Question

Tyler is packing lunches for a local youth program. He has 36 sandwiches, 72 apples, and 108 granola bars. What is the greatest number of lunches he can pack so that each lunch has an equal number of sandwiches, apples, and granola bars? How many of each item will the lunch contain?

Answer 1

Answer:

The greatest number of lunches he can pack so that each lunch has an equal number of sandwiches, apples, and granola bars is 12.

Each lunch will contain 3 sandwiches, 6 apples and 9 granola bars

Step-by-step explanation:

 The greatest common divisor (GCD) of two or more numbers is the largest number by which these numbers can be divided, that is, it is the largest number that divides them all exactly.

In other words, the greatest common divisor of two numbers a and b is the largest number that divides a and divides b.

To find the greatest common divisor of two or more numbers, you start by breaking those numbers down into prime factors. In this case:

36=2²*3²

72 = 2³*3²

108 = 2²*3

The greatest common divisor is obtained by choosing only the prime factors common to the numbers, raised to the smallest exponent. That is, you choose only the common factors and those that are repeated must be raised to the minimum exponent.

The 2 appears as a prime factor in the three decompositions, whose minimum exponent is 2.

3 also appears as a common factor, whose minimum exponent is 1.

By doing the multiplication to get the greatest common divisor, you get:

2²*3=12

So the greatest number of lunches he can pack so that each lunch has an equal number of sandwiches, apples, and granola bars is 12.

To calculate how many lunches each item will contain, you simply divide the amount you have of each item by the most lunches Tyler can pack, 12.

Sandwiches: 36÷12= 3

Apples: 72÷12= 6

Granola bars: 108÷12= 9

Each lunch will contain 3 sandwiches, 6 apples and 9 granola bars