The queen asked her cook how many guests had attended a recent banquet. “I do not know, my lady” he said, “but every two guests shared a plate of fruit between them, every three used a bowl of broth among them, and every four shared a platter of beef. I used 65 serving dishes in all”. The queen was able to compute the number of guests using this information. How many guest were there?
g/2 + g/3 + g/4 = 65
13g/12 = 65
g = 60
im sorry i really don't get how you got that
as in where did you get the 13g/12=65
To solve this problem, we can use the concept of the least common multiple (LCM) to find the number of guests.
Let's start by analyzing the information given:
1. Every two guests shared a plate of fruit.
This means that the number of guests is divisible by 2.
2. Every three guests used a bowl of broth.
This means that the number of guests is divisible by 3.
3. Every four guests shared a platter of beef.
This means that the number of guests is divisible by 4.
We also know that there were 65 serving dishes used in total.
We need to find the LCM of 2, 3, and 4 to determine the smallest number of guests that satisfies all the conditions.
The LCM of 2, 3, and 4 is 12.
Now, let's check if 12 satisfies the number of serving dishes used:
- Every two guests share a plate, which means that there are 12/2 = 6 plates.
- Every three guests use a bowl, which means that there are 12/3 = 4 bowls.
- Every four guests share a platter, which means that there are 12/4 = 3 platters.
- In total, there are 6 plates + 4 bowls + 3 platters = 13 serving dishes.
Since the number of serving dishes used (13) is not equal to the given value (65), 12 is not the correct number of guests.
We need to find the next multiple of 12 that satisfies the given conditions.
The next multiple of 12 is 24.
Now, let's check if 24 satisfies the number of serving dishes used:
- Every two guests share a plate, which means that there are 24/2 = 12 plates.
- Every three guests use a bowl, which means that there are 24/3 = 8 bowls.
- Every four guests share a platter, which means that there are 24/4 = 6 platters.
- In total, there are 12 plates + 8 bowls + 6 platters = 26 serving dishes.
Since the number of serving dishes used (26) is not equal to the given value (65), 24 is not the correct number of guests.
We continue this process until we find the correct number of guests that satisfies the given conditions.
The next multiple of 12 is 36.
Let's check if 36 satisfies the number of serving dishes used:
- Every two guests share a plate, which means that there are 36/2 = 18 plates.
- Every three guests use a bowl, which means that there are 36/3 = 12 bowls.
- Every four guests share a platter, which means that there are 36/4 = 9 platters.
- In total, there are 18 plates + 12 bowls + 9 platters = 39 serving dishes.
Since the number of serving dishes used (39) is not equal to the given value (65), 36 is not the correct number of guests.
The next multiple of 12 is 48.
Let's check if 48 satisfies the number of serving dishes used:
- Every two guests share a plate, which means that there are 48/2 = 24 plates.
- Every three guests use a bowl, which means that there are 48/3 = 16 bowls.
- Every four guests share a platter, which means that there are 48/4 = 12 platters.
- In total, there are 24 plates + 16 bowls + 12 platters = 52 serving dishes.
Since the number of serving dishes used (52) is not equal to the given value (65), 48 is not the correct number of guests.
We continue this process until we find the correct number of guests.
The next multiple of 12 is 60.
Let's check if 60 satisfies the number of serving dishes used:
- Every two guests share a plate, which means that there are 60/2 = 30 plates.
- Every three guests use a bowl, which means that there are 60/3 = 20 bowls.
- Every four guests share a platter, which means that there are 60/4 = 15 platters.
- In total, there are 30 plates + 20 bowls + 15 platters = 65 serving dishes.
Since the number of serving dishes used (65) is equal to the given value (65), 60 is the correct number of guests.
Therefore, there were 60 guests at the recent banquet.