$2\mathrm{.}5$ The following table shows the execution times, in seconds, for five different benchmark programs on three machines.

$\backslash$table$[[$Benchmark$,$Processor$],[$P$,$Q$,$R$],[$A$,10,5,3],[$B$,15,260,22,890,53,410],[$C$,275,150,210],[$D$,3563,3110,3610],[$E$,120,120,83]]$

a$.$ Compute the speed metric for each processor for each benchmark, normalized to machine P$.$ That is$,$ the ratio values for $P$ are all $1\mathrm{.}0.$ Use Equation $(2\mathrm{.}5)$ with $P$ as the reference system to calculate the remaining ratios. Then compute the arithmetic mean value for each system using Equation $(2\mathrm{.}3).$

b$.$ Repeat part $($a$)$ using $Q$ as the reference machine.

c$.$ Which machine is the slowest based on the preceding two calculations?

d$.$ Repeat the calculations for $($a$)$ and $($b$)$ using the geometric mean. Which machine is the slowest based on these calculations?