To find out how much more gasoline Jerry needs to fill his motorcycle tank, we start by calculating the difference between the tank's capacity and the current amount of gasoline.
1. First, we note the tank's capacity:
[tex]\[
6 \frac{1}{2} \text{ gallons}
\][/tex]
To make calculations easier, we convert this mixed number into an improper fraction:
[tex]\[
6 \frac{1}{2} = 6 + \frac{1}{2} = \frac{12}{2} + \frac{1}{2} = \frac{13}{2} \text{ gallons}
\][/tex]
2. Next, we note the current amount of gasoline in the tank:
[tex]\[
3 \frac{3}{4} \text{ gallons}
\][/tex]
Similarly, we convert this mixed number into an improper fraction:
[tex]\[
3 \frac{3}{4} = 3 + \frac{3}{4} = \frac{12}{4} + \frac{3}{4} = \frac{15}{4} \text{ gallons}
\][/tex]
3. Now, we need to find the difference between the tank's capacity and the current amount of gasoline:
[tex]\[
\frac{13}{2} - \frac{15}{4}
\][/tex]
To subtract these fractions, we first find a common denominator, which is 4 in this case:
[tex]\[
\frac{13}{2} = \frac{13 \times 2}{2 \times 2} = \frac{26}{4}
\][/tex]
So the subtraction becomes:
[tex]\[
\frac{26}{4} - \frac{15}{4} = \frac{26 - 15}{4} = \frac{11}{4}
\][/tex]
4. Finally, we need to simplify [tex]\(\frac{11}{4}\)[/tex] into a mixed number:
[tex]\[
\frac{11}{4} = 2 \frac{3}{4}
\][/tex]
Therefore, Jerry needs [tex]\(2 \frac{3}{4}\)[/tex] gallons of gasoline to fill his tank. Thus the correct answer is:
[tex]\[
2 \frac{3}{4} \text{ gallons}
\][/tex]
