Solve the equation: 5/6 a - 4/5 a + 1 = 1/2 a - 2/5

Let's solve the equation:

\[ \frac{5}{6} a - \frac{4}{5} a + 1 = \frac{1}{2} a - \frac{2}{5} \]

Step-by-step solution:

1. Step 1: Move all terms containing \( a \) to the left side and constants to the right side:

   \[ \frac{5}{6} a - \frac{4}{5} a - \frac{1}{2} a = - \frac{2}{5} - 1 \]

2. Step 2: Find a common denominator for the fractions on the left side. The least common multiple of 6, 5, and 2 is 30:

   \[ \frac{25}{30} a - \frac{24}{30} a - \frac{15}{30} a = - \frac{2}{5} - 1 \]

3. Step 3: Combine the fractions on the left side:

   \[ \frac{25 - 24 - 15}{30} a = - \frac{2}{5} - 1 \]

   \[ \frac{-14}{30} a = - \frac{2}{5} - 1 \]

   \[ -\frac{7}{15} a = - \frac{2}{5} - 1 \]

4. Step 4: Find a common denominator for the fractions on the right side. The least common multiple of 5 and 1 is 5:

   \[ -\frac{7}{15} a = -\frac{2}{5} - \frac{5}{5} \]

5. Step 5: Combine the fractions on the right side:

   \[ -\frac{7}{15} a = -\frac{7}{5} \]

6. Step 6: Solve for \( a \) by multiplying both sides by \( -\frac{15}{7} \):

   \[ a = -\frac{7}{5} \cdot -\frac{15}{7} \]

   \[ a = \frac{7 \cdot 15}{5 \cdot 7} \]

   \[ a = \frac{15}{5} \]

   \[ a = 3 \]

Answer: a = 3