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