The price displayed 2.359 at a gas station is the price per gallon. What is the smallest 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? Explain.
To find the smallest number of gallons of gas you would have to buy for the price of the gas to have a whole number of cents, we need to find the least common multiple (LCM) of the numerator and denominator of the price per gallon.
The price per gallon is $2.359, which can be written as the fraction 2359/1000. To find the LCM of 2359 and 1000, we prime factorize each number:
2359 = 7 × 337
1000 = 2³ × 5³
The LCM is found by taking the highest power of each prime factor that appears in either factorization:
LCM = 2³ × 5³ × 7 × 337 = 140840
Therefore, to have a whole number of cents, you would have to buy at least 140840 gallons of gas.
To find the smallest number of gallons of gas you would have to buy for the price of the gas to have a whole number of dollars, we need to look at the denominator of the price per gallon, which is 1000.
Since there are 100 cents in a dollar, we need the denominator to be divisible by 100 to have a whole number of dollars. The smallest number divisible by 1000 and 100 is their LCM, which is 1000.
Therefore, to have a whole number of dollars, you would have to buy at least 1000 gallons of gas.
To determine the smallest number of gallons of gas that would result in a whole number of cents, we need to find the least common multiple (LCM) of the numerator and denominator of the price per gallon.
The price per gallon is $2.359.
To find the LCM, we need to consider the denominator, which is 1. Since 1 is a factor of all numbers, the LCM would simply be 1.
Therefore, to have a whole number of cents, you would need to buy at least 1 gallon of gas.
Now, let's find the smallest number of gallons required to have a whole number of dollars. Since there are no cents in a whole number of dollars, we need to consider the denominator as 100 (since there are 100 cents in a dollar).
To find the LCM, we can ignore the decimal point temporarily. The value becomes 2359/100.
To find the LCM of 2359 and 100, we can break down the numbers into their prime factors:
2359: 17 × 139
100: 2^2 × 5^2
To find the LCM, we need to select the highest power of each prime factor. So the LCM in this case would be 17 × 139 × 2^2 × 5^2.
Multiplying these numbers gives us 47,630.
Therefore, to have a whole number of dollars, you would need to buy at least 47,630 gallons of gas.
To find the smallest number of gallons of gas for the price to have a whole number of cents, we need to determine the smallest common multiple between the price per gallon and 100 cents.
The price per gallon is given as 2.359 dollars. To convert dollars to cents, we multiply by 100: 2.359 dollars * 100 = 235.9 cents.
The smallest common multiple between 235.9 cents and 100 cents will give us the smallest number of gallons of gas needed to have a whole number of cents.
To find the smallest common multiple, we can divide 100 cents by the greatest common divisor (GCD) between 100 and 235.9.
Next, we need to find the GCD of 100 and 235.9 cents. Since 235.9 is a decimal, let's round it to the nearest whole number for simplicity. In this case, rounding 235.9 to the nearest whole number gives us 236.
To find the GCD of 100 and 236, we can use the Euclidean algorithm.
Step 1: Divide 236 by 100: 236 ÷ 100 = 2 remainder 36
Step 2: Divide 100 by 36: 100 ÷ 36 = 2 remainder 28
Step 3: Divide 36 by 28: 36 ÷ 28 = 1 remainder 8
Step 4: Divide 28 by 8: 28 ÷ 8 = 3 remainder 4
Step 5: Divide 8 by 4: 8 ÷ 4 = 2 remainder 0
Since the remainder is 0, the GCD of 100 and 236 is 4.
Now, we can calculate the smallest common multiple (LCM) by dividing the product of 100 and 236 by the GCD of 100 and 236: (100 x 236) / 4 = 5900 / 4 = 1475.
Therefore, the smallest number of gallons of gas you would have to buy for the price of the gas to have a whole number of cents is 1475 gallons.
To calculate the smallest number of gallons for the price to have a whole number of dollars, we divide the price by the smallest common multiple we found, which is 1475: 235.9 cents / 1475 = 0.16 cents.
Since 0.16 cents is less than 1 cent, we need to round it up to the nearest whole number. Therefore, the smallest number of gallons of gas needed for the price to have a whole number of dollars is 1475 gallons.
In summary:
- The smallest number of gallons of gas needed for the price to have a whole number of cents is 1475 gallons.
- The smallest number of gallons of gas needed for the price to have a whole number of dollars is also 1475 gallons.