To simplify the expression
\[ \frac{u}{5u - 5} + \frac{u}{40u - 40}, \]
we first factor the denominators.
-
For \(5u - 5\), we can factor out a 5: \[ 5u - 5 = 5(u - 1). \]
-
For \(40u - 40\), we can similarly factor out a 40: \[ 40u - 40 = 40(u - 1). \]
Now substituting these factored forms back into the original expression, we have: \[ \frac{u}{5(u - 1)} + \frac{u}{40(u - 1)}. \]
Next, we need to find a common denominator to combine these fractions. The common denominator will be the least common multiple (LCM) of \(5(u - 1)\) and \(40(u - 1)\). The LCM is \(40(u - 1)\).
Now we will rewrite each fraction with the common denominator:
-
The first fraction: \[ \frac{u}{5(u - 1)} \cdot \frac{8}{8} = \frac{8u}{40(u - 1)}. \]
-
The second fraction is already: \[ \frac{u}{40(u - 1)}. \]
Now we can combine the two fractions: \[ \frac{8u}{40(u - 1)} + \frac{u}{40(u - 1)} = \frac{8u + u}{40(u - 1)} = \frac{9u}{40(u - 1)}. \]
Thus, the expression simplifies to: \[ \frac{9u}{40(u - 1)}. \]
This fraction is in simplest form, as there are no common factors in the numerator and the denominator.
The final answer is: \[ \frac{9u}{40(u - 1)}. \]
1 answer