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. **Understanding the Ratios**:
We have the ratios:
\[ p:q = 2:3, \quad r:s = 2:3, \quad t:u = 2:3 \]
This means we can express \(p\), \(q\), \(r\), \(s\), \(t\), and \(u\) in terms of a common variable. Let's set:
\[ p = 2x, \quad q = 3x, \quad r = 2y, \quad s = 3y, \quad t = 2z, \quad u = 3z \]
where \(x\), \(y\), and \(z\) are any positive constants.

2. **Substituting into the Expression**:
Now we substitute these values into the expression \((mp + nr + ot) : (mq + ns + ou)\):
\[ mp + nr + ot = m(2x) + n(2y) + o(2z) = 2(mx + ny + oz) \]
\[ mq + ns + ou = m(3x) + n(3y) + o(3z) = 3(mx + ny + oz) \]

3. **Forming the Ratio**:
Now we can form the ratio:
\[ (mp + nr + ot) : (mq + ns + ou) = (2(mx + ny + oz)) : (3(mx + ny + oz)) \]

4. **Simplifying the Ratio**:
Since \(mx + ny + oz\) is common in both terms, we can simplify the ratio:
\[ = \frac{2}{3} \]

5. **Final Answer**:
Therefore, the final ratio is:
\[ (mp + nr + ot) : (mq + ns + ou) = 2 : 3 \]