To determine how many gallons of gas the motorist will have pumped into his car in [tex]\(\frac{1}{2}\)[/tex] of a minute given the pumping rate, we'll go through the calculations step by step.
1. Determine the rate of pumping in gallons per minute:
We know that the motorist is pumping gas at a rate of [tex]\(\frac{5}{12}\)[/tex] of a gallon every [tex]\(\frac{1}{24}\)[/tex] of a minute. To find the rate in gallons per minute, we divide the amount of gas pumped by the fraction of the minute in which it is pumped:
[tex]\[
\text{Rate (gallons per minute)} = \frac{\frac{5}{12} \text{ gallons}}{\frac{1}{24} \text{ minutes}} = \left(\frac{5}{12}\right) \div \left(\frac{1}{24}\right)
\][/tex]
To divide by a fraction, we multiply by its reciprocal:
[tex]\[
\left(\frac{5}{12}\right) \times \left(\frac{24}{1}\right) = \frac{5 \times 24}{12 \times 1} = \frac{120}{12} = 10 \text{ gallons per minute}
\][/tex]
2. Calculate the amount of gas pumped in [tex]\(\frac{1}{2}\)[/tex] of a minute:
Now that we have the rate in gallons per minute, we can find out how much gas is pumped in [tex]\(\frac{1}{2}\)[/tex] of a minute by multiplying the rate by the time:
[tex]\[
\text{Gallons pumped} = \text{Rate} \times \text{Time}
\][/tex]
Given the rate is 10 gallons per minute and the time is [tex]\(\frac{1}{2}\)[/tex] of a minute:
[tex]\[
\text{Gallons pumped} = 10 \text{ gallons per minute} \times \frac{1}{2} \text{ minute} = 10 \times \frac{1}{2} = 5 \text{ gallons}
\][/tex]
So, the motorist will have pumped 5 gallons of gas into his car in [tex]\(\frac{1}{2}\)[/tex] of a minute. Therefore, the correct answer is:
[tex]\[ \boxed{5} \][/tex]
