Certainly! Let’s fill in the missing values in the table step by step.
The given table is:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$x$ & a & 12 & c & 36 & 54 & e \\
\hline
$y$ & 9 & b & 45 & 54 & d & 126 \\
\hline
\end{tabular}
\][/tex]
To start, let's observe that there is a consistent ratio between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. We can determine this ratio using the known values. Here are the calculations based on identified patterns and given steps:
1. Finding [tex]\( b \)[/tex]:
- Given [tex]\( x = 12 \)[/tex] and the ratio [tex]\( x / y \)[/tex]:
- Calculate [tex]\( \frac{12}{b} \)[/tex].
From the ratio derived from another pair [tex]\( (54, 126) \)[/tex]:
[tex]\[
\frac{12}{b} = \frac{4}{15}
\][/tex]
Thus,
[tex]\[
b = 12 \times \frac{15}{4} = 45.0
\][/tex]
2. Finding [tex]\( d \)[/tex]:
- Given [tex]\( x = 54 \)[/tex]:
- Calculate [tex]\( \frac{54}{d} \)[/tex].
Using the consistent ratio:
[tex]\[
\frac{54}{d} = \frac{4}{15}
\][/tex]
Thus,
[tex]\[
d = 54 \times \frac{15}{4} = 202.5
\][/tex]
3. Finding [tex]\( a \)[/tex]:
- Given [tex]\( y = 9 \)[/tex]:
- Calculate [tex]\( \frac{a}{9} \)[/tex].
With the ratio:
[tex]\[
\frac{a}{9} = \frac{4}{15}
\][/tex]
Thus,
[tex]\[
a = 9 \times \frac{4}{15} = 2.4
\][/tex]
4. Finding [tex]\( e \)[/tex]:
- Given [tex]\( y = 126 \)[/tex]:
- Calculate [tex]\( \frac{e}{126} \)[/tex].
Following the ratio:
[tex]\[
\frac{e}{126} = \frac{4}{15}
\][/tex]
Thus,
[tex]\[
e = 126 \times \frac{4}{15} = 33.6
\][/tex]
With these calculations, the completed table is:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$x$ & 2.4 & 12 & 36 & 36 & 54 & 33.6 \\
\hline
$y$ & 9 & 45 & 45 & 54 & 202.5 & 126 \\
\hline
\end{tabular}
\][/tex]
So the missing values in the table are:
- [tex]\( a = 2.4 \)[/tex]
- [tex]\( b = 45.0 \)[/tex]
- [tex]\( d = 202.5 \)[/tex]
- [tex]\( e = 33.6 \)[/tex]
[tex]\[
\boxed{2.4, 45.0, 36, 202.5, 33.6}
\][/tex]
