Rates Problem from CAT exam

[ElizabethS105]

Two musicians, Maria and Perry, work at independent constant rates to tune a warehouse full of instruments. If both musicians start at the same time and work at their normal rates, they will complete the job in 45 minutes. However, if Perry were to work at twice Maria’s rate, they would take only 20 minutes. How long would it take Perry, working alone at his normal rate, to tune the warehouse full of instruments?

a. 1 hr 20 min
b. 1 hr 45 min
c. 2 hr
d. 2 hr 20 min
e. 3 hr

I am having some trouble with this problem. Typically, when I set up work=RT tablets, I use 1 for work and if I am given time, then I can get the rate (1/t), but the explanation from the CAT exam is keeping a variable for work. Why can't it be 1? Can you please explain the Algebra approach to this question?

[RonPurewal]

you can totally use '1' for '1 warehouse' (as a unit of work).
in fact, the way i see it, not only can you do so, but you pretty much have to! after all, there is no indication of how many instruments are in the warehouse, so '1 warehouse' is pretty much the only unit of work available.
(you could pick a 'smart number' for the number of instruments, but that would just add an unnecessary layer of complexity for pretty much zero compensatory benefit.)

does the solution say that you can't do that?

(if you're asking about the solution, by the way, please quote the solution; it's not easy for me to dig such things out of the CAT database.)

[RonPurewal]

in any case, you shouldn't need a lot of heavy-duty algebra here, because the second statement will spit out a numerical rate for maria:
M rate + 2(M rate) = 1/20 warehouse per minute
3(M rate) = 1/20 warehouse per minute
M rate = 1/60 warehouse per minute

now use the other equation to find 'P rate':
M rate + P rate = 1/45 warehouse per minute
1/60 + P = 1/45
3/180 + P = 4/180
1/180 warehouse per minute = P rate

so he takes 180 minutes = 3 hours.

[RonPurewal]

in any case, now i'm interested in the solution, because, honestly, i don't even see how you can EITHER do more algebra OR use a work unit other than '1 warehouse'. like, not even seeing how that's possible.

the only variant that immediately comes to mind is to convert the rates into 'warehouses per hour'. this will make some of the rates nicer (e.g., 1/20 warehouse per minute = 3 warehouses per hour), but others will still be nasty (e.g., 1/45 warehouse per minute = 4/3 warehouse per hour). overall i think it's a wash, so that's neither here nor there.

what does the solution do? tell me!

[ElizabethS105]

Sorry for the delay, I was not "watching" this thread and did not get notified of your response.

here is the CAT solution:
(w/60 + p)45 = w
(3/4)w + 45p = w
45p = (1/4)w
180p = w
180 = w/p

What tripped me out is using so many variables. The way you showed it is exactly the way I would do this problem.

What gets me is when I see "X worked double the rate" Or "X completed the work in half the time"... I never know how to turn that into algebra. I.e "Machine X produced twice as many bottles in 4 hours as Y in 3 hours" .. how would I write that out?

Any tips?

[RonPurewal]

What gets me is when I see "X worked double the rate" Or "X completed the work in half the time"... I never know how to turn that into algebra.

because these examples are so much simpler than the one below them, i'm guessing that you're using them simply as an entry point into the conversation—i.e., that you don't actually have trouble with these ones.

still there's one thing worth noting: if you can do things with one variable... then don't use two.

e.g.,
"X works at twice Y's rate"
--> Y rate = 'r'
--> X rate = '2r'

if your first inclination is to use two different variables, you should try to suppress that—at least for long enough to think about whether you can just use a single variable instead. (if you can't think of a one-variable approach, then of course you should just pick up the metaphorical shovel and start digging.)

[RonPurewal]

I.e "Machine X produced twice as many bottles in 4 hours as Y in 3 hours" .. how would I write that out?

Any tips?

there's one thing you should realize immediately here: this statement is not DIRECTLY telling you anything useful; you need to SIMPLIFY it.
when i say 'immediately', i mean IMMEDIATELY.

if that realization is not immediate, then consider the following (similarly indirect) 'real world' statement:
If you lay seven of these items end to end, they are only 1 inch shorter than this yardstick. (in case you are not from the USA, a 'yardstick' is one yard = 36 inches long.)

if i tell you this, then it should be obvious that IN THIS FORM the statement is useless (unless you are addressing a particular VERY specific issue—e.g., 'if i pack these items into a tube that is 1 yard long, how much room will i still have for other small things?').

also, you should ask yourself, 'What kind of information is (indirectly) presented here?'
hopefully it's also clear that, however indirectly, this statement gives you the length of the item.

thus you must transform the statement so that it communicates this information DIRECTLY:
each item is 5 inches long.

most likely, in the real world, you don't even have to think about doing everything i've written here—you just DO it 'because it's common sense'.
the problem, as always, is that in 'academic' or 'classroom' situations people no longer have that sort of common sense. so, now, you actually have to think through things that are normally instinctive.
THAT not an easy thing to do. (this is why 'self-awareness' is much harder than it sounds; most people do not often, if ever, try to analyze their own thoughts.)

--

the same goes for the statement about machines X and Y.

i'll give one possible algebraic interpretation in the next post.
but, BEFORE you look at the next post...
...think about what kind of information is presented here
...try to SIMPLIFY the statement into a form that communicates this information DIRECTLY.

[RonPurewal]

ok, so let's look at this statement:

Machine X produced twice as many bottles in 4 hours as Y in 3 hours

• WHAT KIND of information is this?
...we are NOT talking about any actual amount of production. rather, we're talking about the two machines' production capabilities.
...'production capability' is measured by rate. here, that means bottles per hour.
...this statement tells us nothing about 'absolute' rates; in other words, it should be clear that you're not going to get any actual numbers out of this statement.**
...so it must be a(n annoyingly indirect) way of giving the RATIO of the rates at which the two machines work.

time to simplify.

• let's consider what happens in 12 hours. this way we only need to look at total production.

• let's define a TEMPORARY variable:
in 4 hours, machine X produces '2n' bottles. in 3 hours, machine Y produces 'n' bottles.
we MUST be able to eliminate this variable, since we need to end up with a ratio that's actually a ratio.

• so, in 12 hours...
...machine X produces '6n' bottles
...machine Y produces '4n' bottles

• so the ratio of rates is 6:4 = 3:2 (with machine X having the faster rate).

--

[RonPurewal]

(**) above
as soon as you realize that it's not possible to find actual numbers here, you should immediately realize that PLUGGING IN SMART NUMBERS is an option.

so, let's do that.

• let's say machine Y produces 12 bottles in 3 hours. (12 is a good choice because it can be easily divided by either '3 hours' or '4 hours'.)
this is 4 bottles per hour.

• then machine X produces twice that much, or 24 bottles, in 4 hours.
this is 6 bottles per hour.

and so you have a ratio, with much less effort than algebra would require.