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I have been trying to find a solution to this problem I just encountered. So, I can split up a decimal multiplication if I had say: $2.4 * 4.3$ I could do $(2*0.3)+(2*4)+(0.4*4)+(0.4*0.3)$. This works fine. But when I try to do the same sort of thing for division it does not work at all.

So if i have $8000 / 39.8$ and I want to make it so that I can do $(8000/40)-(8000/0.2)$. It doesn’t work at all. I have been trying to come up with other solutions to do this such as $(8000/39)-(8000/0.8)$ etc, I am pretty sure I am going about this all wrong, but could maybe someone help or give me a hint in the right direction? Thanks.

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The reason the method works is because for multiplication we have $$a\times(b+c)=(a\times b)+(a\times c)\ .$$

With division you have $$(a+b)\div c=(a\div c)+(b\div c)$$ but not $$a\div(b+c)=(a\div b)+(a\div c)\ .$$

For instance, $10=100\div(5+5)\neq (100\div 5)+(100\div 5)=40$.

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Splitting up multiplication works because multiplication distributes over addition. Division does not. One way to do your example is $$\frac {8000}{39.8}=\frac {8000}{40}\cdot \frac {40}{39.8}=200(1+\frac {0.2}{39.8}) \approx 200(1+\frac {0.2}{40})=200 \cdot 105=210$$ Basically you find something nearby that is easy to divide by, then make the correction.

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