Questions: In due villaggi dell'Amazzonia la popolazione di uno era i 5/6 della popolazione dell'altro. Una grave epidemia costrinse a emigrare 70 abitanti di ogni villaggio, per cui attualmente il primo villaggio ha i 4/5 degli abitanti del secondo. Quanti abitanti ha oggi ciascun villaggio?

Solution Steps

Let \( P_1 \) be the population of the first village and \( P_2 \) be the population of the second village before the epidemic.

From the problem statement, we have the following relationships:

1. The population of the first village is \( \frac{5}{6} \) of the population of the second village: \[ P_1 = \frac{5}{6} P_2 \]
2. After 70 inhabitants emigrated from each village, the populations become \( P_1 - 70 \) and \( P_2 - 70 \). The new population of the first village is \( \frac{4}{5} \) of the new population of the second village: \[ P_1 - 70 = \frac{4}{5} (P_2 - 70) \]

Substituting \( P_1 \) from the first equation into the second equation: \[ \frac{5}{6} P_2 - 70 = \frac{4}{5} (P_2 - 70) \]

Expanding the right side: \[ \frac{5}{6} P_2 - 70 = \frac{4}{5} P_2 - \frac{4}{5} \cdot 70 \] Calculating \( \frac{4}{5} \cdot 70 \): \[ \frac{4}{5} \cdot 70 = 56 \] Thus, the equation becomes: \[ \frac{5}{6} P_2 - 70 = \frac{4}{5} P_2 - 56 \]

To eliminate the fractions, multiply the entire equation by 30 (the least common multiple of 6 and 5): \[ 30 \left( \frac{5}{6} P_2 - 70 \right) = 30 \left( \frac{4}{5} P_2 - 56 \right) \] This simplifies to: \[ 25 P_2 - 2100 = 24 P_2 - 1680 \]

Rearranging the equation gives: \[ 25 P_2 - 24 P_2 = 2100 - 1680 \] \[ P_2 = 420 \]

Now substitute \( P_2 \) back into the first equation to find \( P_1 \): \[ P_1 = \frac{5}{6} \cdot 420 = 350 \]

After the epidemic, the populations of the villages are:

• First village: \( P_1 - 70 = 350 - 70 = 280 \)
• Second village: \( P_2 - 70 = 420 - 70 = 350 \)

Thus, the current populations are:

• First village: \( 280 \)
• Second village: \( 350 \)

Final Answer