Certainly! Let's solve each of the given proportions step by step to find the values of [tex]\( x \)[/tex], [tex]\( d \)[/tex], [tex]\( n \)[/tex], and [tex]\( z \)[/tex].
### Proportion 1: [tex]\( \frac{8}{24} = \frac{x}{42} \)[/tex]
1. To solve for [tex]\( x \)[/tex], we need to cross-multiply.
2. [tex]\( 8 \times 42 = 24 \times x \)[/tex]
3. Simplifying the left side: [tex]\( 336 = 24x \)[/tex]
4. Divide both sides by 24: [tex]\( x = \frac{336}{24} \)[/tex]
5. Simplifying: [tex]\( x = 14 \)[/tex]
So, [tex]\( x = 14.0 \)[/tex].
### Proportion 2: [tex]\( \frac{16}{96} = \frac{4}{d} \)[/tex]
1. To solve for [tex]\( d \)[/tex], we cross-multiply.
2. [tex]\( 16 \times d = 96 \times 4 \)[/tex]
3. Simplifying the right side: [tex]\( 16d = 384 \)[/tex]
4. Divide both sides by 16: [tex]\( d = \frac{384}{16} \)[/tex]
5. Simplifying: [tex]\( d = 24 \)[/tex]
So, [tex]\( d = 24.0 \)[/tex].
### Proportion 3: [tex]\( \frac{8}{n} = \frac{40}{65} \)[/tex]
1. To solve for [tex]\( n \)[/tex], we need to cross-multiply.
2. [tex]\( 8 \times 65 = 40 \times n \)[/tex]
3. Simplifying the left side: [tex]\( 520 = 40n \)[/tex]
4. Divide both sides by 40: [tex]\( n = \frac{520}{40} \)[/tex]
5. Simplifying: [tex]\( n = 13 \)[/tex]
So, [tex]\( n = 13.0 \)[/tex].
### Proportion 4: [tex]\( \frac{z}{15} = \frac{19}{57} \)[/tex]
1. To solve for [tex]\( z \)[/tex], we need to cross-multiply.
2. [tex]\( z \times 57 = 19 \times 15 \)[/tex]
3. Simplifying the right side: [tex]\( 57z = 285 \)[/tex]
4. Divide both sides by 57: [tex]\( z = \frac{285}{57} \)[/tex]
5. Simplifying: [tex]\( z = 5 \)[/tex]
So, [tex]\( z = 5.0 \)[/tex].
In summary, the solutions to the proportions are:
[tex]\[
\begin{tabular}{|c|l|}
\hline Proportion & Solution \\
\hline$\frac{8}{24}=\frac{x}{42}$ & \( x = 14.0 \) \\
\hline$\frac{16}{96}=\frac{4}{d}$ & \( d = 24.0 \) \\
\hline$\frac{8}{n}=\frac{40}{65}$ & \( n = 13.0 \) \\
\hline$\frac{z}{15}=\frac{19}{57}$ & \( z = 5.0 \) \\
\hline
\end{tabular}
\][/tex]
