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Consider the following problems:

  1. $$\frac{14}{4.6\dot{6}}$$

  2. $$20 \cdot 45$$

  3. $$\frac{7}{6}$$

  4. $$28 \cdot 0.28$$

  5. $$\frac{42}{168}$$

I'm looking for some techniques to solve these problems fast without a calculator, and preferably without relying on pen and paper.

I've read most of the multiplication section in the Trachtenberg book, and How To Calculate Quickly by Henry Sticker, but it seems like these books work best with paper. Maybe I just don't have the capacity to do this mentally?

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  • $\begingroup$ "Maybe I just don't have the capacity to do this mentally?" I think you do, but just like any other skill it takes practice. $\endgroup$ – Arthur Aug 13 '19 at 7:00
  • $\begingroup$ Yes quite likely. I think the issue is that when I do these mentally, I visualise it as though I were writing it on paper, hence, I'm very slow. $\endgroup$ – mr-matt Aug 13 '19 at 7:10
  • $\begingroup$ Yes, mental techniques utilise the fact that in your imagination, changing something is a bit faster and easier than it is on paper. For instance, adding two large numbers from the left (because that's the intuitive way) rather than from the right (which is what you do on paper because carries might happen). $\endgroup$ – Arthur Aug 13 '19 at 7:22
  • $\begingroup$ Please clarify what you mean by "solve these problems". Does it mean to find the lowest terms fraction, or decimal rep, or something else? $\endgroup$ – Bill Dubuque Aug 13 '19 at 15:31
  • $\begingroup$ depends on what you call large, and fast. $\endgroup$ – user645636 Aug 13 '19 at 17:50
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  1. should be possible without thinking, since this is pretty basic:

$20\cdot 45=900$

A method here could be to split it into more multiplications, if you struggle:

It is easier when you calculate $45\cdot 2\cdot 10=90\cdot 10=900$

An other example: $14\cdot 15$ might be trickier. But calculating $15\cdot 2\cdot 7=30\cdot 7=210$ is pretty easy.

  1. If you know that $\frac16=0.1\bar{6}$ this is trivial, since $\frac{7}{6}=1+\frac16=1.1\bar{6}$

  2. $28\cdot 0.28$ the trick here is to transfrom $0.28$ into a fraction. $0.28=\frac{28}{100}$ This should also be obvious.

Then we are left with $\frac{28\cdot 28}{100}=\frac{14\cdot 28}{50}=\frac{14\cdot 14}{25}=\frac{196}{25}$

Most people know the quadratics of the numbers 1 to 25, because they had to learn it in school. If you do not know them, you should learn them.

If you want to give a decimal instead of a fraction, it is often a good idea, to add and subtract something. I would consider this more advanced:

$\frac{196}{25}=\frac{200}{25}-\frac{4}{25}=8-\frac{16}{100}=8-0.16=7.84$

If you learned your quadratics, you could also know that $28\cdot 28=784$ which makes the solution trivial. Of course there are many ways leading to an answer, I am just trying to give some examples how it could be done, by showing some commen principles.

  1. $\frac{42}{168}=\frac{1}{4}$

this should be seen immediatly, that $4\times 42=168$. If not it is not hard to cancel commen factors:

$\frac{42}{168}=\frac{21}{84}=\frac{3}{12}=\frac14$

  1. $\frac{14}{4.6\bar{6}}$ is in my opinion the hardest to calculate here. Especially without pen and paper.

I would try this:

$4.\bar{6}=4+\frac23=\frac{14}{3}$

Then $\frac{14}{\frac{14}{3}}=\frac13$

So tips you should take away:

a) It is almost always easier to calculate with fractions.

b) Know some basic multiplications like the quadratics $1^2, 2^2, \dotso, 24^2, 25^2$ or even more. This is also useful if you want to find a root in your head! If you know the root of a five digit number is an integer, then you can 'guess' the roots of numbers in the range of $100^2$ to $250^2$ pretty easy.

Example: $\sqrt{20736}$ You only have to focus on the first three digits $207$ and the last one $6$.

The first three digits help as follows. If you know the quadratics by heart you just have to bound 207 between two quadratics which are the 'closest' to it:

$14^2=196<207<225=15^2$ So $\sqrt{20736}$ is between $1400$ and $1500$

Now you need the last digit. Note that every (besides 0 and 5) 'last digit' can only be created by two multiplications:

$1^2$ and $9^2$ give last digit $1$

$2^2$ and $8^2$ give last digit $4$

$3^2$ and $7^2$ give last digit $9$

$4^2$ and $6^2$ give last digit $6$

$5^2$ gives digit 5 and $0^2$ of course $0$.

Since 20736 ends with 6 we know that the root is either $144$ or $146$. Since 20736 is closer to 19600 we can guess it is $4$.

c) Adding numbers to get higher (or subtracting for lower) easier to factor numbers and use binomial formulas.

Example:

$199^2$ might be hard to calculate by itself, but $(200-1)^2=200^2-400+1$ is trivial.

These methods also help alot. Practice makes perfect and there are nearly unlimited methods to calculate.

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  • $\begingroup$ I endorse all of this. Generally, I think that a lot of "mental math" tips are not worth the brainspace it takes to memorize them. But if you know common squares and reciprocals and have some fluency in manipulating them, I find that you can work out a lot of quick problems in the time it takes others to pull their smart phones out of their pockets. $\endgroup$ – Matthew Daly Aug 13 '19 at 7:40
  • $\begingroup$ Also to learn quadratics you can use other 'mental' methods. Like $18^2=324$. When I learned quadratics with my father he said "When you are 18, then you go 324mph with your new car". This stuck to me. :) Also good to know is that if you know 18^2 for example and you have to find 19^2, then you can find it by adding 18 and 19. So $19^2=324+18+19=324+20+20-3=361$ (Using some techniques from above too). Of course this works for every quadratic. So $20^2=361+19+20=400$ and so on. $\endgroup$ – Cornman Aug 13 '19 at 8:00

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