Question

I need to send a picture of this homework to get help for my son.Find all the positive fractions less than 1 that have denominators of 100 once those fractions have been simplified? In other words, the numerator does not share a common factor other than 1 with the denominator (100). You can think about it as all the possible change (coins) less than $1. An organized list may prove helpful. Be sure to explain your thinking/approach to arriving at your solution

Answer 1

Given that the fractions must be less than 1 but greater than zero, and their denominator must be 100, you need to remember that, by definition:

`(a)/(a)=1`

Therefore, in order for the fraction to be less than 1, the numerator has to be less than the denominator 100.

It is also important to remember that, when the numerator and the denominator do not share a common factor other than 1, the fraction is irreducible. This means that the numerator and the denominator can be both divided only by 1.

Knowing the above, you can make the following list of fractions with those characteristics:

`\begin{gathered} (99)/(100) \\ \\ (97)/(100) \end{gathered}`
`\begin{gathered} (93)/(100) \\ \\ (91)/(100) \\ \\ (89)/(100) \end{gathered}`
`\begin{gathered} (87)/(100) \\ \\ (83)/(100) \\ \\ (81)/(100) \\ \\ (79)/(100) \end{gathered}`
`\begin{gathered} (77)/(100) \\ \\ (73)/(100) \\ \\ (71)/(100) \\ \\ (69)/(100) \\ \\ (67)/(100) \\ \\ (63)/(100) \\ \\ (61)/(100) \end{gathered}`
`\begin{gathered} (59)/(100) \\ \\ (57)/(100) \\ \\ (53)/(100) \\ \\ (51)/(100) \\ \\ (49)/(100) \\ \\ (47)/(100) \\ \\ (43)/(100) \\ \\ (41)/(100) \end{gathered}`
`\begin{gathered} (39)/(100) \\ \\ (37)/(100) \\ \\ (33)/(100) \\ \\ (31)/(100) \\ \\ (29)/(100) \\ \\ (27)/(100) \\ \\ (23)/(100) \\ \\ (21)/(100) \end{gathered}`
`\begin{gathered} (19)/(100) \\ \\ (17)/(100) \\ \\ (13)/(100) \\ \\ (11)/(100) \\ \\ (9)/(100) \\ \\ (7)/(100) \\ \\ (3)/(100) \\ \\ (1)/(100) \end{gathered}`

Hence, the answer is:

`\begin{gathered} (99)/(100) \\ \\ (97)/(100) \end{gathered}`
`\begin{gathered} (93)/(100) \\ \\ (91)/(100) \\ \\ (89)/(100) \end{gathered}`
`\begin{gathered} (87)/(100) \\ \\ (83)/(100) \\ \\ (81)/(100) \\ \\ (79)/(100) \end{gathered}`
`\begin{gathered} (77)/(100) \\ \\ (73)/(100) \\ \\ (71)/(100) \\ \\ (69)/(100) \\ \\ (67)/(100) \\ \\ (63)/(100) \\ \\ (61)/(100) \end{gathered}`
`\begin{gathered} (59)/(100) \\ \\ (57)/(100) \\ \\ (53)/(100) \\ \\ (51)/(100) \\ \\ (49)/(100) \\ \\ (47)/(100) \\ \\ (43)/(100) \\ \\ (41)/(100) \end{gathered}`
`\begin{gathered} (39)/(100) \\ \\ (37)/(100) \\ \\ (33)/(100) \\ \\ (31)/(100) \\ \\ (29)/(100) \\ \\ (27)/(100) \\ \\ (23)/(100) \\ \\ (21)/(100) \end{gathered}`
`\begin{gathered} (19)/(100) \\ \\ (17)/(100) \\ \\ (13)/(100) \\ \\ (11)/(100) \\ \\ (9)/(100) \\ \\ (7)/(100) \\ \\ (3)/(100) \\ \\ (1)/(100) \end{gathered}`