Find five rational numbers between $1$ and $2$.
(A) We can approach this problem in at least two ways.
Method $1$: To find a rational number between $r$ and $s$,we can use the formula $\frac{r+s}{2}$.
First number: $\frac{1+2}{2} = \frac{3}{2}$.
Second number: $\frac{1 + 3/2}{2} = \frac{5/2}{2} = \frac{5}{4}$.
Third number: $\frac{1 + 5/4}{2} = \frac{9/4}{2} = \frac{9}{8}$.
Fourth number: $\frac{3/2 + 2}{2} = \frac{7/2}{2} = \frac{7}{4}$.
Fifth number: $\frac{7/4 + 2}{2} = \frac{15/4}{2} = \frac{15}{8}$.
Method $2$: To find $n$ rational numbers between two numbers,we can express them with denominator $n+1$.
Here $n=5$,so we use denominator $5+1=6$.
$1 = \frac{6}{6}$ and $2 = \frac{12}{6}$.
The five rational numbers are $\frac{7}{6}, \frac{8}{6}, \frac{9}{6}, \frac{10}{6}, \text{ and } \frac{11}{6}$.
Simplifying these,we get $\frac{7}{6}, \frac{4}{3}, \frac{3}{2}, \frac{5}{3}, \text{ and } \frac{11}{6}$.
Method $1$: To find a rational number between $r$ and $s$,we can use the formula $\frac{r+s}{2}$.
First number: $\frac{1+2}{2} = \frac{3}{2}$.
Second number: $\frac{1 + 3/2}{2} = \frac{5/2}{2} = \frac{5}{4}$.
Third number: $\frac{1 + 5/4}{2} = \frac{9/4}{2} = \frac{9}{8}$.
Fourth number: $\frac{3/2 + 2}{2} = \frac{7/2}{2} = \frac{7}{4}$.
Fifth number: $\frac{7/4 + 2}{2} = \frac{15/4}{2} = \frac{15}{8}$.
Method $2$: To find $n$ rational numbers between two numbers,we can express them with denominator $n+1$.
Here $n=5$,so we use denominator $5+1=6$.
$1 = \frac{6}{6}$ and $2 = \frac{12}{6}$.
The five rational numbers are $\frac{7}{6}, \frac{8}{6}, \frac{9}{6}, \frac{10}{6}, \text{ and } \frac{11}{6}$.
Simplifying these,we get $\frac{7}{6}, \frac{4}{3}, \frac{3}{2}, \frac{5}{3}, \text{ and } \frac{11}{6}$.
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