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The ratio of boys to girls in a certain classroom was $2:3$. If boys represented five more than one third of the class, how many people were there in the class room?

I do not seem to get how to solve it. Can somebody please help?

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For now, we don't know how many boys and girls there are, so let's just say there are '$b$' boys and there are '$g$' girls.

Now, the ratio of boys to girls tells us that for every $2$ boys, there are $3$ girls. How can we write this algebraically? Well, if $b$ was $2\cdot k$ for some number $k$, then $g$ would be $3\cdot k$ so we get $b=2k$ and $g=3k$. Putting these two equations together we get $\dfrac{b}{2}=\dfrac{g}{3}$ which we can re-write as $3b=2g$.

What can we tell from the other information in the question: boys represented five more than one third of the class.

"Well, a third of the class is $\dfrac{b+g}{3}$ because $b+g$ is the total number of people in the class. So if boys represent five more than this, then $b=5+\dfrac{b+g}{3}$. We can re-arrange this as well to get $g=2b-15$. So, we have two equations

  1. $3b=2g$

  2. $g=2b-15$

I'll let you solve these simultaneous equations. (Remember to always do a sanity check: if the answers you get aren't whole numbers then something's gone wrong - maybe a boy lost a leg or two - so remember to plug in the answer at the end back in to the information you've been given to make sure it satisfies what you've been told.)

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let 'x' total number of boys and girls, 'b' boys 'g' girls for ratio b:g, i.e 2:3

number of boys (b/(b+g))*x ------- 1 number of girls (g/(b+g))*x ------ 2

from the second condition number of boys 1/3*x + 5 -------3

Eque 1 & 3 should be equal, so

(b/(b+g))*x = 1/3*x + 5,

solve it u will get total boys and girls b+g= x = 75,

now counter check the answers all are correct....

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