Question

Simplify. Express your answer as a single fraction in simplest form.
u
5u–5
+
u
40u–40

Answer 1

To simplify the expression

\[
\frac{u}{5u - 5} + \frac{u}{40u - 40},
\]

we first factor the denominators.

1. For \(5u - 5\), we can factor out a 5:
\[
5u - 5 = 5(u - 1).
\]

2. 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:

1. The first fraction:
\[
\frac{u}{5(u - 1)} \cdot \frac{8}{8} = \frac{8u}{40(u - 1)}.
\]

2. 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:
\[
\frac{9u}{40(u - 1)}.
\]