Divide by 1.04 without dividing

Question

Solve in your head. What is 1 / 1.04, to four decimal places? Aim: under 15 seconds.

Accepted Answer

For numbers just above 1, use the alternating geometric series: \frac{1}{1 + x} = 1 - x + x^2 - x^3 + \dots \quad	ext{with } x = 0.04: \frac{1}{1.04} = 1 - 0.04 + 0.0016 - 0.000064 + \dots = 0.961536\ldots \approx 0.9615. Three terms (1 - 0.04 + 0.0016 = 0.9616) are already close; the fourth term pulls it to 0.961536, giving four correct decimals. True value: 0.9615384\ldots Takeaway. \frac{1}{1+x} \approx 1 - x + x^2 turns "divide by something a bit over 1" into a subtraction and a small correction. This is the move behind discounting a price by a small rate or converting a 1.04 gross-up ba…

Divide by 1.04 without dividing

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