Question

Mathematics

- CESS Construct an Argument Explain how complex fractions can be used to solve problems involving ratios.
- CCSS Reason Inductively Write three different complex fractions that simplify to
. - CGSS Persevere with Problems Use mental math to find the value of
.

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- Complex fractions can be a useful tool for solving problems that involve ratios. Ratios are a way of comparing two quantities, and complex fractions can be used to represent these comparisons in a clear and concise way. For example, if we wanted to find the ratio of the number of boys to the number of girls in a classroom, we could use the complex fraction
. This fraction represents the ratio of boys to girls as a single value, making it easier to work with and compare to other ratios. Additionally, complex fractions can be used to simplify complex proportion problems, where multiple ratios are involved. By representing the ratios as complex fractions, we can use basic algebraic methods to solve for unknowns and simplify equations. - There are many different complex fractions that simplify to
. Three examples are , , and . In each of these cases, the numerator and denominator are multiplied or divided in a way that results in a fraction equivalent to . Complex fractions can be used to represent many different types of ratios and comparisons, and there are many different ways to simplify them to obtain a desired value. - To find the value of
using mental math, we can simplify each fraction by canceling out common factors. First, we can simplify by dividing the numerator and denominator by 5 to get . Similarly, we can simplify by dividing the numerator and denominator by 10 to get . Finally, we can simplify by dividing the numerator and denominator by 2 to get . Then, we can substitute these simplified fractions into the original expression to get . We can further simplify this by canceling out common factors: 3 goes into 3 and 24, 23 goes into 23, and 62 goes into 124 twice. This leaves us with . We can simplify this by dividing 1 by 8 to get and multiplying by the reciprocal of to get . Therefore, .