At the town picnic, 480 hot dogs and 320 hamburgers are barbequed. Find the maximum number of families that can attend if the hot dogs and hamburgers are divided equally for each family.(1 point) Responses 120 families 120 families 160 families 160 families 80 families 80 families 40 families

Answer 1

To find the maximum number of families that can attend the picnic with the hot dogs and hamburgers divided equally, we need to determine how many families can equally share 480 hot dogs and 320 hamburgers.

First, we calculate the greatest common divisor (GCD) of 480 and 320 to find the maximum number of families that can share both types of food equally.

Finding the GCD:

1. Prime Factorization:

   • For 480: \[ 480 = 2^5 \times 3^1 \times 5^1 \]
   • For 320: \[ 320 = 2^6 \times 5^1 \]

2. Identifying common factors:

   • The common prime factors in both 480 and 320 are \(2\) and \(5\).
   • The minimum power of \(2\) in both is \(2^5\), and for \(5\) it's \(5^1\).

3. Calculating the GCD: \[ \text{GCD} = 2^5 \times 5^1 = 32 \times 5 = 160 \]

Now that we have 160 as the GCD, this indicates that the maximum number of families that can equally receive hot dogs and hamburgers is:

\[ \text{Maximum families} = \frac{480 \text{ hot dogs}}{160} = 3 \] \[ \text{Maximum families} = \frac{320 \text{ hamburgers}}{160} = 2 \]

However, since the families can only be equal in the least common distribution, we consider the total items divided against the GCD. Since the GCD of hot dogs and hamburgers is 160, that will set the allocation amount per family to ensure both items can be evenly divided.

Thus, the maximum number of families that can attend and receive an equal share of hot dogs and hamburgers is:

\[ 80 \text{ families} , \text{(both hot dogs and hamburgers can be divided into 80 portions)} \]

Therefore, the correct answer is 80 families.

Answer 2

wrong

Answer 3

I apologize for the confusion in my previous response. Let's solve the problem step by step to find out how many families can equally share the hot dogs and hamburgers.

We have:

• 480 hot dogs
• 320 hamburgers

To find the maximum number of families that can attend with an equal distribution of both hot dogs and hamburgers, we need to determine the highest number that can divide both quantities evenly, which is the greatest common divisor (GCD).

1. Finding the GCD:

We can find the GCD using the prime factorization method or the Euclidean algorithm. Here, I'll use the Euclidean algorithm.

• Divide 480 by 320: \[ 480 \div 320 = 1 \quad \text{(remainder: 160)} \]

• Now apply the algorithm on 320 and the remainder 160: \[ 320 \div 160 = 2 \quad \text{(remainder: 0)} \]

Since the last non-zero remainder is 160, we find: \[ \text{GCD}(480, 320) = 160 \]

2. Calculating the number of families:

To decide how many families can share the food equally, we can divide both the number of hot dogs and hamburgers by their GCD (160):

• For hot dogs: \[ \frac{480 \text{ hot dogs}}{160} = 3 \]

• For hamburgers: \[ \frac{320 \text{ hamburgers}}{160} = 2 \]

At this point, it's clear we need to ensure that the hot dogs and hamburgers can be shared equally among the maximum number of families:

Themaximum number of families that could share all of the items equally would be 160 divided by any number of portions, making a maximum of:

• 160 families.

So the correct answer is 160 families. Thank you for your patience, and I appreciate your understanding.