Question

Which expression does the number line represent?

A number line going from 0 to StartFraction 9 Over 4 EndFraction in increments of one-fourth. Arrows go from 0 to three-fourths, from three-fourths to six-fourths, and from six-fourths to nine-fourths.
3 divided by three-fourths
Three-fourths divided by StartFraction 9 Over 4 EndFraction
3 divided by StartFraction 9 Over 4 EndFraction
StartFraction 9 Over 4 EndFraction divided by three-fourths

Answer 1

To determine which expression the number line represents, we need to analyze how the number line is set up and what it indicates.

The number line goes from 0 to \( \frac{9}{4} \) in increments of one-fourth. The points marked on the number line are:
- 0 (which is \( \frac{0}{4} \))
- \( \frac{3}{4} \) (which is \( \frac{3}{4} \))
- \( \frac{6}{4} \) (which is \( \frac{3}{2} \))
- \( \frac{9}{4} \)

From the increment of one-fourth, we can interpret the total number of increments needed to go from 0 to \( \frac{9}{4} \). Since \( \frac{9}{4} \) can also be expressed as 2.25 or \( 2 \frac{1}{4} \), you reach \( \frac{9}{4} \) in 9 increments (each being \( \frac{1}{4} \)) from 0.

Now let's match this interpretation to the expressions presented:

1. **\( \frac{3}{\frac{3}{4}} \)** - This evaluates to \( 3 \div \frac{3}{4} = 3 \times \frac{4}{3} = 4 \).

2. **\( \frac{3}{4} \div \frac{9}{4} \)** - This evaluates to \( \frac{3}{4} \times \frac{4}{9} = \frac{3}{9} = \frac{1}{3} \).

3. **\( \frac{3}{\frac{9}{4}} \)** - This evaluates to \( 3 \div \frac{9}{4} = 3 \times \frac{4}{9} = \frac{12}{9} = \frac{4}{3} \) which is \( \frac{6}{4} \).

4. **\( \frac{9}{4} \div \frac{3}{4} \)** - This gives \( \frac{9}{4} \times \frac{4}{3} = \frac{9 \times 4}{4 \times 3} = \frac{9}{3} = 3 \).

None of these expressions perfectly match, but if we consider the largest endpoint of the number line, going from 0 to \( \frac{9}{4} \) can be seen as dividing the total extent (in value 2.25) uniformly down to the smallest increments in relation to \( \frac{3}{4} \) or reaching \( \frac{9}{4} \) from 0 distributed across the increments, it implies a relationship like \( \frac{9/4}{3/4} \) being an entire division.

Thus, the expression that best represents the length represented by this number line setup from 0 to \( \frac{9}{4} \) modeled against divisible portions \( \frac{3}{4} \) would be:

**\( \frac{9}{4} \div \frac{3}{4} \)**

So the final answer is \( \frac{9}{4} \div \frac{3}{4} \).