Brazilian National Exam Question (ENEM)

Let us convert the ratios into fractions

For the first half of the journey, José carried $\dfrac{6}{15}$ , Carlos carried $\dfrac{5}{15}$ and Paulo carried $\dfrac{4}{15}$ of the oranges

For the second half of the journey, José carried $\dfrac{4}{10}$ , Carlos carried $\dfrac{4}{10}$ and Paulo carried $\dfrac{2}{10}$ of the oranges

Now, what we need to do is convert these fractions to have the same denominator and compare them.

For José: $\dfrac{6}{15} = \dfrac{12}{30}\quad\quad\quad \dfrac{4}{10} = \dfrac{12}{30}\quad\quad\quad \dfrac{6}{15} = \dfrac{4}{10}$

For Carlos: $\dfrac{5}{15} = \dfrac{10}{30}\quad\quad\quad \dfrac{4}{10} = \dfrac{12}{30}\quad\quad\quad \dfrac{5}{15} < \dfrac{4}{10}$

For Paulo: $\dfrac{4}{15} = \dfrac{8}{30}\quad\quad\quad \dfrac{2}{10} = \dfrac{6}{30}\quad\quad\quad \dfrac{4}{15} > \dfrac{2}{10}$

From here, we can see that Carlos carried more oranges in the second part than the first part. He carried $\dfrac{4}{10} - \dfrac{5}{15} = \dfrac{12}{30} - \dfrac{10}{30} = \dfrac{2}{30}$ more oranges in the second part.

We know that he carried $50$ more oranges in the second part. Let the total number of oranges be $x$ . We have

$\dfrac{2}{30} \times x = 50\\ x=50 \times \dfrac{30}{2}=750$

Lastly, we calculate the actual number of oranges each person carried in the second part:

For José: $\dfrac{4}{10} \times 750 = 300$

For Carlos: $\dfrac{4}{10} \times 750 = 300$

For Paulo: $\dfrac{2}{10} \times 750 = 150$

The number of oranges José, Carlos and Paulo transported are $\boxed{300,\,300,\,150}$