SOLUTION: Please help me solve this equation: A large Thanksgiving banquet served between 500 and 600 people. The dinner rolls came in baker’s dozens (packages of 13). After everyone had
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Question 813809: Please help me solve this equation: A large Thanksgiving banquet served between 500 and 600 people. The dinner rolls came in baker’s dozens (packages of 13). After everyone had
been served a roll, there were a number of empty packages plus one open
package with 10 rolls left. The pies for dessert had each been cut into ninths,
and after everyone had been served a slice there were a number of empty pie
tins plus one pie tin with 4 slices left. The tables in the banquet hall each had
room for 15 guests. All of the tables, except one, were full.
How many people attended the banquet? How many people were seated at
the table that was not full? How many pies were there? How many packages
of rolls?
You can put this solution on YOUR website! Let x = the number of pies and
let y = the number of packages of rolls and
let t = the total number of people at the banquet.
From "The dinner rolls came in baker’s dozens (packages of 13). After everyone had been served a roll, there were a number of empty packages plus one open package with 10 rolls left." we should get that the number of rolls used would be 13 times the number of packages minus the 10 rolls from the last package which were not used. And since one roll was given to each person at the banquet:
t = 13y - 10
From "The pies for dessert had each been cut into ninths, and after everyone had been served a slice there were a number of empty pie tins plus one pie tin with 4 slices left." we should get that the number of pieces of pie that were used would be 9 times the number of pies minus the 4 pieces from the last pie which were not used. And since one piece was given to each person at the banquet:
t = 9x - 4
Since the left sides of these two equations are the same, the right sides must be equal, too:
13y - 10 = 9x - 4
Adding 10:
13y = 9x + 6
Multiplying each side by 1/13:
This equation relates x (the number of pies) with y (the number of packages). Both x and y must be positive integers (since we started with whole pies and packages). With this equation we can try an x and then see if the y turns out to be a positive integer, too. Once we find such a combination of x and y, then we see if those numbers could feed between 500 to 600 people.
We could randomly try positive integers for x. But we can speed things up if we remember that there were between 500 and 600 people. Our x must be large enough so that 9 times x (the number of pieces of pie) must greater than 500:
9x > 500
Dividing by 9:
x > 500/9
or x > 55.555...
And since x must be an integer, we will start with x = 56.
So using and starting with x = 56, we will try x's until we find a y that is also an integer:
y is not an integer so x = 56 doesn't work. We'll try 57 next:
y is not an integer so x = 57 doesn't work. Eventually you get to x = 60:
So x = 60 (i.e. 60 pies were used) and y = 42 (42 packages of rolls). To find the total number of people we can use either t = 13y - 10 or t = 9x - 4. Either one tells us that t = 546 (546 people attended the banquet).
As for the number of people at the last table...
Since 15 people can sit at a table and since 15*36 = 540, we can seat 540 people at 36 full tables. The 37th table will have the 6 people left over.
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