The ratio of spirit and water in three solutions are 2 : 3, 3 : 4 and 4 : 5 respectively and they are mixed in the ratio 5 : 3 : 2. 318 ml of water is added to 1.26 liters of the solution thus obtained. The ratio of spirit and water in the final solution is
1 : 2
Understanding the Spirit and Water Ratio Problem
This problem involves calculating the final ratio of spirit to water in a mixture formed by combining three different solutions in specific proportions and then adding extra water. We need to determine the total amounts of spirit and water in the final mixture.
Initial Solution Compositions
We are given three solutions with the following spirit to water ratios:
- Solution 1: Spirit : Water = 2 : 3
- Solution 2: Spirit : Water = 3 : 4
- Solution 3: Spirit : Water = 4 : 5
Mixing the Solutions
The three solutions are mixed in the ratio 5 : 3 : 2. The total volume of this obtained mixture is 1.26 liters, which is equal to 1260 ml.
The total parts in the mixing ratio are \(5 + 3 + 2 = 10\) parts.
The volume of each solution used in the mixture is:
- Volume of Solution 1 = \(\frac{5}{10} \times 1260 \text{ ml} = 630 \text{ ml}\)
- Volume of Solution 2 = \(\frac{3}{10} \times 1260 \text{ ml} = 378 \text{ ml}\)
- Volume of Solution 3 = \(\frac{2}{10} \times 1260 \text{ ml} = 252 \text{ ml}\)
Calculating Spirit and Water in the Mixed Solution
Now, let's find the amount of spirit and water contributed by each solution based on its volume and ratio:
| Solution | Ratio (Spirit : Water) | Volume (ml) | Spirit Amount (ml) | Water Amount (ml) |
|---|---|---|---|---|
| 1 | 2 : 3 (Total 5 parts) | 630 | \(\frac{2}{5} \times 630 = 252\) | \(\frac{3}{5} \times 630 = 378\) |
| 2 | 3 : 4 (Total 7 parts) | 378 | \(\frac{3}{7} \times 378 = 162\) | \(\frac{4}{7} \times 378 = 216\) |
| 3 | 4 : 5 (Total 9 parts) | 252 | \(\frac{4}{9} \times 252 = 112\) | \(\frac{5}{9} \times 252 = 140\) |
The total amount of spirit in the mixed solution is the sum of spirit from each solution:
Total Spirit = \(252 \text{ ml} + 162 \text{ ml} + 112 \text{ ml} = 526 \text{ ml}\)
The total amount of water in the mixed solution is the sum of water from each solution:
Total Water = \(378 \text{ ml} + 216 \text{ ml} + 140 \text{ ml} = 734 \text{ ml}\)
The total volume of the mixed solution is \(526 \text{ ml} + 734 \text{ ml} = 1260 \text{ ml}\), which matches the given 1.26 liters.
Adding More Water
318 ml of water is added to the 1260 ml of the mixed solution.
The amount of spirit remains the same:
Final Spirit Amount = \(526 \text{ ml}\)
The amount of water increases:
Final Water Amount = Total Water + Added Water
Final Water Amount = \(734 \text{ ml} + 318 \text{ ml} = 1052 \text{ ml}\)
Finding the Final Spirit and Water Ratio
The final ratio of spirit to water is the ratio of the final spirit amount to the final water amount:
Final Ratio (Spirit : Water) = \(526 : 1052\)
To simplify the ratio, we can divide both numbers by their greatest common divisor. We can see that 1052 is twice 526 (\(526 \times 2 = 1052\)).
Simplified Ratio = \(\frac{526}{526} : \frac{1052}{526} = 1 : 2\)
The ratio of spirit and water in the final solution is 1 : 2.
Revision Table: Key Steps
| Step | Description | Calculation/Result |
|---|---|---|
| 1 | Calculate volume of each initial solution in the mix | 630 ml, 378 ml, 252 ml |
| 2 | Calculate spirit & water from each solution | Spirit: 252, 162, 112 ml Water: 378, 216, 140 ml |
| 3 | Find total spirit and water in mixed solution | Total Spirit: 526 ml Total Water: 734 ml |
| 4 | Add extra water to the total water amount | Final Water: 734 + 318 = 1052 ml |
| 5 | Form the final ratio and simplify | 526 : 1052 = 1 : 2 |
Additional Information on Ratio and Mixture Problems
Ratio and mixture problems are common in quantitative aptitude. They often involve combining substances with different proportions and calculating the resulting ratio. Key concepts include:
- Ratio: A comparison of two quantities. A ratio \(a : b\) means for every \(a\) units of one quantity, there are \(b\) units of another. The total parts are \(a + b\).
- Proportion: An equality of two ratios.
- Mixing Solutions: When mixing solutions, the total amount of each component (like spirit and water) in the final mixture is the sum of the amounts contributed by each initial solution.
- Unit Conversion: Ensure all quantities are in the same units (e.g., liters to milliliters) before performing calculations.
Solving these problems systematically by calculating the amount of each component from each source and then summing them up helps avoid errors.
Important Questions from To Make a Mixture from Two Mixtures
In a 90 litre solution, acid and water are in the ratio 2 ∶ 1, To make the ratio of acid and water as 1 ∶ 2, how many litre of water should be added to the solution?
How many litres of acid are there in 12 litres of a 20% solution?
The ratio of number of male and female employees in a company was 5 ∶ 4. Some new males and females joined the company in the ratio 2 ∶ 3. The total number of employees become 1620 and the ratio of males to females become 68 ∶ 67. Initially what was the total number of employees in the company?
A manufacturer has 600 L of a 12% solution of acid. How many litres of a 30% acid solution must be added to it such that acid content in the resulting mixture will be more than 15% but less than 18%?
How many kilograms of pure groundnut oil should be added to 20 kgs of 80% pure groundnut oil, so that the resulting mixture may contain 95% pure groundnut oil.
