Problem 72 - Algebra A Combined Function

Question

You are throwing a barbecue and you want to make sure that you purchase the same number of hot dogs as hot dog buns. Hot dogs come 8 to a package and hot dog buns come 12 to a package. What is the least number of each type of package you should buy?

Step-by-Step Solution

Answer

Buy 3 packs of hot dogs and 2 packs of buns.

Step 1: Understanding the Problem

We need to buy the same number of hot dogs and hot dog buns, but hot dogs come in packs of 8, and buns come in packs of 12. We should find the least number of packs needed where the quantities of hot dogs equal the quantities of buns.

Step 2: Finding the Least Common Multiple

To ensure that the hot dogs and buns match up in number, we must find the least common multiple (LCM) of 8 and 12. The LCM is the smallest number that is a multiple of both 8 and 12.

Step 3: Calculating the LCM using Prime Factorization

First, find the prime factors of both numbers. The prime factorization of 8 is \( 2^3 \), and for 12, it is \( 2^2 imes 3^1 \). To find the LCM, take the highest power of each prime number that appears in the factorizations.

Step 4: Determining the LCM

The LCM will include \( 2^3 \) from 8 since it's the highest power of 2, and \( 3^1 \) from 12. Multiply these together: \( 2^3 \times 3^1 = 8 \times 3 = 24 \). Thus, the LCM of 8 and 12 is 24.

Step 5: Calculating the Number of Packages

Since the LCM is 24, you need a total of 24 hot dogs and 24 buns. Divide the LCM by the package size for each product: For hot dogs, 24 divided by 8 equals 3 packages. For buns, 24 divided by 12 equals 2 packages.

Step 6: Solution Summary

To have an equal number of hot dogs and buns, buy 3 packages of hot dogs and 2 packages of buns.

Key Concepts

Prime Factorization

When dealing with numbers, prime factorization is a super useful concept. It involves breaking down a number into its prime number components. Think of it as zooming into a number and expressing it in terms of its building blocks—prime numbers. Prime numbers are those which are only divisible by 1 and themselves, like 2, 3, and 5.

For instance, the number 8 can be broken down into prime factors: it's 2 multiplied by itself three times, so we have 8 = \(2^3\). Similarly, 12 can be expressed as \(2^2 \times 3^1\). By writing numbers this way, we can easily find common multiples and solve various problems involving divisibility.

Multiples

Multiples are numbers that you get when you multiply a number by integers (whole numbers like 1, 2, or 3). For example, some multiples of 8 are 8, 16, 24, and so on. Similarly, multiples of 12 include 12, 24, 36, etc.

In this problem, we need to have multiples where the count of hot dogs equals the count of buns. Finding these matching multiples efficiently is where the concept of the Least Common Multiple (LCM) comes into play. The LCM is the smallest number that is a multiple of each of the given numbers. It's like a friendly handshake between numbers, where both numbers agree. For 8 and 12, their LCM is 24, meaning 24 is the smallest number that both 8 and 12 can multiply into evenly.

Problem Solving

Problem-solving is a critical skill useful in a wide range of challenges. Let's apply it to our barbecue task! We start by understanding our goal: to buy an equal number of hot dogs and buns. They come in different pack sizes, so we apply mathematical strategies to reach our solution.

First, we decide how to ensure equal amounts by calculating the Least Common Multiple (LCM) of the pack sizes. Using Prime Factorization, we've found that the LCM of 8 (hot dogs) and 12 (buns) is 24. Next, we calculate how many full packs of each are needed to reach this total:

• For hot dogs: 24 ÷ 8 = 3 packs
• For buns: 24 ÷ 12 = 2 packs

Thus, we should buy 3 packs of hot dogs and 2 packs of buns. This step-by-step approach makes problem-solving more approachable and helps ensure you always have the right balance of burgers to buns!