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?
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Understanding the Liquid Mixture Problem
This problem asks us to find the amount of water in a liquid mixture given the total quantity and the fractions of other components (acid and alcohol).
The mixture consists of three parts: acid, alcohol, and water. The total quantity of the mixture is 20 litres.
Given Information
- Total quantity of the mixture = 20 litres
- Fraction of acid in the mixture = \(\frac{1}{5}\)
- Fraction of alcohol in the mixture = \(\frac{2}{5}\)
- The remaining part is water.
Calculating the Fraction of Water
The sum of the fractions of all components in a mixture must equal 1 (representing the whole mixture). Therefore, the fraction of water can be found by subtracting the fractions of acid and alcohol from 1.
Fraction of water = 1 - (Fraction of acid + Fraction of alcohol)
Let's calculate the combined fraction of acid and alcohol:
Now, subtract this combined fraction from 1 to find the fraction of water:
So, the fraction of water in the mixture is \(\frac{2}{5}\).
Calculating the Quantity of Water
To find the actual quantity of water in litres, we multiply the fraction of water by the total quantity of the mixture.
Quantity of water = Fraction of water \(\times\) Total quantity of mixture
Therefore, the mixture contains 8 litres of water.
Verification
Let's check the quantities of all components:
- Quantity of acid = \(\frac{1}{5} \times 20 = 4\) litres
- Quantity of alcohol = \(\frac{2}{5} \times 20 = 8\) litres
- Quantity of water = \(\frac{2}{5} \times 20 = 8\) litres (calculated above)
Total quantity = Quantity of acid + Quantity of alcohol + Quantity of water
Total quantity = 4 litres + 8 litres + 8 litres = 20 litres
This matches the given total quantity, so our calculation for the quantity of water is correct.
Summary of Solution Steps
- Identify the total quantity of the mixture and the fractions of known components (acid and alcohol).
- Calculate the sum of the fractions of acid and alcohol.
- Subtract the sum of fractions from 1 to find the fraction of water.
- Multiply the fraction of water by the total quantity of the mixture to find the quantity of water in litres.
| Component | Fraction | Quantity (in litres) |
|---|---|---|
| Acid | \(\frac{1}{5}\) | \(\frac{1}{5} \times 20 = 4\) |
| Alcohol | \(\frac{2}{5}\) | \(\frac{2}{5} \times 20 = 8\) |
| Water | \(\frac{2}{5}\) | \(\frac{2}{5} \times 20 = 8\) |
| Total | \(1\) | \(4 + 8 + 8 = 20\) |
The amount of water in the mixture is 8 litres.
Revision Table: Liquid Mixture Calculations
| Concept | Description | Formula/Approach |
|---|---|---|
| Total Fraction | The sum of fractions of all components in a mixture is always 1. | \(f_1 + f_2 + \dots + f_n = 1\) |
| Finding Unknown Fraction | If some fractions are known, the unknown fraction is 1 minus the sum of known fractions. | \(f_{\text{unknown}} = 1 - (f_{\text{known1}} + f_{\text{known2}} + \dots)\) |
| Calculating Quantity | The quantity of a component is its fraction multiplied by the total quantity of the mixture. | Quantity = Fraction \(\times\) Total Quantity |
Additional Information: Fractions and Mixtures
Understanding fractions is key to solving mixture problems like this. A fraction represents a part of a whole. In mixtures, the whole is the total quantity of the mixture, and the parts are the quantities of the individual components.
- Fractions as Parts of a Whole: A fraction like \(\frac{1}{5}\) means one part out of five equal parts. In this problem, it means the acid makes up one-fifth of the total 20 litres.
- Sum of Fractions: When dealing with components that make up a complete mixture, the sum of their individual fractions must always add up to 1, representing 100% of the mixture.
- Applying Fractions to Quantities: To find the actual amount of a component, you multiply its fraction by the total amount of the mixture. For example, if water is \(\frac{2}{5}\) of a 20-litre mixture, the quantity of water is \(\frac{2}{5} \times 20\).
These principles are fundamental to solving many problems involving ratios, proportions, and mixtures.
Important Questions from Mixture Problems
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%?
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?
