Evaluate ((1/0.02272) * 4) * (500/75) * (5000/590).

Step-by-step solution

1. 1

   Plan the calculation so rounding cannot accumulate. Three of the four factors are non-terminating decimals ($1/0.02272$, $500/75$, $5000/590$). If each is rounded to four or five digits before multiplying, the small errors multiply too. The safe plan is to keep everything as exact fractions and round only the final number.

2. 2

   Turn every factor into an exact fraction. Writing $0.02272=\dfrac{2272}{100000}=\dfrac{71}{3125}$,

   $\frac{1}{0.02272}=\frac{3125}{71},\qquad \frac{500}{75}=\frac{20}{3},\qquad \frac{5000}{590}=\frac{500}{59}$

   These are the reduced forms; $71$ and $59$ are prime, so nothing further cancels against them.

3. 3

   Multiply the fractions in one go.

   $\frac{3125}{71}\cdot 4\cdot\frac{20}{3}\cdot\frac{500}{59}=\frac{3125\cdot 4\cdot 20\cdot 500}{71\cdot 3\cdot 59}=\frac{125000000}{12567}$

4. 4

   Divide once, at the end.

   $\frac{125000000}{12567}=9946.6857\ldots\approx 9946.69$

   Only this single division introduces any rounding, so the four significant figures shown are all trustworthy.

5. 5

   Compare with the rounded route to see the damage. Using $44.0141\cdot 4=176.0564$, then $6.6667$ and $8.4746$, the rounded chain gives $9946.77$ — about $0.08$ too large, a relative error of roughly $8\times 10^{-6}$ that comes entirely from truncating $5000/590=8.474576\ldots$ and $500/75=6.666\overline{6}$ too early. Small as that is, it already corrupts the sixth significant figure and shifts the two-decimal answer. The correct value is $9946.69$.