Introduction
Complex fractions can be intimidating for many students because of the way they look. However, they are simply fractions that contain fractions within them. In this article, we will explore the concept of complex fractions, their properties, and how to simplify them.
A complex fraction is a fraction that contains one or more fractions in either the numerator or denominator, or both. For example, 2/3 is a simple fraction, but (2/3)/(4/5) is a complex fraction. Complex fractions can be written in various ways, including using horizontal or vertical bars, brackets, or parentheses.
One important property of complex fractions is that they can be simplified by multiplying the numerator and denominator of the fraction in the denominator by the reciprocal of that fraction. For example, consider the complex fraction:
(2/3)/(4/5)
To simplify this fraction, we can multiply the numerator and denominator of the fraction in the denominator by its reciprocal, which is 5/4:
(2/3) x (5/4)/(4/5) x (5/4)
Simplifying this expression, we get:
(2/3) x (5/4)/(4/5) x (5/4) = (2/3) x (5/4) x (5/4)/(4/5)
= (2/3) x (25/16)/(4/5)
= (2/3) x (25/16) x (5/4)
= (2 x 25 x 5)/(3 x 16 x 4)
= 125/192
Therefore, (2/3)/(4/5) simplifies to 125/192.
Another property of complex fractions is that they can be written as a sum of simple fractions. To do this, we first need to rewrite the complex fraction so that it has a common denominator. For example, consider the complex fraction:
(2/3)/(4/5)
To rewrite this fraction so that it has a common denominator, we can multiply the numerator and denominator of the first fraction by 5, and the numerator and denominator of the second fraction by 3:
(2/3)/(4/5) = (2 x 5)/(3 x 5) / (4 x 3)/(5 x 3) = 10/15 / 12/15
Now, we can combine the fractions by adding their numerators:
10/15 + 12/15 = 22/15
Therefore, (2/3)/(4/5) is equal to 22/15.
It’s important to note that not all complex fractions can be simplified in this way, and some may require more advanced techniques such as factoring or partial fraction decomposition.
Complex fractions can also arise in real-world problems, such as those involving rates and ratios. For example, consider the following problem:
A car travels 120 miles in 2 hours. What is its average speed in miles per minute?
To solve this problem, we need to convert miles per hour to miles per minute. We know that there are 60 minutes in an hour, so we can write:
average speed in miles per minute = (120 miles/2 hours) / (60 minutes/1 hour)
= (120/2) / 60
= 1 mile/minute
Therefore, the average speed of the car is 1 mile per minute.
In conclusion, complex fractions are fractions that contain one or more fractions within them. They can be simplified by multiplying the numerator and denominator of the fraction in the denominator by its reciprocal or by writing them as a sum of simple fractions. Complex fractions can also arise in real-world problems, such as those involving rates and ratios. Understanding how to work
Definition
A complex fraction is a fraction that contains one or more fractions in either the numerator or the denominator, or both. Complex fractions are different from simple fractions because they have at least one fraction within a fraction. For example, 2/3 is a simple fraction, whereas (3/4)/(5/6) is a complex fraction.
Examples
- (2/3)/(4/5)
This is an example of a complex fraction. The numerator and denominator both contain fractions. To simplify this complex fraction, we can invert the denominator fraction and multiply the fractions, as shown below:
(2/3)/(4/5) = (2/3) x (5/4) = 10/12 = 5/6
Therefore, the simplified form of the complex fraction (2/3)/(4/5) is 5/6.
- (4/5)/(2/3)
This is another example of a complex fraction. To simplify this complex fraction, we can invert the denominator fraction and multiply the fractions, as shown below:
(4/5)/(2/3) = (4/5) x (3/2) = 12/10 = 6/5
Therefore, the simplified form of the complex fraction (4/5)/(2/3) is 6/5.
- (1/2)/((3/4)/(5/6))
This is a more complicated example of a complex fraction. To simplify this complex fraction, we can first invert the fraction in the denominator, as shown below:
(1/2)/((3/4)/(5/6)) = (1/2) x (5/6)/(3/4)
We can then simplify the fraction in the numerator by multiplying the fractions, as shown below:
(1/2) x (5/6)/(3/4) = (1/2) x (5/6) x (4/3) = 10/36
Therefore, the simplified form of the complex fraction (1/2)/((3/4)/(5/6)) is 10/36.
- ((1/2) + (1/3))/((1/4) + (1/5))
This is another example of a complex fraction. To simplify this complex fraction, we can first add the fractions in the numerator and denominator separately, as shown below:
((1/2) + (1/3))/((1/4) + (1/5)) = (5/6)/(9/20)
We can then simplify this complex fraction by inverting the denominator fraction and multiplying the fractions, as shown below:
(5/6)/(9/20) = (5/6) x (20/9) = 100/54 = 50/27
Therefore, the simplified form of the complex fraction ((1/2) + (1/3))/((1/4) + (1/5)) is 50/27.
Quiz
- What is a complex fraction? Answer: A complex fraction is a fraction where either the numerator, denominator, or both, contain fractions themselves.
- What is the general form of a complex fraction? Answer: The general form of a complex fraction is (a/b)/(c/d), where a, b, c, and d are numbers or expressions.
- How can we simplify a complex fraction? Answer: To simplify a complex fraction, we need to find a common denominator for both the numerator and the denominator, then simplify and reduce the resulting fraction.
- What is the first step in simplifying a complex fraction? Answer: The first step in simplifying a complex fraction is to rewrite it as a division problem, i.e., (a/b)/(c/d) = (a/b) ÷ (c/d).
- What is the second step in simplifying a complex fraction? Answer: The second step in simplifying a complex fraction is to invert the denominator and multiply, i.e., (a/b) ÷ (c/d) = (a/b) x (d/c).
- Can we simplify a complex fraction by multiplying the numerator and denominator by the same number or expression? Answer: Yes, we can simplify a complex fraction by multiplying the numerator and denominator by the same number or expression, as long as it does not change the value of the fraction.
- Can we simplify a complex fraction by adding or subtracting the fractions in the numerator and denominator separately? Answer: No, we cannot simplify a complex fraction by adding or subtracting the fractions in the numerator and denominator separately, as this will change the value of the fraction.
- What is an example of a complex fraction? Answer: An example of a complex fraction is (3/4)/(5/6).
- How can we simplify the complex fraction (3/4)/(5/6)? Answer: We can simplify (3/4)/(5/6) by inverting the denominator and multiplying, i.e., (3/4)/(5/6) = (3/4) x (6/5) = 18/20 = 9/10.
- Can we simplify the complex fraction (2/3)/(4/5) by multiplying the numerator and denominator by 5? Answer: Yes, we can simplify (2/3)/(4/5) by multiplying the numerator and denominator by 5, as this gives us (2/3) x (5/4) = 10/12 = 5/6, which is simplified.
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