1. ## help word problems

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.

2/3 = 8/12

3/4 = 9/12

12 boys and 12 girls for a total of 24 students

8+9 = 17 students passed (8 boys and 9 girls)

3. 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]: .$\displaystyle \frac{\frac{2}{3}B + \frac{2}{3}B}{B + \frac{8}{9}B} \;=\;\frac{\frac{4}{3}B}{\frac{17}{9}B} \;= \;\boxed{\frac{12}{17}}$