b. The price displayed at a gas station is the price per gallon $3.15 What is the smallest

Question

number of gallons of gas you would have to buy for the price of the gas to
have a whole number of cents? A whole number of dollars?

Answer 1

To determine the smallest number of gallons of gas you would have to buy for the total price to be a whole number of cents and a whole number of dollars when the price per gallon is $3.15, we can analyze the situation mathematically.

1. Total Price in Whole Number of Cents

The price per gallon is $3.15, which is equivalent to 315 cents. To find the smallest number of gallons $g$ such that the total price in cents is a whole number, we can express the total price as:

$$\text{Total Price in cents}=315g$$

For $315g$ to be a whole number, $g$ can be any positive integer, as multiplying a whole number (315) by a whole number $g$ will yield a whole number.

Thus, the smallest $g$ is:

$$g=1$$

Conclusion: The smallest number of gallons to have a total price in whole cents is 1 gallon.

2. Total Price in Whole Number of Dollars

To find the smallest number of gallons $g$ such that the total price in dollars is a whole number, we use the same price per gallon:

$$\text{Total Price in dollars}=3.15g$$

For $3.15g$ to be a whole number, we can rewrite $3.15$ as a fraction:

$$3.15=\frac{315}{100}$$

Now, we need $g$ such that:

$$3.15g=\frac{315}{100}g$$

This needs to be an integer. For this fraction to be an integer, $g$ must be a multiple of $100$ divided by the greatest common divisor of $315$ and $100$.

Calculating it, we will find the GCD:

• The factors of $315$ are $3^2\times 5\times 7$.
• The factors of $100$ are $2^2\times 5^2$.

The GCD is $5$.

Now, divide $100$ by $5$:

$$\frac{100}{5}=20$$

Thus, $g$ must be a multiple of $20$ for $3.15g$ to yield a whole number. The smallest multiple of $20$ greater than or equal to $0$ is:

$$g=20$$

Conclusion: The smallest number of gallons to have a total price in whole dollars is 20 gallons.

Summary

• Smallest number of gallons for whole number of cents: 1 gallon
• Smallest number of gallons for whole number of dollars: 20 gallons