Algebra Problem Finding More Reps To Work

Question

I am not a math major so please take it easy on me. I am trying to figure this out for a real situation and could use all help necessary.

Problem: There are $4$ reps. who process $32$ transactions in $3$ minutes ($180$ seconds). I want to finds out how many more reps. do I need to minimize that time down to $0.30$ seconds.

My Working: which does not answer the above question

I know $3$ minutes $=$ $180$ seconds.

We want to find out $w$ $=$ $.30$ seconds.

We have $32$ transactions which processes within $180$ seconds.

We can find out how long $1$ transaction takes: $\frac{32}{1}$ $=$ $\frac{180}{x}$

Solving gives us: $x$ $=$ $5.625$ seconds.

We can also find out how many transactions $1$ rep. can process: $\frac{4}{1}$ $=$ $\frac{32}{y}$

Solving gives us: $y$ $=$ $8$ transactions.

Since we know $1$ rep. processes $8$ transactions and it takes $5.625$ seconds to process $1$ transaction we can say:

($8$ transactions)($5.625$ seconds) $=$ $45$ seconds

Which means: it takes $1$ rep. $45$ seconds to process $8$ transactions$.

However, this does not give me the solution to the problem I am looking for. I basically found out for $1$ rep. instead of finding out how many more reps needed to break down the time to $.30$ seconds.

HELP PLEASE! Please can someone give me a formula for this too once it's shown how to actually solve this problem. I have been working on this all day!

Edit: I got it now. Thanks for the help.

Answer 1

You want the job done 6 times faster, so you need six times the reps. So 24 reps altogether (or 20 "more" than you have now.

Answer 2

First of all, throw out the number of transactions. It stays the same, so it shouldn't matter.

Second; $4$ people do something in $3$ minutes. To do it in half the time, i.e. $1.5$ minutes, how many people do you need? Twice. In other words $4\times3/1.5=8$ people. So you need to find out by what factor your time is decreasing, and you must increase the number of people by the same factor.

So your answer will be $4\times3/0.5=24$ people. Or, since you already have $4$ people, $20$ more people.

This is also called the rule of three by some people.

Edit: For 0.30 seconds, you do the same thing but now you compare $4\times180/0.30=2400$. You need to compare equivalent units, so convert 3 minutes to 180 seconds.