A tube A of radius r is connected to two other tubes B and C with the help of a junction valve. The tubes B and C have radii `(r )/(2)` and `(r )/(3)` and the flow velocity in each is v and 3v respectively. If the flow velocity in tube A is `(n)/(m)v`, when n and m are integers, then what is the value of `n+m` ?

Answer

To solve the problem, we will apply the principle of conservation of mass, which states that the mass flow rate must be constant in a closed system. This can be expressed using the equation of continuity for incompressible fluids:

\[ A_1 v_1 = A_2 v_2 + A_3 v_3 \]

Where:
- \( A_1 \) is the cross-sectional area of tube A
- \( v_1 \) is the flow velocity in tube A
- \( A_2 \) is the cross-sectional area of tube B
- \( v_2 \) is the flow velocity in tube B
- \( A_3 \) is the cross-sectional area of tube C
- \( v_3 \) is the flow velocity in tube C

### Step 1: Calculate the cross-sectional areas
The cross-sectional area \( A \) of a tube is given by the formula:

\[ A = \pi r^2 \]

- For tube A (radius \( r \)):
\[ A_1 = \pi r^2 \]

- For tube B (radius \( \frac{r}{2} \)):
\[ A_2 = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} \]

- For tube C (radius \( \frac{r}{3} \)):
\[ A_3 = \pi \left(\frac{r}{3}\right)^2 = \pi \frac{r^2}{9} \]

### Step 2: Substitute the areas and velocities into the continuity equation
Given that the flow velocity in tube B is \( v \) and in tube C is \( 3v \), we can substitute these values into the continuity equation:

\[ A_1 v_1 = A_2 v + A_3 (3v) \]

Substituting the areas we calculated:

\[ \pi r^2 v_1 = \left(\pi \frac{r^2}{4}\right) v + \left(\pi \frac{r^2}{9}\right) (3v) \]

### Step 3: Simplify the equation
We can cancel \( \pi \) from both sides:

\[ r^2 v_1 = \frac{r^2}{4} v + \frac{3r^2}{9} v \]

Now, simplifying the right-hand side:

\[ r^2 v_1 = \frac{r^2}{4} v + \frac{1}{3} r^2 v \]

Finding a common denominator (which is 12):

\[ r^2 v_1 = \left(\frac{3}{12} + \frac{4}{12}\right) r^2 v \]

\[ r^2 v_1 = \frac{7}{12} r^2 v \]

### Step 4: Solve for \( v_1 \)
Now we can divide both sides by \( r^2 \):

\[ v_1 = \frac{7}{12} v \]

### Step 5: Identify \( n \) and \( m \)
From the expression \( v_1 = \frac{n}{m} v \), we can see that \( n = 7 \) and \( m = 12 \).

### Step 6: Calculate \( n + m \)
Thus, the value of \( n + m \) is:

\[ n + m = 7 + 12 = 19 \]

### Final Answer
The final answer is \( 19 \).

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