5/8y - 1/2y + 1/12y=1/5

To solve the equation \( \frac{5}{8}y - \frac{1}{2}y + \frac{1}{12}y = \frac{1}{5} \), we will first find a common denominator for the fractions on the left side.

The common denominator for \( 8 \), \( 2 \), and \( 12 \) is \( 24 \). Now, we'll convert each term to have this common denominator:

1. Convert \( \frac{5}{8}y \): \[ \frac{5}{8}y = \frac{5 \times 3}{8 \times 3}y = \frac{15}{24}y \]

2. Convert \( \frac{1}{2}y \): \[ \frac{1}{2}y = \frac{1 \times 12}{2 \times 12}y = \frac{12}{24}y \]

3. Convert \( \frac{1}{12}y \): \[ \frac{1}{12}y = \frac{1 \times 2}{12 \times 2}y = \frac{2}{24}y \]

Now, we can rewrite the equation with these conversions: \[ \frac{15}{24}y - \frac{12}{24}y + \frac{2}{24}y = \frac{1}{5} \]

Next, combine the left-hand side: \[ \left(\frac{15}{24} - \frac{12}{24} + \frac{2}{24}\right)y = \frac{1}{5} \] \[ \frac{5}{24}y = \frac{1}{5} \]

Now, we will isolate \( y \) by multiplying both sides by the reciprocal of \( \frac{5}{24} \), which is \( \frac{24}{5} \): \[ y = \frac{1}{5} \cdot \frac{24}{5} \] \[ y = \frac{24}{25} \]

Thus, the solution to the equation is: \[ y = \frac{24}{25} \]