Which number line correctly shows the rational number `((9)/(7) ÷ (24)/(14))` ?

Options

A.
B.
C.
D.

Answer

To solve the problem of identifying which number line correctly shows the rational number \(\frac{9}{7} \div \frac{24}{14}\), we will follow these steps:

### Step 1: Simplify the Division of Rational Numbers
We start with the expression:
\[ \frac{9}{7} \div \frac{24}{14} \]
To divide by a fraction, we multiply by its reciprocal:
\[ \frac{9}{7} \times \frac{14}{24} \]

### Step 2: Multiply the Numerators and Denominators
Now we multiply the numerators and the denominators:
\[ \frac{9 \times 14}{7 \times 24} \]

### Step 3: Simplify the Fractions
Next, we simplify the fractions. We can simplify \(14\) and \(7\) first:
\[ \frac{9 \times 2}{1 \times 24} = \frac{18}{24} \]
Now, we can simplify \(\frac{18}{24}\) by dividing both the numerator and the denominator by their greatest common divisor, which is \(6\):
\[ \frac{18 \div 6}{24 \div 6} = \frac{3}{4} \]

### Step 4: Identify the Position of \(\frac{3}{4}\) on the Number Line
Now we need to find out where \(\frac{3}{4}\) lies on a number line. The number line between \(0\) and \(1\) can be divided into \(4\) equal parts:
- \(0\) (which is \(\frac{0}{4}\))
- \(\frac{1}{4}\)
- \(\frac{2}{4}\)
- \(\frac{3}{4}\)
- \(1\) (which is \(\frac{4}{4}\))

### Step 5: Check the Options
Now we check the given options (A, B, C, D) to see which one correctly represents \(\frac{3}{4}\):
- **Option A**: Shows \(0, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, 1\)
- **Option B**: Shows \(0, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, 1\) (but incorrectly marked)
- **Option C**: Shows \(0, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, 1\) (but incorrectly marked)
- **Option D**: Shows \(0, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, 1\) (but incorrectly marked)

From our analysis, it is clear that **Option A** correctly represents \(\frac{3}{4}\).

### Final Answer
The correct number line that shows the rational number \(\frac{9}{7} \div \frac{24}{14}\) is **Option A**.

---