0
$\begingroup$

Recently came across this technique of multiplying two $2$-digit numbers involving the same digit at the tens place and the sum of digits at units place being $10$. E.g., $73 X 77$ has same digit at the tenth place, viz., $7$ and the sum of digits at the ones place is $7+3 = 10$.

As per the technique the easy way to solve this simply multiply the digits at ones place together which in the above case would be $7. 3 = 21$ and the digit at the tenth place by the next number, i.e., $7.8=56$ in the above. The final answer then becomes after concatenating the two answers thus obtained, and hence $73.77=5621$. Similarly, $68.62=4216$ obtained by multiplying $6.7=42$ and $8.2=16$ and concatenating the two to obtain $4216$.

Can anyone explain the math behind this seemingly simple math trick?

$\endgroup$
  • 2
    $\begingroup$ (70 + 3)(70 + 7) = 70*70 + 70*3 + 70*7 + 3*7 = 70*80 + 3*7 = 7*8*100 + 3*7 $\endgroup$ – mdave16 Oct 5 '17 at 10:53
  • 1
    $\begingroup$ it's fairly easy to know if you know FOIL. $\endgroup$ – user451844 Oct 5 '17 at 10:53
  • $\begingroup$ Thanks. I hate this about mental math trick books...they never explain the math behind things and just give the method. $\endgroup$ – naveen dankal Oct 5 '17 at 10:54
2
$\begingroup$

$(10x + a)(10x +b) = 10x(10x+a+b) + ab$. Furthermore, if $a+b = 10$ , then this simplifies to $10x(10x + 10) + ab = 100x(x+1)+ab$.

That is, what we are doing, is basically the following: given say $72 \times 78$, we get $x(x+1) = 56$, and $ab = 16$, so the answer is $5616$. The reason why you can just join $56$ and $16$ is because $ab$ is always a two digit number, and $100x(x+1)$ has it's last two digits as $00$, so linear addition becomes concatenation.

By the way, this assumes nothing about $x$ at all. For example: $$ 173 \times 177 = (17 \times 18) *(3 \times 7) = (306)*(21) = 30621 $$

So $x$ could be anything. All we require is that $a+b = 10$ and all but the last digit of the numbers are the same.

A special case arises when $a=b=5$ for then $ab=25$, and essentially you are squaring the number $(x5)^2$. For example, $75^2 = (7 \times 8)*25 = 5625$.

$\endgroup$
  • $\begingroup$ I edited because in 173X177 it should have been 3X7 instead of 3X9 $\endgroup$ – naveen dankal Oct 5 '17 at 11:05
  • $\begingroup$ Yes, that is correct. thank you for the edit. By the way, I happen to know many of these techniques. The one you are learning can be generalized to the base method of multiplication, as it is called. $\endgroup$ – астон вілла олоф мэллбэрг Oct 5 '17 at 11:07
  • $\begingroup$ Can you suggest me a good book that explains the techniques too? $\endgroup$ – naveen dankal Oct 5 '17 at 11:07
  • 1
    $\begingroup$ I never read from sources. I always either looked up material online, or was told by somebody. In this case, I was told by a teacher. In the end, though, everything has an algebraic interpretation. For example, from $(x-a)^2 = x^2 -2ax + a^2$, you can make your own rule like this : for $90 \leq y \leq 100$, the square of $y$ is this : find $z = 100 - y$, then form the number $(100 - 2z) * z^2$. For example, $y = 93$ then $z = 7$ and the square is $(100-14) * 7^2 = 8649$, which is correct. So I urge you to experiment, it will bring rich rewards. $\endgroup$ – астон вілла олоф мэллбэрг Oct 5 '17 at 11:13
  • $\begingroup$ Thanks for the suggestion $\endgroup$ – naveen dankal Oct 5 '17 at 11:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.