Question

peanuts are sold in 8-ounce and 12-ounce packages. what is the fewest number of ounces you can buy of each package to have equal amounts of each package size?

Explanation:

Step1: Identify the problem type

This is a problem about finding the least number of packages (8-ounce and 12-ounce) to buy so that the total ounces from each package type are equal. This involves finding the least common multiple (LCM) of 8 and 12.

Step2: Find the LCM of 8 and 12

First, factorize 8 and 12:

• Prime factorization of 8: \(8 = 2\times2\times2 = 2^3\)
• Prime factorization of 12: \(12 = 2\times2\times3 = 2^2\times3\)

To find the LCM, take the highest power of each prime number present. For prime 2, the highest power is \(2^3\) (from 8), and for prime 3, the highest power is \(3^1\) (from 12). Then, LCM \(= 2^3\times3 = 8\times3 = 24\).

Step3: Calculate the number of packages for each size

• For 8-ounce packages: Number of packages \(=\frac{24}{8}=3\)
• For 12-ounce packages: Number of packages \(=\frac{24}{12}=2\)

Step4: Find the total number of packages

Total number of packages \(= 3 + 2 = 5\)

Answer:

The fewest number of packages is 5 (3 eight - ounce packages and 2 twelve - ounce packages).