A system became $20\%$ slower than earlier time $T$ should be modeled as $T*1.2$ or $\frac{T}{0.8}$

Question

I am bit confused in the following question as already states in the topic

A system became $20\%$ slower than earlier time $T$ it used to take to complete an operation. Should it be modeled as $T*1.2$ or $\frac{T}{0.8}$?

Actually what each of these representation imply? i.e, $T*1.2$ and $\frac{T}{0.8}$? And which would be correct modeling?

Also, What if the system became $20\%$ faster? I have been correctly solving modeling $\frac{T}{1.2}$ but I am curious what $T*0.8$ would mean in this case?

Answer 1

Imagine that the task involved is to run (or walk) a distance of one mile.

If you run at speed $S$ miles per hour then the time taken to complete the mile is $T=\frac 1 S$ hours. If you now run $20\%$ slower then your speed is now $S'=0.8S$ and your time to complete the mile is now

$T'=\frac{1}{S'} = \frac{1}{0.8S} = \frac{T}{0.8}$

If instead you have $T'=1.2T$ then we would say you took $20\%$ longer - which is different from running $20\%$ slower.