Select All The Correct Answers.A Sailboat Travels A Distance Of $2 \frac{1}{2}$ Miles In $\frac{1}{8}$ Of An Hour. Which Complex Fraction Represents The Unit Rate In Miles Per Hour?A. $\frac{\frac{1}{1}}{21}$B.
Introduction
In mathematics, unit rates are essential for comparing quantities and understanding rates of change. When dealing with complex fractions, it's crucial to simplify them to obtain the unit rate. In this article, we'll explore how to find the unit rate in miles per hour for a sailboat traveling a certain distance in a given time.
Understanding Complex Fractions
A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given problem, we have a complex fraction representing the distance traveled by the sailboat: $2 \frac1}{2}$ miles. To simplify this fraction, we can convert the mixed number to an improper fraction{2}$ miles.
The Problem
A sailboat travels a distance of $\frac{5}{2}$ miles in $\frac{1}{8}$ of an hour. We need to find the unit rate in miles per hour, which is represented by the complex fraction $\frac{\frac{5}{2}}{\frac{1}{8}}$.
Simplifying Complex Fractions
To simplify a complex fraction, we can multiply the numerator and denominator by the reciprocal of the other fraction. In this case, we can multiply the numerator $\frac{5}{2}$ by the reciprocal of the denominator $\frac{1}{8}$, which is $8$.
The Unit Rate
The unit rate in miles per hour is represented by the simplified complex fraction: $20$ miles per hour.
Conclusion
In conclusion, to find the unit rate in miles per hour for a sailboat traveling a certain distance in a given time, we need to simplify the complex fraction representing the distance traveled and the time taken. By multiplying the numerator and denominator by the reciprocal of the other fraction, we can obtain the unit rate. In this case, the unit rate is $20$ miles per hour.
Answer
The correct answer is:
- A. $\frac{\frac{1}{1}}{21}$ is incorrect because it does not represent the unit rate in miles per hour.
- The correct unit rate is $20$ miles per hour, but it is not listed as an option. However, we can represent it as a complex fraction: $\frac{\frac{5}{2}}{\frac{1}{8}} = \frac{40}{2} = 20$
Unit Rates and Complex Fractions: A Sailboat's Journey ===========================================================
Q&A: Unit Rates and Complex Fractions
Q: What is a unit rate?
A: A unit rate is a rate with a denominator of 1. It's a way to compare quantities and understand rates of change. In the context of the sailboat problem, the unit rate represents the distance traveled per hour.
Q: How do I simplify a complex fraction?
A: To simplify a complex fraction, you can multiply the numerator and denominator by the reciprocal of the other fraction. This will eliminate the complex fraction and leave you with a simpler fraction.
Q: What is the reciprocal of a fraction?
A: The reciprocal of a fraction is obtained by swapping the numerator and denominator. For example, the reciprocal of $\frac{5}{2}$ is $\frac{2}{5}$.
Q: How do I multiply fractions?
A: To multiply fractions, you multiply the numerators and denominators separately. For example, to multiply $\frac{5}{2}$ and $\frac{8}{1}$, you would multiply the numerators (5 and 8) and the denominators (2 and 1) separately.
Q: What is the unit rate in miles per hour for the sailboat?
A: The unit rate in miles per hour for the sailboat is $20$ miles per hour. This is obtained by simplifying the complex fraction $\frac{\frac{5}{2}}{\frac{1}{8}}$.
Q: How do I represent a unit rate as a complex fraction?
A: To represent a unit rate as a complex fraction, you can use the formula: $\frac{\text{distance}}{\text{time}}$. For example, the unit rate of $20$ miles per hour can be represented as the complex fraction $\frac{\frac{5}{2}}{\frac{1}{8}}$.
Q: What is the difference between a unit rate and a rate?
A: A unit rate is a rate with a denominator of 1, while a rate is a comparison of two quantities. A unit rate is a special type of rate that helps us understand rates of change.
Q: How do I use unit rates in real-life situations?
A: Unit rates are used in a variety of real-life situations, such as calculating speed, distance, and time. For example, if you're driving a car and you want to know how far you'll travel in a certain amount of time, you can use a unit rate to calculate the distance.
Q: What are some common applications of unit rates?
A: Unit rates are used in a variety of applications, including:
- Calculating speed and distance
- Understanding rates of change
- Comparing quantities
- Making informed decisions
Conclusion
In conclusion, unit rates and complex fractions are essential concepts in mathematics that help us understand rates of change and compare quantities. By simplifying complex fractions and using unit rates, we can make informed decisions and solve real-life problems.