math homework

Question

when 5 adults joined a group of y dines,the ratio of the number of adults to children in the restaurant became 3:5 . express the number of children in terms of y.

Answer 1

The group of diners has two subgroups:   adults and children

 

The ratio of 3:5 represent the ratio of adults to children, written as adults:children.  So, for each 3 adults, there are 5 children.

 

Now, we must find two integers (for the number of adults and children) such that:

      adults:children   is   3:5     and     (adults - 5):children   are integers

 

The LCM of 3 and 5 is 8, so the ratio of adults:children of (8*3):(8*24) is 24:40  (now, 24 adults for each 40 children).

Before the 5 adults arrived, the number of adults could have been 19 (that is, 24 - 5) if the number of children was 40.

We say, "could have been" because we don't have information about the ratio before the 5 adults arrived -- but the problem doesn't require us to know.

 

The problem asks for the number of children in terms of y, the number of diners.  So, if there were 40 children and a total of 64 (that is, 40+24) diners, then: 

 

       children = (40/64)*y

       children = (5/8)*y              [note that we could have used 5/(3+5) originally]

 

Now, all that is required is that y must be a multiple of 8, the LCM, to make the problem statement true.

 

     y       adults:children                                                 previous number of adults

   ----     -----------------------------------------------          --------------------------------------

     8         24:40               [that is (8*3):(8*5)]                                19

    16        48:80               [that is (16*3):(16*5)]                             43

    24        72:120             [that is (24*3):(24*5)]                             67

  . . .

  [note:  now, we are limited only be the capacity of the dining area]