What would be the best approach to solve 73 x 42 in my head?
I started with 70 x 40 and then 3 x 40 and combined, but at this point I forgot what I had done and ended up getting lost and not figuring it out.
Is there a good method for solving multiplication as such in my head?
On
Many mental calculators do mutliplications from left to right this way : \begin{array}{r} 73\\ \times\; 42 \end{array}
\begin{array}{c l} \text{'cross' computation} &\text{ partial result}\\ 7\cdot 4=28 & 28\\ 7\cdot 2+3\cdot 4=26 & 306\\ 3\cdot 2=6 &3066 \end{array}
Other example : \begin{array}{r} 237\\ \times\;543 \end{array}
\begin{array}{c l} \text{'cross' computation} &\text{ partial result}\\ 2\cdot 5=10 & 10\\ 2\cdot 4+3\cdot 5=23 & 123\\ 2\cdot 3+3\cdot 4 +7\cdot5=53 &1283\\ 3\cdot 3+7\cdot 4=37 & 12867\\ 7\cdot 3=21 & 128691\\ \end{array}
To compure the square of a number use $a^2=(a+b)(a-b)+b^2$.
For example $$78^2=80\cdot 76+2^2$$
Other methods and examples in this MSE thread.
On
Your method is good. But keep the idea that (x10 + y)(a10 + b) = xy*100 + (10a)y + (10x)b + by. In other words there will be four things to do.
So 73*42. 1) 70*40 + 2)70*2 + 3)3*40 + 4) 2*3
which is still hard.
but you can do short cuts to simplify. Try to get close to 50 and 25 because those factor into 100 nicely:
73*42 = 73*(50 - 8); 73*50 is 7300/2 is 3650. Put that in the back burner.
Subract 8*72. That's 4*144 = 2*288 = 560 + 16 = 576
Subtract from, what was it, oh yeah, 3650. 3650 - 576 = (3650-500) - 76 = 3150 - 76 = 3100 - 26 = 3066.
There's also:
73*42 = (75 - 2)*42; 75*42 = 150*21 = 3000+ 150 = 3150. 2*42 = 84. 3150 - 84 = 3100 - 34 = 3066.
I would double $73$ to get $146$, and then double it again to get $292$. I take advantage of the fact that doubling a number in your head is relatively easy since you only have to remember one number at a time. Then I would multiply that result by ten to get $2920$. This is $73\times 40$. Now I just need to add $2\times 73$, which we already calculated as $146$. Adding, we get $2920+146=3066$.