Presenting the algebraic way of the solution

Given:We are given that for a field trip \(\frac{3}{5}\) of the total students boarded the first bus and \(\frac{2}{3}\) of the remaining students boarded the second bus while the rest boarded the van. Also, when the second bus broke down \(\frac{1}{2}\) of the students on the second bus boarded the first bus. We are asked to find the ratio of final number of the students on the first bus to the total students.

ApproachSince, we need to find the ratio of two things, we have to express them in either constants or in the same terms. Since we don't have any information about the exact number of students we will try to express them in the same terms.

We are given relations between the total number of students and the students who boarded the buses and the van. We will use this relation to find out the required ratio.

Working OutLet's assume total number of students to be \(x\).

Students who boarded the first bus = \(\frac{3x}{5}\)

Students who boarded the second bus = \(\frac{2}{3}\) of remaining students = \(\frac{2}{3}\) * (Total Students - Students who boarded the first bus) = \(\frac{2}{3}\) *( \(x- \frac{3x}{5}\)) = \(\frac{4x}{15}\)

We don't need to find the number of students who boarded the van as it is not being used anywhere.

No. of students of the second bus who boarded the first bus = \(\frac{1}{2} *(\frac{4x}{15}) = \frac{2x}{15}\)

Hence, total number of students in the first bus = original number of students in the first bus + students from second bus who boarded the first bus = \(\frac{3x}{5} + \frac{2x}{15} = \frac{11x}{15}\)

Now, we need to find the ratio of number of students in the first bus to that of the total students which can be written as \(\frac{11x}{15x} = \frac{11}{15}\) which is our answer.

Hope its clear!

Regards

Harsh

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