Un-gross a 5% markup in your head

Question

Solve in your head. What is 1 / 1.05, to four decimal places? Aim: under 15 seconds.

Accepted Answer

A price marked up by 5\% is un-marked by dividing by 1.05. Use the alternating series: \frac{1}{1 + x} = 1 - x + x^2 - x^3 + \dots \quad	ext{with } x = 0.05: \frac{1}{1.05} = 1 - 0.05 + 0.0025 - 0.000125 + \dots Two terms give 0.95; three give 0.9525; the fourth term -0.000125 pulls it to 0.952375, which rounds to 0.9524. True value: 0.9523809\ldots Because x = 0.05 is a touch bigger, you need that fourth term to lock the last decimal, but it is still just adding and subtracting shrinking numbers. Takeaway. For a 5\% un-gross, \frac{1}{1.05} \approx 1 - 0.05 + 0.0025 = 0.9525 is good to thre…

Un-gross a 5% markup in your head

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