Speeding up calculations

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#1 5 June, 2017 - 04:28
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Speeding up calculations


Every Sunday in the paper we have a puzzle called the 30 second challenge. It's divided into a beginner, intermediate and advanced categories. To be classified as advanced one must complete both the intermediate and advanced puzzles within 30 secs. Over a number of years I think I've been able to reach advanced status perhaps 5 to 10 percent of the time. The types of calculations that slow me down are where you need to find 90 % of a number or divide by some prime number like 19 or multiple get by 7. How does one speed up their processing? If anyone here has done these or similar challenges successfully how long did it take you?

5 June, 2017 - 07:46
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Quote:

The types of calculations that slow me down are where you need to find 90 % of a number or divide by some prime number like 19 or multiple get by 7.

Division by 19 is easy when I tell you how I divide by 109: http://artofmemory.com/forums/analyzing-ruediger-gamms-performance-divis...

For multiplication by 7 there are not many short cuts that I know.

For taking 90% I use 100% minus 10%.
So 90% of let's say 81 is 81-8.1=81-9+0.9=72+.9=72.9.
Accidentially, you now have also calculated 9^3. Think about it.

After this there is mostly practice. For a week take 90% of every number you come across. Then divide the number by 19, until you get proficient.

11 June, 2017 - 21:01
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Thanks Kinma some good tips there. I thought I might put up an example of what has to be done in 30 secs.

Intermediate: 18, x 7, 5/9 of it, x 40, 75% of it, 3/100 of it, 4/9 of it, +27, x 20, 25%

Advanced: 100, times itself, 37.5% of it, 2% of this, x 40, less 10%, 3% of it, - 18, plus 33 and1/3%, -29

I can do the Intermediate one easily enough, both need to be completed within 30 secs to be classified as Advanced.

How many people can actually do this, and I'm wondering what memory strategies could be utilised in order for this to happen as I don't see how you can be an advanced user without resorting to some memory tricks.

12 June, 2017 - 00:33
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with the advanced one I got 55 after 26,7 seconds.

Basically, you can skip half of the equation, starting right at the start. 100 times any percentage is basically just the percentage. Put mathematically, the start is 100 * 100 * 0.375 * 0.02 which is just 37.5 * 2, just like the two percentages written. So 75.
Times 40, that is easy, 3000. Minus 10%, easy too, 2700. 3% might be a bit tough to some, but all you have to do is 27*3, which is 81.
Next step is minus 18, just do minus 20 and plus 2, resulting in 63. Plus 33.3% is just devide by three and multiply by four, so 84. Minus 29 is minus 30 plus 1.
55.

My bet is just that you shouldnt make it harder than it has to be, it is a really easy one

12 June, 2017 - 00:58
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Same thing with the intermediate seemingly.

18, x 7, 5/9 of it, x 40, 75% of it, 3/100 of it, 4/9 of it, +27, x 20, 25%

the first bit, turn it around. 5/9 of 18 is 10, then times 7 is an easy 70.
next it is times 40 and then 75% of the result, all you have to do is times 30. You could even read a bit further ahead and see 3% at the next step, and just decide to make it 7*9 instead of 70 * 30 * 0.03. That way you won't get confused by the zeros.
Depending on how far ahead you can read, you can make it one step further, as the next step is times 4/9, so instead of 7*9 you could do 7*4. I myself only realized that after I already did the 7*9 though. We are now at 28.
The next bit is add 27, easy addition, 55.
The times 20 is just times 2 and add a 0 to the end, 1100. And finally 25%, which is 275.

what I noticed myself is that it works better if I can write it, rather than think it. If I have to think it, I have to do it all in my conscious mind. If I can write it, I feel like I can use my subconscious mind for "instinctive calculations".

12 June, 2017 - 01:19
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As Mayarra says, it is good to look through the whole puzzle first to spot shortcuts. It seems that the puzzles are designed to make you do that.

Mayarra found a great one with the 2 percentages.
In the intermediate one the first two steps - 18 x 7 x 5 / 9 - are easier if you rearrange this to: 18 / 9 x 5 x7 = 70.

Tiger, where do you lose the most time if you solve the puzzle?

12 June, 2017 - 02:26
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The approach used is interesting. While the instructions don't explicitly forbid looking ahead in the challenge I have not really attempted to do it that way. I have also the 30 sec challenge app which will only show you one step at a time so you cannot use that strategy.
Having completed a few of these in my time, I would say that overall the 2 calculations that feature regularly that slow me down are when I need to find 3/8 or 5/8 of a number like 192 or 296. The other calculation that slows me down is where I need to divide 144 by 3, or 162 divide by 9 or 119 divided by 7. If you knew your tables up to 30 by 30 that would help, I usually try and split the number up but is slows me down. I can generally complete the challenges, it's just difficult to do everything within the time limit.

12 June, 2017 - 02:49
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Knowing your tables can indeed help.
Like for 144 divided by 3, I know that 144 divided by 12 is 12, so 144 divided by 3 has to be 48.

Looking at the end digit can also help. Should I need 3/8 of 192, it is easiest to know 1/8 first. It ends on a 2 so (given that it is fully dividable by 8) it will be something-4 or something-9. In this case it is 24. Same for 296. It is something-2 or something-7. Knowing your tables it is easy to spot within a second that it is either 32 or 37, logic then tells you to go with 37 without any calculation needed.
Multiply by 3 and you got your answers, 72 and 111 respectively.

