Fastest way to perform arithematic calculations

Question

This is one of the questions asked for Junior Trader position at prop. trading firm.

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Perform the following operation.

4.3 * 0.58 + 2.0E-5 - 0.9

How can one perform calculations so fast in their mind? I tried learning vedic maths tricks (which are supposed to be the shortcuts for calculations) but they also seems to take time (which won't work for such time crunch calculations).

Any other way to simplify multiplication, division of two digit numbers?

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PS: Tests are rigorous. Time limit per question was max 10 sec.

EDITED: No pen, pencil and paper. No calculator. All has to be done mentally.

Answer 1

Heres how I would do it, though I can't guarantee that it's fast, and whether you can do it in your mind totally depends on you:

$4.3\cdot0.58+2\cdot10^{-5}-0.9$

The only really tricky part is the fist term. I would calculate $43\cdot 58$, which is $40\cdot 58+3\cdot 58=10\cdot 232+174=2494$

Of course we divide by $10$ three times to get back to $2.494$. Subtract the $0.9$ to get $1.594$, and add $0.00002$ to get $1.59402$.

I'm sorry if that's not satisfactory, but I really don't see any shortcut (at least in general). I don't think insanely fast calculation can necessarily be 'taught' any more than simple practice, and tricks such as the one here where we made the problem a bit simpler by multiplying integers first. Frankly I also don't see the utility in spending time multiplying numbers by yourself, although thats totally a personal choice if that's what you want.

Hope that is of some use.

Answer 2

My first attempt at a lightning-fast calculation of this expression was as follows:

4.3 * 0.58 + 2.0E-5 - 0.9

Of the two numbers to multiply, $4.3$ is a few percent larger than $4$ while $0.58$ is a few percent smaller than $0.6$. Round $4.3$ to $4$ and $0.58$ to $0.6$ and hope that the errors cancel well enough within the required precision of the problem. This produces

$$ 4 \times 0.6 + 2 \times 10^{-5} - 0.9 = 2.4 + 2 \times 10^{-5} - 0.9. $$

The term $2 \times 10^{-5}$ is so much smaller than the others that I ignore it, so all that remains is

$$ 2.4 - 0.9 = 1.5. $$

But that's not accurate enough. My error was less than $6\%$ of the exact result, but it was enough to change the second decimal digit after rounding. The desired answer is $1.6$.

My mistake was in counting too heavily on the approximately $3\%$ reduction from $0.6$ to $0.58$ to balance out the $7.5\%$ increase from $4$ to $4.3$. If I had been more cautious and a little more practiced at very fast estimation, I might have tried to actually estimate how much error each rounding produced and how much of the errors canceled.

By increasing $0.58$ to $0.6$ I introduced an error of $$ 0.02 \times 4.3 \approx 0.02 \times 4 \approx 0.1.$$

By decreasing $4.3$ to $4$ I introduced an error of $$ -0.3 \times 0.58 \approx -0.3 \times 0.6 \approx -0.2.$$

So altogether the error was approximately $0.1 - 0.2 = -0.1$, and to compensate for that I should have added $0.1$ to my estimate of the product, so $4.3 \times 0.58 \approx 2.5$.

This takes much longer to explain than to do. Under time pressure I might even look at the numbers, notice that I rounded one number down by a larger percentage than I rounded the other up, and make a guess that this would be enough to cause an error in the second digit of my product, so the true product would be closer to $2.5$ than to $2.4$. With practice, you may be able to recognize these patterns quickly enough.

Decades ago, when my mind was a little more nimble in this way and gasoline pumps had mechanical (not electronic) displays showing only the price per gallon and the number of gallons dispensed, I developed a practice of predicting the cost of a tank of gasoline in the few seconds between when the pump stopped and when the attendant said how much money was due. This was usually a product of a three-digit number (always ending in a $9$) with a four-digit number, with four digits of precision required in the output (since the payment was rounded to the nearest penny). The trick was to make an approximation based on round numbers, then make successive corrections by estimating the round-off errors until the remaining corrections were less than the required precision. So at least some people should be able to do these tricks.

