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How we arrived for for following:

$$\frac{441.66}{958.33}=\frac{441\cdot 3 +2}{958\cdot 3 +1}$$

I understand that $0.66$ is $\frac{2}{3}$ but want to know how we arrived at above relation.

Michael Rybkin
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Manuj Pandey
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    And $958.\overline 3=958+\frac13$. –  Aug 18 '19 at 08:46
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    $0.66\ne\frac{2}{3}.$ That's wrong. $\frac{2}{3}=0.\bar 6$. In fact, you don't have an equality there because there is no way to make the left-hand side look like the right-hand side. – Michael Rybkin Aug 18 '19 at 08:58

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The $2$ parts in the original "equation" are not actually equal to each other. Instead, assuming the "=" was meant to be "$\approx$", you get

$$\frac{441.66}{958.33} \approx \frac{441 + \frac{2}{3}}{958 + \frac{1}{3}} = \frac{441 \times 3 + 2}{958 \times 3 + 1} \tag{1}\label{eq1}$$

The approximation's numerator and denominator was multiplied by $3$, with $\frac{3}{3} = 1$, so the overall value doesn't change.

John Omielan
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  • It's not a good idea to propagate the OP's error into an answer without any mention of such (esp. since it is not clear if it is a notational or conceptual error). – Bill Dubuque Aug 18 '19 at 15:36
  • @BillDubuque Thanks for the feedback. By using the approximation symbol in the answer's equations and explicitly stating in the next line that it's involving an approximation was, I thought, sufficient to show it's only an approximation. However, I have made this more definitive by stating this issue of the OP should have not used an equals sign but, instead, an approximation symbol, at the very beginning of the answer. – John Omielan Aug 18 '19 at 15:41
  • Yes, but I think one should be much more explicit about such in an answer (improved in your edit) – Bill Dubuque Aug 18 '19 at 15:45
  • My bad for not being explicit in mentioning the correct equation. My doubt was regarding how we could have arrived at this approximation. – Manuj Pandey Aug 20 '19 at 05:48