Use the results of exercises 18 and 19 to prove that given any two rational numbers r and s with r , s, there is another rational number between r and s. An important consequence is that there are infinitely many rational numbers in between any two distinct rational numbers.

Step-by-step explanation:

Let r and s be two rational numbers

Without loss of generality assume that r<s, because one number has to be necessarily less than the other otherwise two would be equal.

Then find mid value of r and s as

So we have atleast one rational number between r and s. Note that t is rational because it is sum of two rational numbers r/2 and s/2

Now using r and t we find one rational number say u between r and t.  

Again with r and u we find another rational number between them

This process can be repeated infinitely

Thus we conclude there are infinitely many rational numbers in between any two distinct rational numbers.

0 and 5 minutes

step-by-step explanation:

given that one morning eight buses arrive late by the number of minutes given below:

0 8 2 6 9 2 0 7 8

write these in ascending order

0   0 2 6 7 8 8 9

in the afternoon two more buses arrive at the bus stop say by x and y minutes late.

total number of buses = 10

median = average of 4th and 5th after arranging in ascending order

since 4th entry is 6, the values of x and y should be below 6

also given 0 is the mode so either x or y or both should be 0 because then only 0 gets the maximum frequency and hence mode

thus we have

0 0 0 2 6 7 8 8 9 and another unknown entry which is below 6.

since median is 3.5, we have the unknown entry to be 5 only because then only median =( 2+5)/2

thus the after noon buses were late by 0 and 5 minutes.