A bottle of water is `(3)/(7)` th full. This water is poured into another bottle which is now `(2)/(3)` rd full. Find the ratio of the volume of the first bottle to the volume of the second bottle.

Answer

To find the ratio of the volume of the first bottle to the volume of the second bottle, we can follow these steps:

### Step 1: Understand the problem
We know that the first bottle is \( \frac{3}{7} \) full and the second bottle is \( \frac{2}{3} \) full after pouring the water from the first bottle into the second.

### Step 2: Set up the equation
Let the volume of the first bottle be \( V_1 \) and the volume of the second bottle be \( V_2 \). The amount of water in the first bottle can be expressed as:
\[ \text{Water in first bottle} = \frac{3}{7} V_1 \]
After pouring this water into the second bottle, the amount of water in the second bottle becomes:
\[ \text{Water in second bottle} = \frac{2}{3} V_2 \]
Since the amount of water transferred from the first bottle to the second bottle is the same, we can set these two expressions equal to each other:
\[ \frac{3}{7} V_1 = \frac{2}{3} V_2 \]

### Step 3: Rearrange the equation
To find the ratio \( \frac{V_1}{V_2} \), we can rearrange the equation:
\[ \frac{V_1}{V_2} = \frac{\frac{2}{3}}{\frac{3}{7}} \]

### Step 4: Simplify the right-hand side
To simplify \( \frac{\frac{2}{3}}{\frac{3}{7}} \), we multiply by the reciprocal of the denominator:
\[ \frac{V_1}{V_2} = \frac{2}{3} \times \frac{7}{3} = \frac{2 \times 7}{3 \times 3} = \frac{14}{9} \]

### Step 5: State the final ratio
Thus, the ratio of the volume of the first bottle to the volume of the second bottle is:
\[ \frac{V_1}{V_2} = \frac{14}{9} \]

### Final Answer
The ratio of the volume of the first bottle to the volume of the second bottle is \( 14:9 \).

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