To solve the equation [tex]\( \frac{1}{m} + \frac{1}{t} = \frac{1}{c} \)[/tex] for [tex]\(c\)[/tex], follow these steps:
1. Combine the fractions on the left-hand side: To add the two fractions, you need a common denominator. The common denominator for [tex]\( \frac{1}{m} \)[/tex] and [tex]\( \frac{1}{t} \)[/tex] is [tex]\( m \cdot t \)[/tex].
[tex]\[
\frac{1}{m} + \frac{1}{t} = \frac{t}{m \cdot t} + \frac{m}{m \cdot t}
\][/tex]
2. Add the fractions: Combine the numerators over the common denominator.
[tex]\[
\frac{t}{m \cdot t} + \frac{m}{m \cdot t} = \frac{t + m}{m \cdot t}
\][/tex]
3. Set the equation with a single fraction: Replace the left-hand side with the combined fraction.
[tex]\[
\frac{t + m}{m \cdot t} = \frac{1}{c}
\][/tex]
4. Invert both sides of the equation: To isolate [tex]\( c \)[/tex], take the reciprocal of both sides.
[tex]\[
\frac{m \cdot t}{t + m} = c
\][/tex]
Therefore, the solution for [tex]\( c \)[/tex] is:
[tex]\[
c = \frac{m \cdot t}{m + t}
\][/tex]
