Question

A school is preparing fruit baskets for a local nursing home. There are 162 apples, 108 oranges, and 180 bananas. If the baskets are identical and there is no leftover fruit, what is the greatest number of baskets that can be made? How many apples, oranges, and bananas are in each basket?

Answer 1

Number of baskets = 18
 Apples = 9
 Oranges = 6
 Bananas = 10

   For this problem, you need to find the greatest common factor (GCF) of all three numbers. Thankfully, you can do that one pair at a time.
 For 162 and 108, the GCF is 54.
 Then find the GCF of 54 and 180, which is 18.

   Now you know that the greatest number of baskets you can have with all baskets being identical and no fruit left over is 18. To determine how much of each fruit in a basket, simply divide each quantity by 18, so:

   Apples: 162/18 = 9
 Oranges: 108/18 = 6
 Bananas: 180/180 = 10

   Note: One method for determining the GCF of 2 numbers is Euclid's Algorithm which is quite simple: Take 2 positive numbers A and B and calculate C = A mod B. (mod is simply the remainder after performing an integer division. For example 9 mod 3 = 0, 10 mod 3 = 1, 11 mod 3 = 2, 12 mod 3 = 0 and so on). If the result is 0, then B is the GCF of A and B. If the result is not 0, then assign A the value of B, then assign B the value of C and perform the modulo operation again until you finally get a result of 0.

Some examples: 162 mod 108 = 54, 108 mod 54 = 0; Therefore the GCF of 162 and 108 is 54.
 180 mod 54 = 18, 54 mod 18 = 0; Therefore the GCF of 180 and 54 is 18. 210 mod 54 = 48, 54 mod 48 = 6, 48 mod 6 = 0; Therefore the GCF of 210 and 54 is 6.

Answer 2

To solve this exercise, we must calculate the Maximum Common Divisor (M.C.D.), which is the greatest common divisor of 62,108 and 180. We apply factor decomposition, and we obtain:

 162= (3^4)(2)

 108= (2^2)(3^3)

 180= (2^2)(3^2)(5)

 We choose the common numbers with their lowest exponent:

 M.C.D.= (2)(3^2)

 M.C.D.= 2x9

 M.C.D.= 18

 The greatest number of baskets that can be made is: 18 baskets

 To know how many apples, oranges and bananas are in each basket, we must divide the greatest number of them that can be made between the total amount of apples, oranges and bananas, as shown below:

 162/18= 9 apples

 108/18= 6 oranges

 180/18= 10 bananas

 So, there are 9 apples, 6 oranges and 10 bananas in each basket.