If p : q = r : s = t : u = 2 : 3, then (mp + nr + ot) : mq + ns + ou) is equal to :

Options

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

Answer

To solve the problem, we need to find the ratio \((mp + nr + ot) : (mq + ns + ou)\) given that \(p : q = r : s = t : u = 2 : 3\).

### Step-by-Step Solution:

1. **Understand the Ratios**:
We know that \(p : q = 2 : 3\), \(r : s = 2 : 3\), and \(t : u = 2 : 3\). This means we can express \(p\), \(r\), and \(t\) in terms of \(q\), \(s\), and \(u\) respectively.
- From \(p : q = 2 : 3\), we can write:
\[ p = \frac{2}{3}q \]
- From \(r : s = 2 : 3\), we can write:
\[ r = \frac{2}{3}s \]
- From \(t : u = 2 : 3\), we can write:
\[ t = \frac{2}{3}u \]

2. **Substitute Values into the Expression**:
We need to substitute these values into the expression \(mp + nr + ot\) and \(mq + ns + ou\).
- Substitute \(p\), \(r\), and \(t\):
\[ mp + nr + ot = m\left(\frac{2}{3}q\right) + n\left(\frac{2}{3}s\right) + o\left(\frac{2}{3}u\right) \]
This simplifies to:
\[ = \frac{2}{3}(mq + ns + ou) \]

3. **Write the Denominator**:
The denominator remains as:
\[ mq + ns + ou \]

4. **Form the Ratio**:
Now we can form the ratio:
\[ (mp + nr + ot) : (mq + ns + ou) = \left(\frac{2}{3}(mq + ns + ou)\right) : (mq + ns + ou) \]

5. **Simplify the Ratio**:
We can simplify this ratio:
\[ = \frac{2}{3} : 1 \]
This can also be expressed as:
\[ = 2 : 3 \]

### Final Answer:
Thus, the final ratio is:
\[ \boxed{2 : 3} \]