Which Expression is Equivalent to the Following Complex Fraction?
In mathematics, simplifying complex fractions is a fundamental skill that helps in understanding the underlying concepts of algebra. A complex fraction is a fraction within a fraction, and finding an equivalent expression can sometimes be challenging. This article aims to explore different methods and techniques to determine which expression is equivalent to a given complex fraction. By doing so, we can enhance our problem-solving skills and deepen our understanding of mathematical operations.
Complex fractions can be simplified by applying various algebraic properties and rules. One of the most common methods is to multiply both the numerator and the denominator by the conjugate of the denominator. This process is known as rationalizing the denominator. Let’s consider an example to illustrate this concept.
Example: Simplify the complex fraction (2/3) / (4/5).
To find an equivalent expression, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is (5/4). By doing so, we get:
(2/3) / (4/5) = (2/3) (5/4) / (4/5) (5/4)
Now, we can simplify the expression by canceling out common factors:
= (2 5) / (3 4) / (4 5) / (4 5)
= 10 / 12 / 20 / 20
= 10 / 240
= 1 / 24
Therefore, the equivalent expression to the given complex fraction (2/3) / (4/5) is 1/24.
Another method to simplify complex fractions is by converting them into single fractions. This can be achieved by multiplying the numerator of the outer fraction by the denominator of the inner fraction and vice versa. Let’s consider another example to demonstrate this technique.
Example: Simplify the complex fraction (3/4) / (2/3) + (4/5) / (1/2).
To convert the complex fraction into a single fraction, we can multiply the numerators and denominators accordingly:
= (3/4) (3/2) + (4/5) (2/1)
= 9/8 + 8/5
Now, we need to find a common denominator to add the fractions:
= (9/8) (5/5) + (8/5) (8/8)
= 45/40 + 64/40
= 109/40
Therefore, the equivalent expression to the given complex fraction (3/4) / (2/3) + (4/5) / (1/2) is 109/40.
In conclusion, finding an equivalent expression to a complex fraction requires a solid understanding of algebraic properties and rules. By applying techniques such as rationalizing the denominator and converting complex fractions into single fractions, we can simplify these expressions and gain a deeper insight into mathematical operations. Practice and perseverance are key to mastering this skill, as it is a fundamental building block in higher-level mathematics.