12 June, 2017 - 05:47
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Quote:

Having completed a few of these in my time, I would say that overall the 2 calculations that feature regularly that slow me down are when I need to find 3/8 or 5/8 of a number like 192 or 296.

I understand these are tough. You need to do 2 things in one step. First divide by 8, then multiply. You need to keep all intermediate results in short term memory.

If you don't know the tables up to these numbers, do a quick divide: 8x20=160, and 192 - 160 = 32. so you should see a quick 24 coming up.

296 / 8
30 x 8 = 240, which leaves 56 to divide.

Quote:

The other calculation that slows me down is where I need to divide 144 by 3, or 162 divide by 9 or 119 divided by 7.

144 /3. Like Mayarra says, it helps to realize that a gross = a dozen dozen = 12 x 12 = 144 items.

If you play the app, do your times improve?

12 June, 2017 - 17:43
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Thanks Mayarra and Kinma (btw I love that stuff you posted about logs and am finding myself revisiting them after many years) for your insights. Thinking more and more about where I lose time it seems clear that it's division that is setting me back. 496/8 or 182/7 slow me down a bit, I came across 8% of 625 on the last one I attempted and that had me flummoxed until I worked out that 8 x 625 was 5000. I am seriously thinking about committing at least 1 x 2 multiplications up to 50 ie up to 9 by 50. It's a bit of work though so I'll have to weigh it up.
In regards to the app, my speed has improved but I cannot hand on heart say that I'm in the Advanced category yet.

13 June, 2017 - 00:58
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I think that if you would focus on your tables and combine that with looking at the last number, your divisions can go faster.

For example the two you mentioned, with 496 / 8 I almost instantly see it is 62 because the last number is a 6 and the highest multiplication of 8 fitting into 49 is 48, which is 6 times. 182 / 7 goes the same way, the 2 means it has to be a 6, and 14 is the highest multiplication of 7 that fits in 18, so 26.

As you mentioned, that is an easy way to do low percentages. 8% of 625 is 8 times 6.25, which is a lot easier. You could even split that up like (8*6)+(8*0.25) which might make the calculation easier.

13 June, 2017 - 01:06
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Quote:

I love that stuff you posted about logs and am finding myself revisiting them after many years

Thanks for the compliment!

Quote:

496/8

Try to see 496 as 400 + 80 = 16.
In other words, when dividing by 8, try to see numbers in terms of their 8-ness.
Thus, factors of 8.

Quote:

182/7

Try to see 182 as 140 + 42.

Quote:

8% of 625

Sometimes it is quicker to double 3 times. If you can see 1250 when you think of doubling 625, then 2500 and 5000 will also quickly come up.
It is usually quick to double numbers.

Also for me I do 6.25 x 8 quicker than 625 x 8.
Maybe in my head I immediately see 'a quarter times 8' as 2.

I am doing some research on the 30 Second Challenge and it seems there is usually at least one way to combine 2 steps together.

Take a look at the advanced one here:

30 second challenge

'10% of it' and then 'double it' is '20% of it' or '1/5 of it'.
Next 2 steps: 'x 1000' and '1% of it' can of course be combined as 'x 10'.

Another tip.
I see ' 17 1/2% of it' or 'plus 17 1/2%' often. Example:

30 seconds

The number is probably even. Half it and use 35%. Maybe even half it again and use 70% if that is easier.

13 June, 2017 - 03:15
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Thanks again guys, those tips really help.
Kinma, some excellent research and those excerpts are indeed 30 sec challenges. I have found that the degree of difficulty can vary between issues. The first one displayed I completed inter and adv in 29 secs. The second one took twice as long, the addition at the start really slowed me down. Also you are correct that 17&1/2% appears a lot, I've found a neat workaround for it and that is x by 7/40 but I like your approach as well.

13 June, 2017 - 07:44
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Quote:

Having completed a few of these in my time, I would say that overall the 2 calculations that feature regularly that slow me down are when I need to find 3/8 or 5/8 of a number like 192 or 296.

I have a different approach for these which you might find helpful. Whenever I see 3/8 or 5/8, I always first divide by 2. So 192/2 (which is the same as 192 * 4/8) = 96. Then I divide that by 4 so 96/4 = 24.

3/8 * 192 = 96-24 = 72
5/8 * 192 = 96+24 = 120

At least for me, this is the fastest way to calculate these in my head.

13 June, 2017 - 16:55
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Thanks Ptken, I have been working through some more of these and your tip was helpful, thank you for sharing.

14 June, 2017 - 03:00
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I've found a neat workaround for it and that is x by 7/40

Excellent find!

15 June, 2017 - 16:45
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The other calculation that sometimes slows me down are atypical percentage calculations. I came across one yesterday which required 24% of 625. I managed to work it out but it still took too long, I treated the 625 as 6&1/4 and then multiplied it by 24, I'm sure there must be an easier way. I'm fine with standard percentages like 10, 20, 5 etc. it's the 12%, or 28% that slow me down.

16 June, 2017 - 02:10
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I would do this:
24% of 625 =
12% of 1250 =
6% of 2500 =
6 X 25 =
150

These are small steps, so easy on the brain.
One side halves, the other doubles.

16 June, 2017 - 02:39
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Kinma's way would indeed be easy, multiply 6.25 by four and dividing 24 by four would make that a bit faster if you can do both quickly.

another way could be to divide 625 by 25, which is rather an easy step as 625 is 25 squared, and multiply by 6, you will end up with the same result.

16 June, 2017 - 03:40
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And that's why I love this site, both ideas are an improvement on my method. Thanks.

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