The ratio of those included to those excluded is 4 to 7. If 5x the # of excluded is 62 greater than the # included, how many were included/excluded?

Question

Need by morning of 11/5 please

Answer 1

Hi Jon,

 

We can use the information given to set up a system of equations, then rearrange and substitute to solve for both of the unknown variables.

 

First, "the ratio of those included to those excluded is 4 to 7". We can put this into mathematical terms using an equation:

 

    x = # included

 

    y = # excluded

 

    x/y = 4/7

 

    7x = 4y or y = 7x/4

 

Second, "5x the # of excluded is 62 greater than the # included". Using the same definitions of x and y from before, we could write this as an equation:

 

    5y = x + 62

 

We now have two equations, each with the same two unknown variables, so we can substitute what we know from one equation into the other.

 

    I know that y = 7x/4 (from the first equation), so I will replace the "y" in the second equation with "7x/4":

 

    5(7x/4) = x + 62

 

I know have an equation with only one variable (x), so I can use algebra to solve for it:

 

    35x/4 = x + 62

 

    35x/4 - x = 62

 

    x(35/4 -1) = 62

 

    x(35/4 - 4/4) = 62

 

    x(31/4) = 62

 

    x = 62(4/31)

 

    x = 8

 

So I know that the number included must be 8. I can now use this number in the "y=" equation that I obtained above to figure out the y value:

 

    y = 7x/4

 

    y = 7(8)/4

 

    y = 14

 

So the number included is 8, and the number excluded is 14 - we can verify these numbers using the original equations that we wrote based on the word problem. I hope that helps!

Answer 2

x=# included

y=# excluded

x/y=4/7

5y=62+x

from x/y=4/7, cross multiply to get 4y=7x and x=(4/7)y

substitute into the second equation

5y=62+((4/7)y

5y-(4/7)y=62

(4 3/7)y=62

(31/7)y=62

multiply both sides by (7/31)

(7/31)(31/7)y=(7/31)(62)

y=(7*2)  (I factored out 31 with 62 to get 2)

y=14

x/14=4/7

2*7=14 and 2*4=8

x=8 and y=14, 8 included, 14 excluded

 

another solution...

4x and 7x are the number of included and excluded for some number x

5(7x)=62+4x

35x=62+4x

35x-4x=62

31x=62

x=62/31

x=2

4x=4*2=8 included

7x=7*2=14 excluded