Divide by 0.98 without dividing

Question

Solve in your head. What is 1 / 0.98, to four decimal places? Aim: under 15 seconds.

Accepted Answer

Use the geometric series for reciprocals just below 1, where every term adds: \frac{1}{1 - x} = 1 + x + x^2 + x^3 + \dots \quad	ext{with } x = 0.02: \frac{1}{0.98} = 1 + 0.02 + 0.0004 + 0.000008 + \dots = 1.020408\ldots \approx 1.0204. Because x = 0.02 is small, the series collapses fast: after three terms you already have 1.020408, matching the true value 1.0204081\ldots to four decimals. Takeaway. \frac{1}{1-x} \approx 1 + x + x^2 converts "divide by a number just under 1" into additions. Grossing up a 2\% fee or reversing a 2\% discount is exactly this: the first-order answer is "add 2\%,…

Divide by 0.98 without dividing

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