how solve this problem by one variable

Question

from a group of boys and girls. 15 girls leave there are then left 2 boys for each girl. after this 45 boys leave. there are then 5 girls for each boy find the number of girls in the beginning

 

 

 

 

Answer 1

Let's let g = number of girls and b = number of boys. We can write some equations to help us solve for g, the number of girls in the beginning:

 

 

"15 girls leave, and then there are 2 boys left for each girl"

After 15 girls leave, there are g - 15 girls. There are also 2 boys left for each of those g - 15 girls. So...

 

b = 2(g - 15) (since the number of boys is twice the number of girls after 15 girls leave)

 

 

"After this 45 boys leave, and then there are 5 girls left for each boy"

After 45 boys leave, there are b - 45 boys and there are still g - 15 girls. There are then 5 girls left for each of those b - 45 boys. So...

 

g - 15 = 5(b - 45)  (since the number of girls after 15 girls leave is five times the number of boys after 45 boys leave)

 

 

Now you have a system of equations:

 

b = 2(g - 15)

g - 15 = 5(b - 45)

 

 

Plug b = 2(g - 15) into the second equation and solve for g:

 

g - 15 = 5(2(g - 15) - 45)

g - 15 = 5(2g - 30 - 45)

g - 15 = 5(2g - 75)

g - 15 = 10g - 375

g = 10g - 360

-9g = -360

g = 40

 

 

There were 40 girls in the beginning.

 

 

Let's figure out how many boys there were in the beginning and check our work. Plug g = 40 into the first equation:

 

b = 2(40 - 15)

b = 2(25)

b = 50

 

There were 50 boys in the beginning. So if 15 of the 40 girls left, there would be 25 remaining, which is half of the number of boys (50). Then, if 45 boys left, there would be 5 remaining, which is one fifth of the remaining girls (25). That checks out!