To find the expression that best estimates \(-18 \frac{1}{4} \div 2 \frac{2}{3}\), let's go through the problem step-by-step.
### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers \(-18 \frac{1}{4}\) and \(2 \frac{2}{3}\) into improper fractions.
1. \(-18 \frac{1}{4}\):
[tex]\[
-18 \frac{1}{4} = -18 - \frac{1}{4} = -18.25
\][/tex]
2. \(2 \frac{2}{3}\):
[tex]\[
2 \frac{2}{3} = 2 + \frac{2}{3} = 2.6667 \, (\text{approximately})
\][/tex]
### Step 2: Perform the Division
Now, we perform the division of these two numbers:
[tex]\[
\frac{-18.25}{2.6667} \approx -6.84375
\][/tex]
### Step 3: Identify the Closest Fraction from Choices
We compare the computed value with the given choices:
1. \(18 \div 3\):
[tex]\[
18 \div 3 = 6.0
\][/tex]
2. \(-18 \div 3\):
[tex]\[
-18 \div 3 = -6.0
\][/tex]
3. \(-18 \div (-3)\):
[tex]\[
-18 \div (-3) = 6.0
\][/tex]
4. \(18 \div (-3)\):
[tex]\[
18 \div (-3) = -6.0
\][/tex]
### Step 4: Find the Closest Estimate
Now, we determine which one of the results above is closest to \(-6.84375\). We compare the absolute differences:
- Difference from \(6.0\):
[tex]\[
|6.0 - (-6.84375)| = 12.84375
\][/tex]
- Difference from \(-6.0\):
[tex]\[
|-6.0 - (-6.84375)| = 0.84375
\][/tex]
Since \(|-6.0 - (-6.84375)| = 0.84375\) is the smallest difference, the choice \(-18 \div 3 = -6.0\) is the closest estimate.
Thus, the best estimate for the expression \(-18 \frac{1}{4} \div 2 \frac{2}{3}\) is:
[tex]\[
18 \div (-3) = -6.0
\][/tex]
