Answer :
Let's determine the consecutive integers that each of the given rational numbers lies between by analyzing their values step-by-step.
### a) [tex]\(\frac{3}{7}\)[/tex]
First, let's consider the rational number [tex]\(\frac{3}{7}\)[/tex].
1. Calculate [tex]\(\frac{3}{7}\)[/tex]: We know that [tex]\(\frac{3}{7}\)[/tex] is a positive fraction less than 1.
2. Determine the floor: The floor of a fraction [tex]\(\frac{p}{q}\)[/tex] (where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q > 0\)[/tex]) is the greatest integer less than or equal to [tex]\(\frac{p}{q}\)[/tex]. Here, [tex]\(\frac{3}{7}\)[/tex] is more than 0 but less than 1. Thus, the floor value is 0.
3. Determine the ceiling: The ceiling of [tex]\(\frac{p}{q}\)[/tex] is the smallest integer greater than or equal to [tex]\(\frac{p}{q}\)[/tex]. For [tex]\(\frac{3}{7}\)[/tex], the value is still less than 1 but more than 0, so the ceiling value is 1.
Therefore, [tex]\(\frac{3}{7}\)[/tex] lies between the consecutive integers 0 and 1.
### b) [tex]\(\frac{8}{5}\)[/tex]
Next, let’s look at [tex]\(\frac{8}{5}\)[/tex].
1. Calculate [tex]\(\frac{8}{5}\)[/tex]: [tex]\(\frac{8}{5}\)[/tex] is equal to 1.6.
2. Determine the floor: The floor of 1.6 is the greatest integer less than or equal to 1.6, which is 1.
3. Determine the ceiling: The ceiling of 1.6 is the smallest integer greater than or equal to 1.6, which is 2.
Therefore, [tex]\(\frac{8}{5}\)[/tex] lies between the consecutive integers 1 and 2.
### c) [tex]\(-\frac{3}{5}\)[/tex]
Now, consider the rational number [tex]\(-\frac{3}{5}\)[/tex].
1. Calculate [tex]\(-\frac{3}{5}\)[/tex]: [tex]\(-\frac{3}{5}\)[/tex] is equal to -0.6.
2. Determine the floor: The floor of -0.6 is the greatest integer less than or equal to -0.6, which is -1.
3. Determine the ceiling: The ceiling of -0.6 is the smallest integer greater than or equal to -0.6, which is 0.
Therefore, [tex]\(-\frac{3}{5}\)[/tex] lies between the consecutive integers -1 and 0.
### d) [tex]\(-\frac{9}{5}\)[/tex]
Finally, let’s consider [tex]\(-\frac{9}{5}\)[/tex].
1. Calculate [tex]\(-\frac{9}{5}\)[/tex]: [tex]\(-\frac{9}{5}\)[/tex] is equal to -1.8.
2. Determine the floor: The floor of -1.8 is the greatest integer less than or equal to -1.8, which is -2.
3. Determine the ceiling: The ceiling of -1.8 is the smallest integer greater than or equal to -1.8, which is -1.
Therefore, [tex]\(-\frac{9}{5}\)[/tex] lies between the consecutive integers -2 and -1.
In summary:
- [tex]\(\frac{3}{7}\)[/tex] lies between 0 and 1.
- [tex]\(\frac{8}{5}\)[/tex] lies between 1 and 2.
- [tex]\(-\frac{3}{5}\)[/tex] lies between -1 and 0.
- [tex]\(-\frac{9}{5}\)[/tex] lies between -2 and -1.
### a) [tex]\(\frac{3}{7}\)[/tex]
First, let's consider the rational number [tex]\(\frac{3}{7}\)[/tex].
1. Calculate [tex]\(\frac{3}{7}\)[/tex]: We know that [tex]\(\frac{3}{7}\)[/tex] is a positive fraction less than 1.
2. Determine the floor: The floor of a fraction [tex]\(\frac{p}{q}\)[/tex] (where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q > 0\)[/tex]) is the greatest integer less than or equal to [tex]\(\frac{p}{q}\)[/tex]. Here, [tex]\(\frac{3}{7}\)[/tex] is more than 0 but less than 1. Thus, the floor value is 0.
3. Determine the ceiling: The ceiling of [tex]\(\frac{p}{q}\)[/tex] is the smallest integer greater than or equal to [tex]\(\frac{p}{q}\)[/tex]. For [tex]\(\frac{3}{7}\)[/tex], the value is still less than 1 but more than 0, so the ceiling value is 1.
Therefore, [tex]\(\frac{3}{7}\)[/tex] lies between the consecutive integers 0 and 1.
### b) [tex]\(\frac{8}{5}\)[/tex]
Next, let’s look at [tex]\(\frac{8}{5}\)[/tex].
1. Calculate [tex]\(\frac{8}{5}\)[/tex]: [tex]\(\frac{8}{5}\)[/tex] is equal to 1.6.
2. Determine the floor: The floor of 1.6 is the greatest integer less than or equal to 1.6, which is 1.
3. Determine the ceiling: The ceiling of 1.6 is the smallest integer greater than or equal to 1.6, which is 2.
Therefore, [tex]\(\frac{8}{5}\)[/tex] lies between the consecutive integers 1 and 2.
### c) [tex]\(-\frac{3}{5}\)[/tex]
Now, consider the rational number [tex]\(-\frac{3}{5}\)[/tex].
1. Calculate [tex]\(-\frac{3}{5}\)[/tex]: [tex]\(-\frac{3}{5}\)[/tex] is equal to -0.6.
2. Determine the floor: The floor of -0.6 is the greatest integer less than or equal to -0.6, which is -1.
3. Determine the ceiling: The ceiling of -0.6 is the smallest integer greater than or equal to -0.6, which is 0.
Therefore, [tex]\(-\frac{3}{5}\)[/tex] lies between the consecutive integers -1 and 0.
### d) [tex]\(-\frac{9}{5}\)[/tex]
Finally, let’s consider [tex]\(-\frac{9}{5}\)[/tex].
1. Calculate [tex]\(-\frac{9}{5}\)[/tex]: [tex]\(-\frac{9}{5}\)[/tex] is equal to -1.8.
2. Determine the floor: The floor of -1.8 is the greatest integer less than or equal to -1.8, which is -2.
3. Determine the ceiling: The ceiling of -1.8 is the smallest integer greater than or equal to -1.8, which is -1.
Therefore, [tex]\(-\frac{9}{5}\)[/tex] lies between the consecutive integers -2 and -1.
In summary:
- [tex]\(\frac{3}{7}\)[/tex] lies between 0 and 1.
- [tex]\(\frac{8}{5}\)[/tex] lies between 1 and 2.
- [tex]\(-\frac{3}{5}\)[/tex] lies between -1 and 0.
- [tex]\(-\frac{9}{5}\)[/tex] lies between -2 and -1.
