A tin contains a mixture of Dew and Sprite in the ratio of `7:3` and another tin contain the Dew and Sprite in the ratio of 5:4. In what proportion should the solution of two tins be mixed to achieve a perfect proportion of 2:1 (in which Dew is 2 times that of sprite).

Options

A. `10:3`
B. `4:1`
C. `3:10`
D. `3:1`

Answer

To solve the problem of mixing two tins containing Dew and Sprite in specific ratios to achieve a desired ratio, we can follow these steps:

### Step 1: Identify the Ratios
- Tin 1 contains Dew and Sprite in the ratio of 7:3.
- Tin 2 contains Dew and Sprite in the ratio of 5:4.
- We want to mix them to achieve a final ratio of Dew to Sprite of 2:1.

### Step 2: Calculate the Fraction of Dew in Each Tin
For Tin 1:
- Total parts = 7 (Dew) + 3 (Sprite) = 10
- Fraction of Dew in Tin 1 = \( \frac{7}{10} \)

For Tin 2:
- Total parts = 5 (Dew) + 4 (Sprite) = 9
- Fraction of Dew in Tin 2 = \( \frac{5}{9} \)

### Step 3: Calculate the Fraction of Dew in the Desired Mixture
For the desired mixture (2:1 ratio):
- Total parts = 2 (Dew) + 1 (Sprite) = 3
- Fraction of Dew in the mixture = \( \frac{2}{3} \)

### Step 4: Set Up the Allegation Method
Using the allegation method, we can find the ratio in which the two tins should be mixed.

- Dew fraction in Tin 1 = \( \frac{7}{10} \)
- Dew fraction in Tin 2 = \( \frac{5}{9} \)
- Dew fraction in the mixture = \( \frac{2}{3} \)

### Step 5: Apply the Allegation Formula
Using the formula:
- Difference between the Dew fraction in the mixture and Tin 1:
\[ \frac{2}{3} - \frac{7}{10} \]
To calculate this, we need a common denominator (which is 30):
\[ \frac{2 \times 10}{30} - \frac{7 \times 3}{30} = \frac{20}{30} - \frac{21}{30} = -\frac{1}{30} \]

- Difference between the Dew fraction in Tin 2 and the mixture:
\[ \frac{5}{9} - \frac{2}{3} \]
Again, using a common denominator (which is 9):
\[ \frac{5}{9} - \frac{6}{9} = -\frac{1}{9} \]

### Step 6: Find the Ratio
Now we have:
- The difference for Tin 1: \( \frac{1}{30} \)
- The difference for Tin 2: \( \frac{1}{9} \)

The ratio of Tin 1 to Tin 2 is given by:
\[ \text{Ratio} = \frac{\frac{1}{9}}{\frac{1}{30}} = \frac{30}{9} = \frac{10}{3} \]

### Final Answer
Thus, the two tins should be mixed in the ratio of **10:3**.