If `t=(1)/(r)+(1)/(s)`, then what is `(1)/(t)` in terms of r and s?

Answer

To find \( \frac{1}{t} \) in terms of \( r \) and \( s \) given that \( t = \frac{1}{r} + \frac{1}{s} \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ t = \frac{1}{r} + \frac{1}{s} \]

2. **Find a common denominator**:
The common denominator for the fractions \( \frac{1}{r} \) and \( \frac{1}{s} \) is \( rs \). Therefore, we can rewrite \( t \) as:
\[ t = \frac{s}{rs} + \frac{r}{rs} \]

3. **Combine the fractions**:
Now, combine the two fractions:
\[ t = \frac{s + r}{rs} \]

4. **Reciprocate to find \( \frac{1}{t} \)**:
To find \( \frac{1}{t} \), we take the reciprocal of \( t \):
\[ \frac{1}{t} = \frac{rs}{s + r} \]

### Final Answer:
Thus, the expression for \( \frac{1}{t} \) in terms of \( r \) and \( s \) is:
\[ \frac{1}{t} = \frac{rs}{s + r} \]