Sure, let's solve the given problem step by step.
### Part a: Finding the value of [tex]\( x \)[/tex]
We are given a table that shows the number of children of each age in terms of [tex]\( x \)[/tex]. The total number of children is 36. We can form an equation with this information.
Given:
- Number of children aged 7: [tex]\( 2x \)[/tex]
- Number of children aged 8: [tex]\( 3x \)[/tex]
- Number of children aged 9: [tex]\( 4x - 1 \)[/tex]
- Number of children aged 10: [tex]\( x \)[/tex]
- Number of children aged 11: [tex]\( x - 2 \)[/tex]
- Number of children aged 12: [tex]\( x - 3 \)[/tex]
The sum of all these children is 36, so we can write:
[tex]\[ 2x + 3x + (4x - 1) + x + (x - 2) + (x - 3) = 36 \][/tex]
Simplifying the equation:
[tex]\[ 2x + 3x + 4x - 1 + x + x - 2 + x - 3 = 36 \][/tex]
[tex]\[ 12x - 6 = 36 \][/tex]
Adding 6 to both sides:
[tex]\[ 12x = 42 \][/tex]
Dividing both sides by 12:
[tex]\[ x = \frac{42}{12} = \frac{7}{2} \][/tex]
So the value of [tex]\( x \)[/tex] is [tex]\( \frac{7}{2} \)[/tex].
### Part b: Calculating the mean age of the children
Next, we calculate the total sum of the ages of all the children and then find the mean age by dividing by the total number of children (36).
First, let's express the number of children for each age in terms of [tex]\( x \)[/tex]:
- Number of children aged 7: [tex]\( 2x = 2 \times \frac{7}{2} = 7 \)[/tex]
- Number of children aged 8: [tex]\( 3x = 3 \times \frac{7}{2} = \frac{21}{2} \)[/tex]
- Number of children aged 9: [tex]\( 4x - 1 = 4 \times \frac{7}{2} - 1 = 14 - 1 = 13 \)[/tex]
- Number of children aged 10: [tex]\( x = \frac{7}{2} \)[/tex]
- Number of children aged 11: [tex]\( x - 2 = \frac{7}{2} - 2 = \frac{7}{2} - \frac{4}{2} = \frac{3}{2} \)[/tex]
- Number of children aged 12: [tex]\( x - 3 = \frac{7}{2} - 3 = \frac{7}{2} - \frac{6}{2} = \frac{1}{2} \)[/tex]
Now, we calculate the total sum of the ages of all the children:
[tex]\[ \text{Total Ages} = 7 \times 7 + 8 \times \frac{21}{2} + 9 \times 13 + 10 \times \frac{7}{2} + 11 \times \frac{3}{2} + 12 \times \frac{1}{2} \][/tex]
Breaking it down:
[tex]\[ = 49 + 8 \times 10.5 + 117 + 10 \times 3.5 + 11 \times 1.5 + 12 \times 0.5 \][/tex]
[tex]\[ = 49 + 84 + 117 + 35 + 16.5 + 6 \][/tex]
[tex]\[ = 307.5 \][/tex]
Finally, calculate the mean age:
[tex]\[ \text{Mean Age} = \frac{\text{Total Ages}}{\text{Total Number of Children}} = \frac{307.5}{36} \approx 8.54 \][/tex]
Rounding the mean age to the nearest whole number:
[tex]\[ \text{Mean Age} \approx 9 \][/tex]
Therefore, the mean age of the children at the birthday party, to the nearest whole number, is 9.
