Simplify the following complex fraction: [tex]$\frac{1}{\frac{1}{P}+\frac{1}{S}}$[/tex]
Answer (2)
Find a common denominator for the fractions in the denominator: P 1 + S 1 = PS S + P .
Substitute this back into the original expression: PS S + P 1 .
Invert and multiply to simplify: S + P PS .
The simplified expression is P + S PS .
Explanation
Understanding the Problem We are asked to simplify the complex fraction P 1 + S 1 1 . This involves algebraic manipulation to combine the fractions in the denominator and then simplify the entire expression.
Combining Fractions First, we need to find a common denominator for the two fractions in the denominator of the complex fraction. The common denominator for P 1 and S 1 is PS . So we rewrite the denominator as: P 1 + S 1 = PS S + PS P = PS S + P
Substituting Back Now we substitute this back into the original expression: P 1 + S 1 1 = PS S + P 1
Simplifying the Fraction To simplify the complex fraction, we invert and multiply: PS S + P 1 = S + P PS Thus, the simplified expression is P + S PS .
Final Answer Therefore, the simplified form of the complex fraction P 1 + S 1 1 is P + S PS .
Examples
Complex fractions appear in various fields, such as physics and engineering, when dealing with combinations of resistances or capacitances in electrical circuits. For example, if you have two resistors in parallel with resistances P and S, the total resistance of the parallel combination is given by the formula P 1 + S 1 1 . Simplifying this expression to P + S PS allows for easier calculation of the total resistance. Understanding how to simplify complex fractions is therefore crucial for solving practical problems in circuit analysis and other related areas.
The complex fraction P 1 + S 1 1 simplifies to P + S PS . This is achieved by finding a common denominator for the fractions in the denominator and then inverting and multiplying. This method makes working with complex fractions more manageable.
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