Sure, let's go through the problem step by step:
1. Understanding the Problem:
- The car's entire fuel tank capacity is [tex]\(8 \frac{1}{4}\)[/tex] gallons.
- It currently contains [tex]\(4 \frac{5}{6}\)[/tex] gallons of fuel.
- We need to determine how much more fuel is required to fill up the tank completely.
2. Convert Mixed Numbers to Improper Fractions:
- First, we need to convert the mixed numbers into improper fractions for easier calculation.
For [tex]\(8 \frac{1}{4}\)[/tex]:
[tex]\[
8 \frac{1}{4} = \frac{8 \times 4 + 1}{4} = \frac{32 + 1}{4} = \frac{33}{4}
\][/tex]
For [tex]\(4 \frac{5}{6}\)[/tex]:
[tex]\[
4 \frac{5}{6} = \frac{4 \times 6 + 5}{6} = \frac{24 + 5}{6} = \frac{29}{6}
\][/tex]
3. Find a Common Denominator:
- To subtract these fractions, we need a common denominator. Here, the denominators are 4 and 6. The least common denominator (LCD) of 4 and 6 is 12.
4. Convert Fractions to have a Common Denominator:
- Convert [tex]\(\frac{33}{4}\)[/tex] to a fraction with a denominator of 12:
[tex]\[
\frac{33}{4} = \frac{33 \times 3}{4 \times 3} = \frac{99}{12}
\][/tex]
- Convert [tex]\(\frac{29}{6}\)[/tex] to a fraction with a denominator of 12:
[tex]\[
\frac{29}{6} = \frac{29 \times 2}{6 \times 2} = \frac{58}{12}
\][/tex]
5. Subtract the Fractions:
- Now subtract [tex]\(\frac{58}{12}\)[/tex] from [tex]\(\frac{99}{12}\)[/tex]:
[tex]\[
\frac{99}{12} - \frac{58}{12} = \frac{99 - 58}{12} = \frac{41}{12}
\][/tex]
6. Convert Result Back to a Mixed Number:
- Convert [tex]\(\frac{41}{12}\)[/tex] back to a mixed number:
[tex]\[
\frac{41}{12} = 3 \frac{5}{12}
\][/tex]
So, the additional fuel needed to fill the car’s tank is [tex]\(3 \frac{5}{12}\)[/tex] gallons.
