There are 100 students at the end of year school bar-b-que. the ratio of boys to girls is 4: 6. the ratio of vegetarian girls to omnivorous girls is 1: 1. how many girls and how many boys are there at the bar-b-que?

There are 60 girls and 40 boys at the bar-b-que



Explanation:

b = number of boys
g = number of girls

"The ratio of boys to girls is 4:6" so 
b = 4*x
g = 6*x
where x is some positive whole number

Add up the counts of boys and girls. This count should equal 100
b+g = 100

Plug in b = 4x and g = 6x. Then solve for x
b+g = 100
4x+6x = 100
10x = 100
10x/10 = 100/10 ... divide both sides by 10
x = 10

If x = 10, then
b = 4*x = 4*10 = 40 ---> there are 40 boys
g = 6*x = 6*10 = 60 ---> there are 60 girls

we can see that the ratio
40 boys: 60 girls
reduces to
4 boys: 6 girls
In other words, for every 4 boys, there are 6 girls. The ratio 4:6 can be fully reduced to 2:3 which means for every 2 boys, there are 3 girls. 

Side note: if the ratio of vegetarian girls to omnivorous girls is 1:1, then there are 30 of each
we can solve y+y = 60 to find that y = 30, or we can simply divide 60 in half to find this count. Saying "a ratio of 1:1" means that there are the same number of vegetarians as omnivorous girls.