Maylin and Nina are making fruit baskets. They have 36 apples, 27 bananas, and 18 oranges. They want each basket to contain the same amount of each fruit. Maylin believes the greatest number of baskets they can make is 6, and Nina believes the greatest number of baskets they can make is 9.
Determine who is correct. Explain how you know.
The GCF of 36, 27, and 18 is 9
so she can make 9 baskets, each containing
4 apples, 3 bananas, and 2 oranges
6 baskets would not work, since the 27 bananas can not
be evenly placed in the 6 baskets.
3 baskets would also work, but it wouldn't be the largest number
of baskets, as required.
To determine who is correct, we need to find the greatest common divisor (GCD) of the given quantities of fruits (apples, bananas, and oranges).
The prime factors of the given quantities are:
36 = 2^2 * 3^2
27 = 3^3
18 = 2 * 3^2
Now, let's find the highest power of each prime factor that is present in all three quantities:
- The factor 2 is present only once in 18, twice in 36, and is not present in 27.
- The factor 3 is present twice in 18, twice in 36, and three times in 27.
Therefore, the GCD of 36, 27, and 18 is 2^1 * 3^2 = 2 * 9 = 18.
This means that we can make a maximum of 18 fruit baskets with each basket containing the same amount of each fruit.
Since Maylin believes they can make 6 baskets (which is less than the maximum of 18), her claim is incorrect. However, Nina's belief of being able to make 9 baskets is also incorrect, as it is still less than the maximum possible of 18 baskets.
Therefore, neither Maylin nor Nina is correct. The maximum number of fruit baskets they can make with each basket containing the same amount of each fruit is 18.
To determine who is correct, we need to find the common factors of 36, 27, and 18. The common factors will represent the maximum number of baskets that can be made with an equal amount of each fruit. Let's calculate the factors separately for each number.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 27: 1, 3, 9, 27
Factors of 18: 1, 2, 3, 6, 9, 18
Now, let's find the common factors:
Factors common to 36, 27, and 18: 1, 3, 9
From the common factors, we can see that the greatest number of baskets they can make with an equal amount of each fruit is 9. This means Nina is correct, and they can make a maximum of 9 baskets.