(To be honest, I only remember doing this trick when my father was driving the car, and the main purpose may not have been to avoid being cheated by the gas station, but rather to annoy my father.)

Answer 3

This is what I will do in my head, but definitely not within $10$ seconds.

• Start from $4.3 \times 0.58$,
• multiply $4.3$ by $0.5$ is easy, you just divide it by half to get $2.15$.
• multiply $43$ by $8$ is too big in my head, so I split it out as $$4 \times 8 \to 32\quad\text{ and }\quad3 \times 8 \to 24$$
• To patch the two results together, we need to match the end of $32$ to beginning to $24$.
  This gives us $32\leftrightarrow 24 \to 344$.

• Where should the decimal point for the piece $344$?

  The $8$ in $0.58$ is at the $2^{nd}$ slot after the decimal places. Since the part corresponding to $344$ should be slightly more than the part for $8$. The $344$ really corresponds to the number $0.344$.

• Adding $2.15$ and $0.344$ directly is again too big for my head.

  • I will first add the $2^{nd}$ decimal place of $0.344$ to the $1^{st}$ number. This give us $2.194$.
  • I then add the $1^{st}$ decimal place of $0.344$ to the $1^{st}$ number. This give us $2.494$
• Subtracting the 0.9 is easy, it is just $1.594$.
• Finally, add the small piece $2\times 10^{-5}$. The $2$ should be in the $5^{th}$ decimal place.
  Since $1.594$ already have $3$, we just patch "02" to the end of $1.594$ to get $1.59402$.

If you want to be quick in mental calculation, one crucial thing is to have a good memory.

These sort of calculations requires a lot of temporary memory. If you can keep all the intermediates calculations in your memory, you just need to know how to multiply two single digits numbers (or two double digits numbers) and add all the pieces in your head.

Please note that different people has different means to extend their memory.
In the documented cases of mental calculators,

• Some relied mainly to the ears and articulation.
• Some partly on the ear, and when he visualized a number, it was not as a collection of digits, but as a concrete collection of units divisible.
• Some relied mainly on visualizing a number.

You need to experiment with yourself. You need to figure out what sort of action allow you to remember more numbers temporarily. For me, If my brain is running out of slot for a number, I will mumble the number a loud to remember it temporarily. You mileage will vary.

Answer 4

This question aroused my interest, I did a bit of research. I don't know where the original poster is from, but mental arithmetic seems like a important discipline in Singapore, so perhaps the test is assuming the candidate has had the basic training over the years at junior school. There's many tricks to learn, it takes a while. Anyway, here's my sketch of the problem

Multiplying two two-digit numbers mentally can be simplified using the cross-casting method, in our example: ( we ignore the decimal point for now and adjust later). In cross-casting we essentially multiply each pair of digits in the number and add together like we do in traditional long multiplication,

         43
       x 58
    ---------
         24    3x4
        32     8x4
        15     5x3
       20      4x5
    -------------
       2494

Doing this mentally, we would write down '4' as the right most digit of the result, carry 2 and add this to the sum of 32 and 15 = 47, write down 7 as the next (10s) digit, and carry 4, add this to the sum 20, so the final (leftmost) digits are 24, giving the intermediate result 2494. We have to adjust the decimal point, so the final result would be 2.494. With some practice, it is possible to do this process mentally with any pair of two digit numbers. To finish the expression, we need to subtract .9, giving 1.594 and then add .00002, giving 1.59402

Can I do this in 10 seconds ? No. But with a lot of practice, I guess multiplying two 2=digit numbers would become fluent and perhaps develop the neurons to do this. To answer the original poster's question, there's no magic here, its like learning a language, you can't expect to know it overnight, it takes a lot of work.