Word problems with ratios.

Question

The ratio of girls to boys in the band was 9:8. Half of the boys left for practice which left 15 more girls than boys. How many students were in the band before the boys left?

Answer 1

This question basically says that for every 9 girls in the band, there are 8 boys, and that half the number of boys in the band is 15 less than the number of girls.

This is where we start working with variables

Let's call the number of girls G and the number of boys B.

We can start with the following relations to help us build an equation with one variable:

G=(9/8)B

or

B=(8/9)G

Now we can build the function accordingly, taking the difference of 15 as the end result. Let's use the definition of G in this case

(9/8)*B - (1/2)*B = 15

Working out the fractions to create one instance of the variable B:

(9/8)*B - (4/8)*B = 15

(5/8)*B = 15

Now divide 15 by 5/8 or multiply by 8/5

B = 15*8/5

B = 3*8

B = 24

Now we know how many boys are in the band, but is it the correct number? Let's find out by first multiplying it by 9/8

24*9/8 = 3*9 = 27

So this means that there are 27 boys and 24 girls in the band theoretically.

Now...

What is half the number of boys?

24/2 = 12

Add 15...

12 + 15 = 27

Now we know that the number of boys we found earlier was correct, and therefore there are 27 girls and 24 boys in the band. With this information, we can determine the total number of students in the band:

27 girls + 24 boys = 51 students

So there are 51 students in the band, which answers the question.