Which number line correctly shows the rational number `- (2)/(9) + (1)/(9) ?`

Options

A.
B.
C.
D.

Answer

To solve the question, we need to find the correct number line that represents the rational number \(-\frac{2}{9} + \frac{1}{9}\). Here’s the step-by-step solution:

### Step 1: Write the expression
We start with the expression given in the question:
\[ -\frac{2}{9} + \frac{1}{9} \]

### Step 2: Combine the fractions
Since both fractions have the same denominator, we can combine them by adding the numerators:
\[ -\frac{2}{9} + \frac{1}{9} = \frac{-2 + 1}{9} \]

### Step 3: Simplify the numerator
Now, simplify the numerator:
\[ -2 + 1 = -1 \]
So, we have:
\[ \frac{-1}{9} \]

### Step 4: Identify the location on the number line
The rational number \(-\frac{1}{9}\) is negative, and it lies between 0 and -1 on the number line.

### Step 5: Divide the interval between 0 and -1
To accurately place \(-\frac{1}{9}\) on the number line, we need to divide the interval between 0 and -1 into 9 equal parts (since the denominator is 9). Each part will represent a fraction of \(-\frac{1}{9}\).

### Step 6: Mark the points on the number line
The points on the number line will be:
- \(0\) (which can be represented as \(\frac{0}{9}\))
- \(-\frac{1}{9}\)
- \(-\frac{2}{9}\)
- \(-\frac{3}{9}\) (which is \(-\frac{1}{3}\))
- \(-\frac{4}{9}\)
- \(-\frac{5}{9}\)
- \(-\frac{6}{9}\) (which is \(-\frac{2}{3}\))
- \(-\frac{7}{9}\)
- \(-\frac{8}{9}\)
- \(-1\) (which can be represented as \(-\frac{9}{9}\))

### Step 7: Choose the correct number line
Now, we look for the number line that accurately shows these points, particularly \(-\frac{1}{9}\) between 0 and -1.

### Conclusion
The correct number line that shows the rational number \(-\frac{1}{9}\) is option A, as it correctly divides the interval between 0 and -1 into 9 equal parts.

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