# Multiplying multiple relative rates - figuring out what it is asking.

This problem really confuses me:

$$\frac{\ 542}{1 wk} . \frac{\ 1 wk}{5 days} . \frac{\ 1 day}{8 hr}$$

where "wk" = weekday week
"days" = work day

So what it's saying is, "five hundred forty-two per week, times 1 week per 5 days, times 1 day per 8 hours". I do not understand this at all. I'm not really looking to solve it as I think I can do that with my tutor, but wrapping my head around this is something that was left up to me until friday. Some one please help!

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I guess "wk" should be interpreted as "weekday week" and "day" should be interpreted as "work day," or else we live on different planets :) – rschwieb Feb 25 '13 at 17:02
Yep! I'll edit. – randmath Feb 25 '13 at 17:32
homework should not be used as a standalone tag; see tag-wiki and meta. – Martin Sleziak Feb 26 '13 at 7:34

This is a way to do conversions that works on any sort of problem.

If you look at this in words, we want to know what the hourly pay is for the 5 day work week based on the total earning for an 8-hour work day.

Look at how "wk", and "day" cancel.

Now, if you multiply across the numerator you get $542$.

If you multiply across the denominator, you get $40$.

All other units cancel out, so we have pay/hr.

So, we are left with:

$$\frac{\ 542}{40} \text{per hour}$$

Does that make sense?

As another example, use the same method to convert $24$ hours into seconds to see if it makes sense.

$$\begin{array}{c|c|c|c} \text{24 hours}& 60 ~minutes & 60 ~seconds\\\hline \text{}& 1 hour& 1 minute\\ \end{array}$$

See how the "hours" and "minutes" cancel.

Now just multiply numerators and divide by the multiplication of denominators and there is your conversion.

So, we are left with $\displaystyle \frac{24 \times 60 \times 60}{1 \times 1} = 86400$ seconds.

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Ahhh. Okay! Thanks for the enlightenment. I will study this a little more. Thanks again. – randmath Feb 25 '13 at 16:54
@randmath: You are very welcome! – Amzoti Feb 25 '13 at 17:00
Nice work! I like your tables, and the follow-up from the OP! – amWhy Apr 30 '13 at 0:41
@amWhy: That simple method saved me many a time in Physics classes! :-) Just know what the units are supposed to be at the end, using the given information, is very useful indeed! Thx – Amzoti Apr 30 '13 at 0:43