A person has two solutions of sugar with 30% and 50% con centration respectively. In what ratio should he mix two solutions to get 45% concentration in the resulting mixture?

Options

A. `1:3`
B. `2:3`
C. `2:5`
D. `5:2`

Answer

To solve the problem of mixing two sugar solutions with concentrations of 30% and 50% to achieve a final concentration of 45%, we can use the method of alligation. Here’s the step-by-step solution:

### Step 1: Identify the Concentrations
We have two solutions:
- Solution A: 30% sugar concentration
- Solution B: 50% sugar concentration
- Desired concentration: 45%

### Step 2: Set Up the Alligation
To find the ratio in which the two solutions should be mixed, we can use the alligation method. We will calculate the differences between the concentrations of the solutions and the desired concentration.

### Step 3: Calculate the Differences
- Difference between Solution B (50%) and Desired Concentration (45%):
\[ 50\% - 45\% = 5\% \]

- Difference between Solution A (30%) and Desired Concentration (45%):
\[ 45\% - 30\% = 15\% \]

### Step 4: Set Up the Ratio
According to the alligation rule, the ratio of the two solutions can be determined by the differences calculated:
- The ratio of Solution A to Solution B is given by the differences:
\[ \text{Ratio} = \frac{\text{Difference of B}}{\text{Difference of A}} = \frac{5\%}{15\%} \]

### Step 5: Simplify the Ratio
Now, simplify the ratio:
\[ \frac{5}{15} = \frac{1}{3} \]

### Conclusion
Thus, the two solutions should be mixed in the ratio of 1:3 to achieve a 45% concentration in the resulting mixture.

### Final Answer
The required ratio of mixing the two solutions is **1:3**.

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