Gross up $7.00 net of a 4% rebate

Question

A quote of \7.00 is net of a 4\% rebate, so the gross amount is 7 / 0.96. Solve in your head. What is 7 / 0.96, to four decimal places? Aim: under 20 seconds.

Accepted Answer

First get the reciprocal with the geometric series, then multiply by 7. With x = 0.04: \frac{1}{0.96} = \frac{1}{1 - 0.04} = 1 + 0.04 + 0.0016 + 0.000064 + \dots \approx 1.041667. The series converges to a clean fraction, a useful check: 0.96 = 	frac{24}{25}, so \frac{1}{0.96} = 	frac{25}{24} = 1.041666\ldots Now multiply by 7: \frac{7}{0.96} = 7 	imes 1.041667 \approx 7.291667 \approx 7.2917. As an exact cross-check, \frac{7}{0.96} = \frac{175}{24} = 7.29166\ldots, confirming 7.2917. Takeaway. Un-netting an amount is "reciprocal by series, then scale." \frac{1}{1-x} \approx 1 + x + x^2 ha…

Gross up $7.00 net of a 4% rebate

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