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Please excuse me if this question is too vanilla. What's a faster way to do mental arithmetic involving powers of ten? I've always had to do this and I do it using scientific notation which I'm equivocal about, but am finding myself roaringly slow.

Here's what I do. Suppose I need to calculate in my head $1.25\%$ of $75 \text{ billion.}$ I start by converting $1.25\%$ of $75$ billion to scientific notation.
$\because 75E9 = 7.5(10^{10}) \,\, \& \,\, 1.25\% = 1.25(10^{-2}) $
$\therefore 1.25\% \text{ of } 75 \text{ billion} =1.25(10^{-2}) \times 7.5(10^{10}) = 1.25 \times 7.5 \times 10^8.$
$\because 1.25 \times 7.5 = (1 + 0.25) \times (7 + 0.5) = 7 + 0.5 + 1.75 + 0.125 = 9.375$, $\therefore 1.25 \times 7.5 \times 10^8 = 9.375E8 = 0.9375E9 = \text{ 937 million & 500 hundred thousand.}$

I'll devour Books or site/guides about calculations by hand and mental tricks?, Mental math tip needed; moving decimal around on larger and smaller numbers?, Is it possible to practice mental math too often?, & Fast arithmetic, without a calculator?.

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  • $\begingroup$ Solution: don't do mental arithmetic. Have the internet at your fingertips. $\endgroup$ – Chris Gerig Jul 15 '13 at 5:23
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    $\begingroup$ Note: 0.125=1/8, thus simply divide 75 billion by 10 and then divide 3 times by 2. $\endgroup$ – Raskolnikov Jul 15 '13 at 5:25
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As my father would do it:

Take $\,75\,$ , calculate the easy $\;\frac{75}4=18.75\;$ , add now this to $\;75\;:\;\;75+18.75=93.75\;$ , and now go to the billions:

$$1.25\%\;\;\text{of}\;\;75\;\;\text{billion is}\;\;937.5\;\;\text{million}$$

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    $\begingroup$ $1.25 \%$ of $75$ billion is $0.9375$ billion. $\endgroup$ – Alraxite Aug 8 '13 at 18:07
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My train of thought would be:

  • What is $10\%$ of 75 billion? 7.5 billion (very quick)
  • What is $10\%$ of 7.5 billion? 750 million (very quick)
  • What is $\frac{1}{4}$ = ($25\%$) of 750 [million]? 187.50 [million] (slower part)
  • What is the sum of 750 and 187.5 [million]? $937.5$ million
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