Ch. 1 Lesson 2 (Complex Fractions & Unit Rates)
Mrs. Schock's Flipped Classroom・5 minutes read
A complex fraction involves fractions within a fraction, as demonstrated by the simplification of 5/6 divided by 7/8 to 20/21. The lesson emphasizes finding unit rates with fractions by placing money in the numerator and time in the denominator, exemplified by Horace's consumption of hot dogs which leads to a rate of 15 dozen in three hours.
Insights
- A complex fraction includes fractions within fractions, as demonstrated by the example of dividing 5/6 by 7/8, which simplifies to 20/21 through multiplication of the outer and inner numbers and division by the greatest common factor.
- The lesson emphasizes calculating unit rates with fractions by placing money in the numerator and time in the denominator, as shown in Horace's example where he eats 2.5 dozen hot dogs in a quarter hour, leading to a calculation of 5 dozen hot dogs per hour, ultimately illustrating the practical application of these concepts over an extended time frame.
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Recent questions
What is a complex fraction?
A complex fraction is a type of fraction that contains one or more fractions in either its numerator, denominator, or both. For example, if you have a fraction like 5/6 divided by 7/8, this is considered a complex fraction because it involves the division of two fractions. To simplify a complex fraction, you typically multiply the outer numbers and the inner numbers, which helps in finding a simpler equivalent. In the case of 5/6 divided by 7/8, you would multiply 5 by 8 to get 40 and 6 by 7 to get 42, leading to a simplified result of 20/21 after dividing by the greatest common factor. Understanding complex fractions is essential in various mathematical contexts, especially when dealing with more advanced calculations.
How do I find unit rates?
Finding unit rates involves determining how much of one quantity corresponds to a single unit of another quantity, often expressed as a fraction. The key to setting up a unit rate is to place the relevant terms correctly in the fraction, typically with money in the numerator and time in the denominator. This setup allows for a clear understanding of the relationship between the two quantities. For instance, if you are calculating the cost of items over a specific time period, you would express the total cost as the numerator and the total time as the denominator. This method ensures that you can easily identify and interpret the unit rate, which is crucial for making comparisons and informed decisions in various scenarios, such as budgeting or analyzing speed.
What is a unit rate example?
A unit rate example can be illustrated through a scenario involving a competitive eater named Horace. Suppose Horace consumes 2.5 dozen hot dogs in a quarter of an hour. To find out how many hot dogs he can eat in one hour, you would set up the rate with 2.5 dozen in the numerator and 1/4 hour in the denominator. By converting the quarter hour into a fraction and simplifying, you can determine that Horace eats 5 dozen hot dogs per hour. This example highlights the process of calculating unit rates, demonstrating how to convert mixed numbers to improper fractions and simplify them to find a clear rate of consumption over time.
How to calculate total from unit rates?
To calculate the total from unit rates, you simply multiply the unit rate by the number of units you are interested in. For example, if you know that Horace can eat 5 dozen hot dogs per hour, and you want to find out how many he can eat in three hours, you would multiply the unit rate of 5 dozen by 3. This calculation results in a total of 15 dozen hot dogs that Horace can consume in three hours. This method of multiplying the unit rate by the desired time or quantity is a straightforward way to apply unit rates in practical situations, allowing for quick and accurate estimations of totals based on known rates.
What is the greatest common factor?
The greatest common factor (GCF) is the largest positive integer that divides two or more numbers without leaving a remainder. It is a crucial concept in simplifying fractions and performing operations involving whole numbers. To find the GCF, you can list the factors of each number and identify the largest one they share. For example, if you are simplifying the fraction 20/42, you would find the GCF of 20 and 42, which is 2. Dividing both the numerator and the denominator by the GCF simplifies the fraction to its lowest terms, making it easier to work with in calculations. Understanding the GCF is essential for effective fraction manipulation and simplification in various mathematical contexts.
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Summary
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Understanding Complex Fractions and Unit Rates
- A complex fraction is defined as a fraction that contains one or more fractions within it, exemplified by the calculation of 5/6 divided by 7/8, which simplifies to 20/21 after multiplying the outer numbers (5 times 8 equals 40) and the inner numbers (6 times 7 equals 42), followed by dividing by the greatest common factor (GCF) of 2.
- In this lesson, the focus is on finding unit rates that involve fractions, with the key instruction being to place money in the numerator and time in the denominator when setting up rates, ensuring clarity in identifying relevant terms such as "money words" and "time words."
- An example problem involves Horace, a competitive eater, who consumes 2.5 dozen hot dogs in a quarter of an hour. To find out how many hot dogs he can eat in one hour, the rate is set up as 2.5 dozen in the numerator and 1/4 hour in the denominator, leading to the calculation of 5 dozen hot dogs per hour after converting the mixed number to an improper fraction and simplifying.
- To determine how many hot dogs Horace can eat in three hours, the unit rate of 5 dozen per hour is multiplied by 3, resulting in a total of 15 dozen hot dogs in three hours, illustrating the application of unit rates with fractions and the use of complex fractions for division.
