Sure, let's solve these proportions step-by-step.
### Part (a):
We are given the proportion: [tex]\(\frac{x}{6} = \frac{55}{11}\)[/tex].
1. First, we'll set up the equation:
[tex]\[\frac{x}{6} = \frac{55}{11}\][/tex]
2. Cross-multiply to solve for [tex]\(x\)[/tex]:
[tex]\[x \cdot 11 = 6 \cdot 55\][/tex]
3. Calculate the right-hand side of the equation:
[tex]\[6 \cdot 55 = 330\][/tex]
4. Now solve for [tex]\(x\)[/tex]:
[tex]\[x \cdot 11 = 330 \implies x = \frac{330}{11}\][/tex]
5. Simplify the fraction:
[tex]\[x = 30\][/tex]
So, the value of [tex]\(x\)[/tex] in part (a) is [tex]\(30.0\)[/tex].
### Part (b):
We are given the proportion: [tex]\(\frac{16}{18} = \frac{x}{108}\)[/tex].
1. First, we'll set up the equation:
[tex]\[\frac{16}{18} = \frac{x}{108}\][/tex]
2. Cross-multiply to solve for [tex]\(x\)[/tex]:
[tex]\[16 \cdot 108 = 18 \cdot x\][/tex]
3. Calculate the left-hand side of the equation:
[tex]\[16 \cdot 108 = 1728\][/tex]
4. Now solve for [tex]\(x\)[/tex]:
[tex]\[1728 = 18 \cdot x \implies x = \frac{1728}{18}\][/tex]
5. Simplify the fraction:
[tex]\[x = 96\][/tex]
So, the value of [tex]\(x\)[/tex] in part (b) is [tex]\(96.0\)[/tex].
### Part (c):
We are given the proportion: [tex]\(\frac{4}{x} = \frac{x}{16}\)[/tex].
1. First, we'll set up the equation:
[tex]\[\frac{4}{x} = \frac{x}{16}\][/tex]
2. Cross-multiply to solve for [tex]\(x\)[/tex]:
[tex]\[4 \cdot 16 = x \cdot x\][/tex]
3. Simplify the left-hand side of the equation:
[tex]\[64 = x^2\][/tex]
4. Solve for [tex]\(x\)[/tex] by taking the square root of both sides:
[tex]\[x = \sqrt{64}\][/tex]
5. Calculate the square root:
[tex]\[x = 8\][/tex]
So, the value of [tex]\(x\)[/tex] in part (c) is [tex]\(8.0\)[/tex].
In summary, the solutions for the given proportions are:
- [tex]\(x = 30.0\)[/tex] in part (a)
- [tex]\(x = 96.0\)[/tex] in part (b)
- [tex]\(x = 8.0\)[/tex] in part (c)
