To find the remaining five numbers given their ratio and the mean of the observations, follow these steps:
1. Calculate the total sum of all observations:
The mean of observations is given as 50, and there are 9 observations in total.
[tex]\[
\text{Total sum} = (\text{Mean}) \times (\text{Number of observations}) = 50 \times 9 = 450
\][/tex]
2. Calculate the sum of the given observations:
The four given observations are 20, 21, 22, and 23.
[tex]\[
\text{Sum of given observations} = 20 + 21 + 22 + 23 = 86
\][/tex]
3. Find the sum of the remaining observations:
Subtract the sum of the given observations from the total sum of all observations to find the sum of the remaining five observations.
[tex]\[
\text{Sum of remaining observations} = \text{Total sum} - \text{Sum of given observations} = 450 - 86 = 364
\][/tex]
4. Analyze the ratio of the remaining five numbers:
The ratio given for the remaining five numbers is [tex]\(2:4:5:6:9\)[/tex]. Let's denote these five numbers as [tex]\(2x, 4x, 5x, 6x,\)[/tex] and [tex]\(9x\)[/tex]. The total parts of the ratio are:
[tex]\[
2 + 4 + 5 + 6 + 9 = 26
\][/tex]
This means each part of the ratio represents a fraction of the sum of the remaining observations.
5. Determine each remaining number based on the ratio:
Since the sum of the remaining observations is 364, we can find each number corresponding to its ratio part of the total sum. So,
[tex]\[
2x = \frac{2}{26} \times 364 = \frac{2 \times 364}{26} = 28
\][/tex]
[tex]\[
4x = \frac{4}{26} \times 364 = \frac{4 \times 364}{26} = 56
\][/tex]
[tex]\[
5x = \frac{5}{26} \times 364 = \frac{5 \times 364}{26} = 70
\][/tex]
[tex]\[
6x = \frac{6}{26} \times 364 = \frac{6 \times 364}{26} = 84
\][/tex]
[tex]\[
9x = \frac{9}{26} \times 364 = \frac{9 \times 364}{26} = 126
\][/tex]
So, the remaining five numbers are [tex]\(28, 56, 70, 84,\)[/tex] and [tex]\(126\)[/tex].
