Which expression is equivalent to the following complex fraction 1+1/y/1-1/y – Embarking on an exploration of complex fractions, we delve into the intricacies of simplifying the expression 1+1/y/1-1/y. This mathematical odyssey unveils the equivalence of complex fractions, shedding light on their algebraic operations and practical applications.
Venturing beyond mere simplification, we uncover alternative methods and explore the broader landscape of related concepts. Our journey culminates in a comprehensive understanding of complex fractions, empowering problem-solving in diverse fields.
Complex Fractions: Which Expression Is Equivalent To The Following Complex Fraction 1+1/y/1-1/y
Complex fractions involve fractions within fractions. To simplify a complex fraction, we can apply algebraic operations to obtain an equivalent expression that is simpler to work with.
Simplifying the Complex Fraction 1+1/y/1-1/y, Which expression is equivalent to the following complex fraction 1+1/y/1-1/y
We can simplify this complex fraction as follows:
- Multiply both the numerator and denominator by (1-1/y):
- Simplify the numerator:
- Invert the denominator:
- Simplify the product:
- Combine like terms:
- Multiply both the numerator and denominator of the first fraction by y:
- Multiply both the numerator and denominator of the second fraction by (y-1):
- Substitute the simplified fractions into the original expression:
- Physics: Calculating the magnification of optical instruments.
- Engineering: Designing electrical circuits.
- Economics: Modeling supply and demand.
- Rational expressions: Expressions that involve fractions of polynomials.
- Algebraic fractions: Fractions involving algebraic expressions.
- Simplify: (2/x – 1/y) / (3/x + 4/y)
- Simplify: (1 + 1/x) / (1 – 1/x) – (1 – 1/x) / (1 + 1/x)
- Simplify: (x^2 – 1) / (x + 1) / (x – 1) / (x^2 + 1)
- Simplify: (3/x + 2/y) / (1/x – 5/y)
- Simplify: (x + 1/x) / (x – 1/x) – (x – 1/x) / (x + 1/x)
- Simplify: (x^2 + y^2) / (x – y) / (x + y) / (x^2 – y^2)
- (3y – 10x) / (5x + 3y)
- 2 / (x^2 – 1)
- 1 / (x + y)
“` (1+1/y) – (1-1/y) / (1-1/y) – (1-1/y) = (1-1/y+1) / (1-1/y) “`
“` (1-1/y+1) / (1-1/y) = (2-1/y) / (1-1/y) “`
“` (2-1/y) / (1-1/y) = (2-1/y) – (1-1/y) “`
“` (2-1/y) – (1-1/y) = 2 – 2/y + 1/y – 1/y^2 “`
“` 2 – 2/y + 1/y – 1/y^2 = 2 – 1/y – 1/y^2 “`
Therefore, the simplified expression is 2 – 1/y – 1/y^2.
Alternative Methods
An alternative method to simplify complex fractions is to use a common denominator. In this case, the common denominator is y(y-1).
“` (1+1/y) – y / (1-1/y) – y = (y+1) / (y-1) “`
“` (1-1/y) – (y-1) / (1-1/y) – (y-1) = (y-1) / (y-1) = 1 “`
“` (y+1) / (y-1) / 1 = (y+1) / (y-1) “`
This method yields the same simplified expression as the previous method.
Applications
Complex fractions have various applications in real-world scenarios, including:
Related Concepts
Complex fractions are related to the following concepts:
Understanding complex fractions contributes to a broader understanding of algebraic operations and their applications in various fields.
Examples
Practice Exercises
Answer Key:
Helpful Answers
What is the significance of simplifying complex fractions?
Simplifying complex fractions allows for easier manipulation, comparison, and evaluation, making them more manageable in problem-solving.
How do alternative methods contribute to simplifying complex fractions?
Alternative methods, such as using a common denominator, provide diverse approaches to simplification, offering flexibility and efficiency in solving complex fraction problems.
What are the practical applications of complex fractions?
Complex fractions find applications in various fields, including physics, engineering, and economics, where they model real-world scenarios and facilitate problem-solving.
