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Mathematics
  1. CESS Construct an Argument Explain how complex fractions can be used to solve problems involving ratios.
  2. CCSS Reason Inductively Write three different complex fractions that simplify to 1 4 (1)/(4).
  3. CGSS Persevere with Problems Use mental math to find the value of 15 124 230 30 ÷ 230 124 (15)/(124)*(230)/(30)-:(230)/(124).
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  1. 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 number of boys total number of students number of girls total number of students (("number of boys")/("total number of students"))/(("number of girls")/("total number of students")). 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.
  2. There are many different complex fractions that simplify to 1 4 (1)/(4). Three examples are 2 8 1 4 ((2)/(8))/((1)/(4)), 5 20 3 12 ((5)/(20))/((3)/(12)), and 6 12 3 6 ((6)/(12))/((3)/(6)). In each of these cases, the numerator and denominator are multiplied or divided in a way that results in a fraction equivalent to 1 4 (1)/(4). 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.
  3. To find the value of 15 124 230 30 ÷ 230 124 (15)/(124)*(230)/(30)-:(230)/(124) using mental math, we can simplify each fraction by canceling out common factors. First, we can simplify 15 124 (15)/(124) by dividing the numerator and denominator by 5 to get 3 24 (3)/(24). Similarly, we can simplify 230 30 (230)/(30) by dividing the numerator and denominator by 10 to get 23 3 (23)/(3). Finally, we can simplify 230 124 (230)/(124) by dividing the numerator and denominator by 2 to get 115 62 (115)/(62). Then, we can substitute these simplified fractions into the original expression to get 3 24 23 3 ÷ 115 62 (3)/(24)*(23)/(3)-:(115)/(62). 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 1 8 1 ÷ 115 31 (1)/(8)*1-:(115)/(31). We can simplify this by dividing 1 by 8 to get 1 8 (1)/(8) and multiplying by the reciprocal of 115 31 (115)/(31) to get 31 920 (31)/(920). Therefore, 15 124 230 30 ÷ 230 124 = 31 920 (15)/(124)*(230)/(30)-:(230)/(124)=(31)/(920).
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