Expert: Chanda Walker - 3/19/2009

Question
I have a few questions, and I know the answers from the answer key but I don't understand how to get the answers for these type of questions.

1) 5 boys and 2 girls ate 24 sandwiches. 7 boys and 4 girls ate 36 sandwiches.  Each girl ate the same amount of whole sandwiches.  Each boy ate the same amount of whole sandwiches.  How many sandwiches did each boy and each girl eat?

Answer = boys:4  girls:2

2)A baker makes 9 dozen cookies.  He divides them into 6-cookie and 2-cookie packages.  If he wants to make the greatest number of each size package and the greatest number of total packages, how many packages of each size can he make?
Answer = 13 6-cookie packages and 15 2-cookie packages

3)Jane either ran 5 or 7 miles each day for 30 consecutive days.  Her total mileage during this period was 166.  How many days did she run 5 miles? How many days did she run 7 miles?
Answer = 22 days (running 5 miles) and 8 days (7 miles)

Answer
5b + 2g = 24
7b + 4g = 36

Multiply the top equation by 2 and subtract the equations.

(10b + 4g = 48)
-(7b + 4g = 36)
______________
3b = 12
b = 4

There were four boys.

5b + 2g = 24
5*4 + 2g = 24
2g = 4
g = 2

There are two girls.

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cookies = 9 *12 = 108 cookies

we want as many of each as we can make.  Think of the packages in groups of one little and one big package that has 8 cookies in it.

108/8 = 13.5

So let's round that down to a whole number and use 13 big and 13 little packages.

Now we have used 13*8 = 104 cookies  There are 4 cookies left.  Let's put that into 2 little packages making the total:

13 big
15 little

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x*5 + y*7=166

x + y = 30

x = 30 - y
(30 - y)*5 + 7y = 166
150 + 2y = 166
2y = 16
y = 8

So there were 8 days that she ran 7 miles.  Thus there must have been 22 days that she ran 5 miles.