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This is a very trivial question, but I can't seem to reason out why $$30\times 50= 15\times 100$$

As a kid, I never really thought about why it works, but now I can't figure it out and the idea is really troubling me. I understand that we can break up the problem like this: $$3\times 10\times 5\times 10$$ but at this point I feel like I've lost the intuitive aspect of the problem. Can someone plz help and provide some intuition?

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    $\begingroup$ +1 for the title $\endgroup$ – clathratus Oct 9 '18 at 3:58
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    $\begingroup$ Also, is it ok to ask such questions? I sometimes feel like I am going insane (or being stupid) when I think of these problems, but I feel like I accepted these facts as a kid rather than pondering them too deeply. Ever since I took Calculus, I can't seem to accept any fact without a formal proof. $\endgroup$ – Dude156 Oct 9 '18 at 4:10
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\begin{array}{|c|c|c|c|c|} \hline 1 & 2 & 3 & 4 & 5 \\ \hline 6 & 7 & 8 & 9 & 10 \\ \hline11 & 12 & 13 & 14 & 15 \\ \hline \end{array}

You own this piece of land, you have $15$ squares in total, the size of the square is $10$ m by $10$m. Each square is $100m^2$. What is the total area? $15 \times 100$

One dimension of the land is $3\times 10$. The other dimension if $5 \times 10$.

If you have land of $a \times c$ number of rectangles land of size $b \times d $ each, size of each rectangle is $b \times d$. Total area would be $(a \times c)(b \times d)$.

One dimension of your land is $a \times b$ and the other dimension would be $c \times d.$ Hence total area is $(a \times b)(c \times d)$.

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    $\begingroup$ Now I say, Eureka! $\endgroup$ – Dude156 Oct 9 '18 at 3:51
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    $\begingroup$ Also, is it ok to ask such questions? I sometimes feel like I am going insane (or being stupid) when I think of these problems, but I feel like I accepted these facts as a kid rather than pondering them too deeply. Ever since I took Calculus, I can't seem to accept any fact without a formal proof. $\endgroup$ – Dude156 Oct 9 '18 at 4:16
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    $\begingroup$ The question is fine. mine is not a formal proof though. I was just trying to give an intuition. to come up with a proof, we need to know which set are we working on and what is a multiplication. Check the commutative and associative property. $\endgroup$ – Siong Thye Goh Oct 9 '18 at 4:27
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    $\begingroup$ @Dude156 I think that it is a sign of you thinking about things more deeply. Mathematicians aren't supposed to assume anything, and there's nothing wrong with questioning the basics that we've always just assumed. $\endgroup$ – clathratus Oct 9 '18 at 18:01
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$$30\cdot50=(3\cdot10)(5\cdot10)=(3\cdot5)(10\cdot10)=15\cdot100$$ It's all just the property that $$(ab)(cd)=(ac)(bd)$$

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  • $\begingroup$ Can you please provide a proof (or a link to the proof)? $\endgroup$ – Dude156 Oct 9 '18 at 3:57
  • $\begingroup$ $$(ab)(cd)=abcd=acbd=(ac)(bd)$$ I think it's just like reliant on an extension of the associative property for multiplication $\endgroup$ – clathratus Oct 9 '18 at 3:58
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If you're looking for an intuitive explanation, maybe consider it like a change of units. $30\cdot50$ is, say, the area of a rectangle 30 millimeters by 50 millimeters in square millimeters, but you can switch back and forth between units. So Instead you think of it as a rectangle 3 centimeters by 5 centimeters, and so the area is just 15 square centimeters, which, converting back to millimeters, is $15 \text{cm}^2 \cdot \left(\frac{10\text{mm}}{1 \text{cm}}\right)^2=1500\text{mm}$

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Intuitively why your particular problem works is because you are doubling one of the numbers and halving the other with no net change in the product. This always works for any two numbers, say $8$ and $13$, whereby $8\times 13 = 4\times 26 = 104$

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