The price displayed at a gas station is the price per gallon. What is the smallest

Question

number of gallons of gas you would have to buy for a whole number of dollars?

Answer 1

To find the smallest number of gallons of gas you would need to buy for a whole number of dollars, we should analyze the price per gallon. Let's denote the price per gallon as $P$ dollars.

For a given price per gallon $P$:

• If you buy $g$ gallons, the total cost will be $P\times g$ dollars.
• We want $P\times g$ to be a whole number of dollars.

In order for $P\times g$ to be an integer, $g$ must be a multiple of the denominator of $P$ when $P$ is expressed as a fraction in the simplest form.

For instance:

• If the price per gallon $P$ is $3.29$, it can be expressed as $\frac{329}{100}$. Here, the denominator is 100. Therefore, $g$ must be a multiple of 100. This means the smallest number of gallons you could buy for a whole number of dollars is 100 gallons.
• If $P$ were $2.50$, it can be expressed as $\frac{250}{100}$. Again, the smallest $g$ would be 100 gallons.

Thus, to determine the smallest $g$, it is necessary to find the denominator when the price per gallon is expressed as a fraction. In any case, the smallest $g$ that ensures $P\times g$ is an integer will typically be the least common multiple of the denominators corresponding to the decimal representation of fuel prices.

If the price $P$ has $n$ decimal places, you would need to buy ${10}^n$ gallons for $P\times g$ to yield a whole number amount in dollars.

To conclude, for most fuel prices represented in dollars and cents (up to two decimal places), the smallest number of gallons you would have to buy for a whole number of dollars is typically:

$$\boxed{{\displaystyle 100}}$$

assuming the price has two decimal places.