To solve the equation ( \frac{5}{8}y - \frac{1}{2}y + \frac{1}{12}y = \frac{1}{4} ), let's first find a common denominator for the fractions involving ( y ).
The denominators are 8, 2, and 12. The least common multiple (LCM) of these numbers is 24.
Now, we can rewrite each term:
- ( \frac{5}{8}y = \frac{5 \cdot 3}{8 \cdot 3}y = \frac{15}{24}y )
- ( \frac{1}{2}y = \frac{1 \cdot 12}{2 \cdot 12}y = \frac{12}{24}y )
- ( \frac{1}{12}y = \frac{1 \cdot 2}{12 \cdot 2}y = \frac{2}{24}y )
Now, substitute these back into the equation:
[
\frac{15}{24}y - \frac{12}{24}y + \frac{2}{24}y = \frac{1}{4}
]
Combining the fractions on the left side:
[
\left( \frac{15 - 12 + 2}{24} \right)y = \frac{1}{4}
]
Simplifying the left side:
[
\frac{5}{24}y = \frac{1}{4}
]
To isolate ( y ), multiply both sides by the reciprocal of ( \frac{5}{24} ), which is ( \frac{24}{5} ):
[
y = \frac{1}{4} \cdot \frac{24}{5}
]
Calculating the right side:
[
y = \frac{24}{20} = \frac{6}{5}
]
So, the solution is:
[
\boxed{\frac{6}{5}}
]
