Question 1005201
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A group of boys and girls sit a test. Exactly 2/3 of the boys and 3/4 of the girls pass the test. 
If an equal number of boys and girls passed the test, what fraction of the entire group passed the test? 
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Let &nbsp;b = the number of boys and &nbsp;g = the number of girls in the class.

Then we have &nbsp;{{{(2/3)*b}}} = {{{(3/4)*g}}}, &nbsp;according to the condition.

Let us write it using the common denominator: {{{(8/12)*b}}} = {{{(9/12)*g}}}, or, which is the same, {{{(8b)/12}}} = {{{(9g)/12}}}. 

It implies that 8b = 9g.     (1)

Next, since the numbers 8 and 9 are <U>relatively primes</U>, (1) implies that b is multiple of 9 and g is multiple of 8:

b = 9*n, g = 8*m             (2)

with integer n and m. 

Then you can re-write (1) in the form

8*9n = 9*8m,   or   72n = 72m.

It implies that n = m and, hence, 

b = 9n, g = 8n               (3)

with some integer n.

Now, the number of those students who passed the test is 


{{{(2/3)*b + (3/4)*g}}} = {{{(8b)/12 + (9g)/12}}} = {{{(8*9n)/12 + (9*8n)/12}}} = {{{(72n)/12 + (72*n)/12}}} = {{{(144/12)*n}}} = 12n.  (5)

The total number of students in the class is 

b + g = 9n + 8n = 17n.                                               (6)

Now it is easy to calculate the ratio of those who passed the test to the total number of students in the class. It is (5) divided by (6):

{{{(12n)/(17n)}}} = {{{12/17}}}.

<U>Answer</U>.  The ratio of those who passed the test to the total number of students in the class is {{{12/17}}}.
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