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Every week I hop on a treadmill and figure out how long I need to run. I am given the pace:

Pace = 7:41/mile = (7 minutes + 41 seconds) per mile

I need to add this up to calculate how long I should run to run 1.5 miles. I use the formula

7:41 + (7:41/2) = ? 1.5 mile

I find this somewhat difficult to calculate in my head, especially while I am starting my warmup. Converting to seconds doesn't make it any easier. Do you have any suggestions as to how I can do this more efficiently?

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If you run 1.5 miles every day... then there's no need to keep track of the time it takes for you to run a single mile, just keep track of how long it takes you to run the 1.5 miles. –  Nicolas Villanueva Jun 3 '11 at 20:55
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The pace is given to me by the treadmill itself. The pace at which I run changes weekly. I go up roughly 1 unit every week (7 seconds quicker mile). After my warmup I usually forget to see how far I've run. The only thing I know for sure is my warmup is 2 minutes. I could stop the treadmill and start over again to reset the clock, but that would kill my pace. Adding the time in my head is the best I can think to do. –  P.Brian.Mackey Jun 3 '11 at 20:57
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Relearn all you division in sexagismal. –  Eric Naslund Jun 3 '11 at 20:58
    
@Eric: very funny - perhaps something for another round of New Math. +1 –  mixedmath Jun 3 '11 at 21:00
    
@Eric Naslund - +1 Wow, base 60. Never even heard of it. That's a pretty cool idea. –  P.Brian.Mackey Jun 3 '11 at 21:03
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3 Answers 3

up vote 10 down vote accepted

I understand your question as this: "How do I efficiently divide numbers by $2$ in sexagismal. (Base 60)"

Suppose you have $a*60+b$ as your time. In your case, $a=7$, $b=41$. To divide by two, just do it the way you normally would, but carrying is by $60$ instead of $10$. (Base $60$ instead of base $10$)

Divide $a$ by two. If it is even, no problem. If it is odd, then you "carry" a 60 over to $b$. So when dividing $7:41$, we get $3: \frac{41+60}{2}$. Then you just divide $b$ by $2$ (or $b+60$ if we had to carry).

So to divide $7:41$ by two, we get $3:50.5$. Lets try another. How about $16:56$? Well the first term is even so we just get $8:28$. What about $27:32$? Well, the first part will be $13$, we add $60$ to $32$ to get $92$, then divide this by two, so the answer is $13:46$.

You try one: What is $9:51$ divided by two? (answer at end)

I hope this was helpful,

Important Note: Notice that this works for other numbers besides just dividing by $2$. Dividing by any number in base $60$ is the same, we just carry $60$'s instead of $10$'s.
Even more generally, notice this works for any base $b$. Division base $b$ is just done by carrying $b$ instead of $10$.

Answer: $9:51$ divided by two is $4:55.5$. We divide $9$ by two, and get $4$, and carry the $60$ over to the $51$ to get $111$, which is $55.5$ after division by $2$.

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This is going to take a little practice, but this is what I am looking for. Thanks all –  P.Brian.Mackey Jun 3 '11 at 21:23
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@P.Brian: For the full problem (how to multiply by 1.5), it may be easier to multiply by 3 and then divide by 2 as above. For instance, with 7:41, you could first multiply by 3 to get 21:123 = 23:03, then divide by 2 to get 11:(60+3)/2 = 11:31.5. This may be easier than first dividing by 7:41 by 2 to get 3:50.5, and adding 3:50.5 to 7:41. Also note that you can do the "carry" operations whenever you feel like it. For instance you could see that 7:41 * 3 = 21:123 = 22:63 (carried only one 60), and dividing this by 2 to get 11:31.5 is easier. Try everything for a while and see what is fastest. –  ShreevatsaR Jun 4 '11 at 4:16
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I recommend splitting it up in your head. 7:41 is 7 minutes and 41 seconds. We know that half of 7 minutes is 3 minutes and a half. Half of 41 is about 20. So we add these to get 3:50 or so, and add that to 7:41. Even when I add them, I take 10 seconds from 7:41 (getting 7:31) and add it to 3:50 (getting 4:00) so that 11:31 is that much easier.

That's how I process it mentally, anyway.

Alternately, I estimate a little. I know that 8 minutes is 480 seconds, and that 7:41 is therefore about 5% less than 480 (I think that 20 seconds less than a minute is about half of the 48 seconds that is 10% of 480 - obviously, we could be more precise, but I'm just giving an example of estimation). So if I want to go a mile and a half, I want to go about 5% less than 12 minutes. As 12 minutes is 720 seconds, 10% of 720 is 72, so 5% is 36. So I would approximate the amount of time I have to run as about 12:00 - 0:36 = 11:24.

Both yield things that are close enough, but if I were worried, I would just run for an additional 20 seconds or so to make up any shortcomings. Is that what you were looking for?

NOTE: thanks to amWhy for pointing out my incapability of multiplying 6 by 8.

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oops: 7 minutes is 420 seconds! –  amWhy Jun 3 '11 at 21:17
    
@amWhy: very embarassing! I'm fixing it - thank you. –  mixedmath Jun 3 '11 at 21:19
    
Upvoted because this is the way I do it in my head too! Seems like the most natural way. –  Sputnik Jun 4 '11 at 18:15
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My answer is going to be more specific to the calculation you're doing (a + 1/2 * a), so if you have the pace 7:41 and you want to find 7:41 + 1/2 * 7:41.

First you do 7/2 = 3.5, and add it to the original time 7:41+3.5 = 10.5:41, then if necessary, normalize the .5 to 10:71

Second you add the seconds 10:71 + 41/2 = 10:91.

Finally, normalize it to 11:31.

An example run of it in your head:

A    B    C
7    ...  ... (read the minute from the panel)
7    3.5  ... (divide by 2)
10.5 ...  ... (add it to the minute)
10.5 41   ... (read the panel again for the second)
10   71   ... (normalize the .5 by adding 30 to the seconds)
10   71   41  (read the panel again for the second)
10   71   20  (divide by 2)
10   91   ... (add to the seconds)
11   31   ... (normalize) 

This might be easier for some people than doing a base-60 division first as the steps are IMO simpler to work with in (my) head. So the algorithm is basically:

  1. Read the time
  2. Add the minute part
  3. Add the seconds part
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