For $20$ values of $y,$ it is given that $\Sigma \left(ax-b\right)=400$ and $\Sigma \left(bx-a\right)=545.$ Given also that $\overline{x}=6.25,$ find the value of $a$ and of $b.$

The midpoint of the line segment between $A$ and $B$ is at $\left(5.2, -1.2\right).$

Find the coordinates of the midpoint after the following transformations have been applied to $A$ and to $B.$

$T:$ Translation by the vector $\left(\begin{array}{r}-7\\ 4\end{array}\right).$

The midpoint of the line segment between $A$ and $B$ is at $\left(5.2, -1.2\right).$

Find the coordinates of the midpoint after the following transformations have been applied to $A$ and to $B.$

$E:$ Enlargement through the origin with scale factor $5.$

The midpoint of the line segment between $A$ and $B$ is at $\left(5.2, -1.2\right).$

Find the coordinates of the midpoint after the following transformations have been applied to $A$ and to $B.$

Transformations $T$ and $E$ are carried out one after the other. Investigate whether the location of the midpoint of $AB$ is independent of the order in which the transformations are carried out.

Five investors are repaid, each with their initial investment increased by $p\%$ plus a fixed 'thank you' bonus of $\$q.$ The woman who invested $\$20000$ is repaid double her investment and the man who invested $\$7500$ is repaid triple his investment. Find the total amount that the five people invested, given that the mean amount repaid to them was $\$33000.$

Do you think the method of repayment is fair? Give a reason for your answer.

The number of patients treated each day by a dentist during a $20$-day period is shown in the following stem-and-leaf diagram.

Find the median number of patients.

The number of patients treated each day by a dentist during a $20$-day period is shown in the following stem-and-leaf diagram.

On eight of these $20$ days, the dentist arrived late to collect their son from school. If they decide to use their average number of patients as a reason for arriving late, would they use the median or the mean? Explain your answer.

The number of patients treated each day by a dentist during a $20$-day period is shown in the following stem-and-leaf diagram.

Describe a situation in which it would be to the dentist's advantage to use a mode as the average.