To determine which of the given expressions can be modeled using the number line, let's carefully evaluate and interpret each one:
1. [tex]\(\frac{1}{10} \div 7\)[/tex]:
- This expression means "one-tenth divided by seven".
- Evaluating this, [tex]\(\frac{1}{10} \div 7 = \frac{1}{10} \times \frac{1}{7} = \frac{1}{70}\)[/tex].
- The result is approximately [tex]\(0.014285714\)[/tex].
2. [tex]\(\frac{7}{10} \div \frac{1}{10}\)[/tex]:
- This expression means "seven-tenths divided by one-tenth".
- Evaluating this, [tex]\(\frac{7}{10} \div \frac{1}{10} = \frac{7}{10} \times \frac{10}{1} = 7\)[/tex].
- The result is [tex]\(7\)[/tex].
3. [tex]\(\frac{1}{10} \div \frac{7}{10}\)[/tex]:
- This expression means "one-tenth divided by seven-tenths".
- Evaluating this, [tex]\(\frac{1}{10} \div \frac{7}{10} = \frac{1}{10} \times \frac{10}{7} = \frac{1}{7}\)[/tex].
- The result is approximately [tex]\(0.142857143\)[/tex].
4. [tex]\(7 \div \frac{7}{10}\)[/tex]:
- This expression means "seven divided by seven-tenths".
- Evaluating this, [tex]\(7 \div \frac{7}{10} = 7 \times \frac{10}{7} = 10\)[/tex].
- The result is [tex]\(10\)[/tex].
Next, we consider which of these results can be modeled easily using a number line.
- [tex]\(\frac{1}{70}\)[/tex] and [tex]\(\frac{1}{7}\)[/tex] are very small fractions and would be less convenient to represent on a number line in a clear and precise way, given their decimal representations are non-integers.
- The value [tex]\(7\)[/tex] can certainly be placed on a number line, but as it is a whole number.
- The value [tex]\(10\)[/tex] is also a whole number and sits very comfortably on the number line.
Since we are looking for the expression that can be most easily modeled on a number line, the expression that results in a whole number is most straightforward. Therefore, the most suitable expression for modeling on a number line is:
[tex]\[ 7 \div \frac{7}{10} \][/tex]
This expression evaluates to [tex]\(10\)[/tex], which can be represented clearly on a number line. Therefore, the expression [tex]\(7 \div \frac{7}{10}\)[/tex] is the correct answer.
