The mean of two samples is 45. The mean of `1^(st)` samples and `2^(nd)` samples ware 30 and 50 respectively. Determine the ratio of the number of observations of the two samples.

Answer

To determine the ratio of the number of observations of two samples given their means, we can follow these steps:

### Step 1: Understand the Given Information
We know:
- The combined mean of the two samples (Xc) = 45
- The mean of the first sample (X1) = 30
- The mean of the second sample (X2) = 50

### Step 2: Use the Formula for Combined Mean
The formula for the combined mean of two samples is given by:

\[ X_c = \frac{m \cdot X_1 + n \cdot X_2}{m + n} \]

where:
- \(m\) = number of observations in the first sample
- \(n\) = number of observations in the second sample

### Step 3: Substitute the Known Values into the Formula
Substituting the known values into the formula:

\[ 45 = \frac{m \cdot 30 + n \cdot 50}{m + n} \]

### Step 4: Cross Multiply to Eliminate the Denominator
Cross multiplying gives us:

\[ 45(m + n) = 30m + 50n \]

### Step 5: Expand the Equation
Expanding the left side:

\[ 45m + 45n = 30m + 50n \]

### Step 6: Rearrange the Equation
Now, rearranging the equation to group like terms:

\[ 45m - 30m = 50n - 45n \]

This simplifies to:

\[ 15m = 5n \]

### Step 7: Solve for the Ratio of m to n
Dividing both sides by 5 gives:

\[ 3m = n \]

Thus, we can express the ratio of \(m\) to \(n\):

\[ \frac{m}{n} = \frac{1}{3} \]

### Step 8: Conclusion
Therefore, the ratio of the number of observations in the first sample to the second sample is:

\[ 1:3 \]

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