If Amelie worked alone, she would have worked for 4 hours.
We are going to be establishing mathematical relationships to solve this question.
Let t represent the time we are looking for
r to represent Ricardo's rate
a to represent Amelie's rate
we can then say that
from the first equation, we can agree that, r = 1/(t+8)
from the second we can also agree that, a = 1/t
If we substitute these two values in the third equation, we have
1 = {1/(t+8) + 1/t} 3, from this we have
t(t + 8) = 3t + 3(t+8)
t² + 8t = 3t + 3t + 24
t² + 8t = 6t + 24, if we collect like terms, we have
t² +8t -6t - 24 = 0
t² +2t -24 = 0, if we factorize, we have
t² + 6t - 4t - 24 = 0
t(t+6) -4(t+6) = 0
(t+6) (t-4) = 0
t = 4 hours
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