Word Problems "per" Clarification

[RyanS540]

I understand that "per" usually means division and this usually works out in the problems I work through. However I see that this doesn't work sometimes and I'm not sure why. For example in the Word Problems guide, Chapter 2, p. 22, there's a question: "Machine X produces cartons at a uniform rate of 90 every 3 minutes, and Machine Y produces cartons at a uniform rate of 100 every 2 minutes. Working simultaneously, how many minutes would it take for the two machines to produce a total of 560 cartons?"

I set this up as (90c/3m) + (100c/2m) = 560c
Simplify to: (30c/m) + (50c/m) = 560c.

Multiply each side by (m/c) = 30 + 50 = 560m
80 = 560m
80/560 = 1/7 = m

However the answer is actually 7. Setting up the problem like 30cm + 50cm = 560c leads the right answer of m = 7. Why do we use "per" as multiplication here and how can I corroborate that with the also accurate representations of "per" as division? Such as miles per hour, and many other examples.

[Sage Pearce-Higgins]

You're on the right line here. However, it looks like you've made an equation without really understanding what it means. This is dangerous as you can make the kind of mistake you did here, but, more seriously, you'll have real trouble on harder problems unless you grasp the deeper concept.

The key idea is that, with machines working together, you can add the rates. Think about why this is. You basically do this in this part of the equation: (90c/3m) + (100c/2m). However, I encourage you to actually work out the number before moving further. Then you can apply the formula Rate x Time = Work, putting in 560 for the work and the rate that you've calculated.

Takeaways: know why you're doing what you're doing, and break strategies down into small steps where possible rather than trying to compile a mega equation.

[RyanS540]

I wouldn't say I made an equation without knowing what it means, but now I see I was missing the T component in RT=D. I was solving for the time variable in the denominator of the R component. Thanks for posting.

[Sage Pearce-Higgins]

You're welcome.