Express `-5/9` as a rational number with numerator 40

Answer

To express \(-\frac{5}{9}\) as a rational number with a numerator of 40, we can follow these steps:

### Step 1: Identify the relationship
We want to find a rational number that has a numerator of 40 while maintaining the same value as \(-\frac{5}{9}\). This means we need to find a multiplier that will change the numerator from -5 to 40.

### Step 2: Set up the equation
Let \(x\) be the multiplier. We can set up the equation:
\[ -\frac{5}{9} \times x = \frac{40}{9} \]

### Step 3: Solve for \(x\)
To isolate \(x\), we can rearrange the equation:
\[ x = \frac{40}{9} \div -\frac{5}{9} \]
This can be rewritten as:
\[ x = \frac{40}{9} \times \frac{9}{-5} \]

### Step 4: Simplify the expression
Now, we can simplify:
\[ x = \frac{40 \times 9}{9 \times -5} = \frac{40}{-5} \]
This simplifies to:
\[ x = -8 \]

### Step 5: Multiply the original rational number
Now that we have \(x = -8\), we can multiply both the numerator and the denominator of \(-\frac{5}{9}\) by -8:
\[ -\frac{5 \times -8}{9 \times -8} = \frac{40}{-72} \]

### Step 6: Final expression
Thus, we can express \(-\frac{5}{9}\) as:
\[ -\frac{40}{72} \]
This means that \(-\frac{5}{9}\) is equivalent to \(-\frac{40}{72}\) with a numerator of 40.