proportion: An equality between two ratios. If a:b = c:d, then ad = bc (cross-multiplication property). This property is often used to solve ratio-related equations.
- Mixture Problems: These problems often involve combining substances or changing the composition of a mixture, and ratios are used to describe the proportions of the components. The key is often to identify which quantity remains constant and which changes, and then set up an equation based on the ratios.
Always read the question carefully to identify what is being added or removed and how it affects the quantities of the different components in the mixture.
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Important Questions from Mixture Problems
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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.
