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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 b = the number of boys and g = the number of girls in the class.
Then we have = , according to the condition.
Let us write it using the common denominator: = , or, which is the same, = .
It implies that 8b = 9g. (1)
Next, since the numbers 8 and 9 are relatively primes, (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
= = = = = 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):
= .
Answer. The ratio of those who passed the test to the total number of students in the class is .