Answer :
Let's complete the table by finding the missing entries step-by-step.
1. For the first fraction [tex]\(\frac{-3 \times 3}{8 \times 3}\)[/tex]:
- The numerator is [tex]\(-3 \times 3\)[/tex].
[tex]\[ -3 \times 3 = -9 \][/tex]
So, the numerator is [tex]\(-9\)[/tex].
- The denominator is [tex]\(8 \times 3\)[/tex], which is already given as 24.
- To find the sign of the rational number, note that the numerator is [tex]\(-9\)[/tex] (negative) and the denominator is [tex]\(24\)[/tex] (positive).
[tex]\[ \text{Sign} = \text{Numerator} < 0 \rightarrow \text{Negative} \][/tex]
Therefore, the sign is "Negative".
2. For the second fraction [tex]\(\frac{6}{7}\)[/tex]:
- The given entry shows that the fraction should be equivalent to having a numerator of 84. To find the denominator:
[tex]\[ \frac{6}{7} \text{ equivalent to } \frac{84}{x} \][/tex]
We scale the original fraction by a factor that transforms 6 to 84:
[tex]\[ 6 \times 14 = 84 \][/tex]
Therefore,
[tex]\[ x = 7 \times 14 = 98 \][/tex]
So, the denominator is [tex]\(98\)[/tex].
- The sign of the rational number is given as "Positive".
3. For the third fraction:
- We are given the numerator as [tex]\(-5\)[/tex] and the denominator as [tex]\(-8\)[/tex].
- To find the sign of the rational number, note that both the numerator and the denominator are negative.
[tex]\[ \frac{\text{Negative}}{\text{Negative}} = \text{Positive} \][/tex]
Therefore, the sign is "Positive".
4. For the fourth fraction [tex]\(\frac{-21 \times 2}{71 \times -2}\)[/tex]:
- The numerator is [tex]\(-21 \times 2\)[/tex].
[tex]\[ -21 \times 2 = -42 \][/tex]
So, the numerator is [tex]\(-42\)[/tex].
- The denominator is [tex]\(71 \times -2\)[/tex], which is already given as [tex]\(-142\)[/tex].
- To find the sign of the rational number, note that both the numerator and the denominator are negative or positive. However, we consider its simplified version:
[tex]\[ \frac{\text{Negative}}{\text{Positive or Negative}} = \text{Positive} \rightarrow \text{Negative} \rightarrow Positive \][/tex]
Therefore, the answer, considering all, is "Positive".
```
Let's fill in the table with the completed entries:
\begin{tabular}{|l|c|c|c|}
\hline \begin{tabular}{c}
Fraction in \\
Standard
\end{tabular} & Numerator & Denominator & \begin{tabular}{c}
Sign of the \\
rational number
\end{tabular} \\
\hline 1) [tex]$\frac{-3 \times 3}{8 \times 3}$[/tex] & -9 & 24 & Negative \\
\hline 2) [tex]$\frac{6}{7}$[/tex] & 84 & 98 & Positive \\
\hline 3) - & -5 & -8 & Positive \\
\hline 4) [tex]$\frac{-21 \times 2}{71 \times -2}$[/tex] & -42 & -142 & Positive \\
\hline
\end{tabular}
"`
1. For the first fraction [tex]\(\frac{-3 \times 3}{8 \times 3}\)[/tex]:
- The numerator is [tex]\(-3 \times 3\)[/tex].
[tex]\[ -3 \times 3 = -9 \][/tex]
So, the numerator is [tex]\(-9\)[/tex].
- The denominator is [tex]\(8 \times 3\)[/tex], which is already given as 24.
- To find the sign of the rational number, note that the numerator is [tex]\(-9\)[/tex] (negative) and the denominator is [tex]\(24\)[/tex] (positive).
[tex]\[ \text{Sign} = \text{Numerator} < 0 \rightarrow \text{Negative} \][/tex]
Therefore, the sign is "Negative".
2. For the second fraction [tex]\(\frac{6}{7}\)[/tex]:
- The given entry shows that the fraction should be equivalent to having a numerator of 84. To find the denominator:
[tex]\[ \frac{6}{7} \text{ equivalent to } \frac{84}{x} \][/tex]
We scale the original fraction by a factor that transforms 6 to 84:
[tex]\[ 6 \times 14 = 84 \][/tex]
Therefore,
[tex]\[ x = 7 \times 14 = 98 \][/tex]
So, the denominator is [tex]\(98\)[/tex].
- The sign of the rational number is given as "Positive".
3. For the third fraction:
- We are given the numerator as [tex]\(-5\)[/tex] and the denominator as [tex]\(-8\)[/tex].
- To find the sign of the rational number, note that both the numerator and the denominator are negative.
[tex]\[ \frac{\text{Negative}}{\text{Negative}} = \text{Positive} \][/tex]
Therefore, the sign is "Positive".
4. For the fourth fraction [tex]\(\frac{-21 \times 2}{71 \times -2}\)[/tex]:
- The numerator is [tex]\(-21 \times 2\)[/tex].
[tex]\[ -21 \times 2 = -42 \][/tex]
So, the numerator is [tex]\(-42\)[/tex].
- The denominator is [tex]\(71 \times -2\)[/tex], which is already given as [tex]\(-142\)[/tex].
- To find the sign of the rational number, note that both the numerator and the denominator are negative or positive. However, we consider its simplified version:
[tex]\[ \frac{\text{Negative}}{\text{Positive or Negative}} = \text{Positive} \rightarrow \text{Negative} \rightarrow Positive \][/tex]
Therefore, the answer, considering all, is "Positive".
```
Let's fill in the table with the completed entries:
\begin{tabular}{|l|c|c|c|}
\hline \begin{tabular}{c}
Fraction in \\
Standard
\end{tabular} & Numerator & Denominator & \begin{tabular}{c}
Sign of the \\
rational number
\end{tabular} \\
\hline 1) [tex]$\frac{-3 \times 3}{8 \times 3}$[/tex] & -9 & 24 & Negative \\
\hline 2) [tex]$\frac{6}{7}$[/tex] & 84 & 98 & Positive \\
\hline 3) - & -5 & -8 & Positive \\
\hline 4) [tex]$\frac{-21 \times 2}{71 \times -2}$[/tex] & -42 & -142 & Positive \\
\hline
\end{tabular}
"`
