A group of boys and girls took a test. Exactly 2/3 of the boys and exactly 3/4 of the girls passed the test. If an equal number of boys and girls passed the test, what fraction of the entire group passed? help would be appreciated, thanks.
A group of boys and girls took a test. Exactly 2/3 of the boys and exactly 3/4 of the girls passed the test. If an equal number of boys and girls passed the test, what fraction of the entire group passed? help would be appreciated, thanks.
Hello, jarny!
A group of boys and girls took a test.
Exactly 2/3 of the boys and exactly 3/4 of the girls passed the test.
If an equal number of boys and girls passed the test,
what fraction of the entire group passed?
Let .$\displaystyle \begin{array}{cc} B = \text{number of boys} \\ G = \text{number of girls}\end{array}$
$\displaystyle \frac{2}{3}$ of the boys passed: .$\displaystyle \frac{2}{3}B$ boys passed.
$\displaystyle \frac{3}{4}$ of the girls passed: .$\displaystyle \frac{3}{4}G$ girls passed.
. . Hence, a total of $\displaystyle \frac{2}{3}B + \frac{3}{4}G$ students passed.
There was a total of $\displaystyle B + G$ students.
. . Hence, the fraction that passed is: .$\displaystyle \frac{\frac{2}{3}B + \frac{3}{4}G}{B + G}$ [1]
We are told that: .$\displaystyle \frac{2}{3}B\,=\,\frac{3}{4}G$ [2]
. . and hence: .$\displaystyle 8B \,=\,9G\quad\Rightarrow\quad G\,=\,\frac{8}{9}B$ [3]
Substitute [2] and [3] into [1]: .$\displaystyle \frac{\frac{2}{3}B + \frac{2}{3}B}{B + \frac{8}{9}B} \;=\;\frac{\frac{4}{3}B}{\frac{17}{9}B} \;= \;\boxed{\frac{12}{17}}$