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Imagine a financial report in a certain currency, indicating the following numbers (and below them, indicating the amount after conversion to USD, from the same report).

1,394,278
(USD46,743)

225,283
(USD7,553)

111,518
(USD3,739)

62,938
(USD2,110)

38,777
(USD1,300)

As you can see, the conversion rate seems to have been off slightly each time:

  • 1,394,278/46,743 = 29.8285946559

  • 225,283/7,553 = 29.8269561764

  • 111,518/3,739 = 29.825621824

  • 62,938/2,110 = 29.828436019

  • 38,777/1,300 = 29.8284615385

However, an important detail is that the numbers are all listed in 1000s (they may thus have been rounded), meaning that they need to me multiplied by one thousand to find their real order of magnitude.

My question is: how much can the conversion rate estimate (see how we have calculated it above) differ between the estimates to be able to discover whether or not the source has actually used multiple conversion rates on the actual un-rounded numbers?

Community
  • 1
O0123
  • 143

1 Answers1

2

Assuming integer amounts and rounding to nearest thousands, the range of the exchange rate $\,x\,$ in the first case would be:

$$ 29.828\color{red}{26}\ldots \simeq \frac{1,394,278,000 - 500}{46,743,000 + 500} \lt x \lt \frac{1,394,278,000 + 500}{46,743,000 - 500} \simeq 29.828\color{red}{92}\ldots $$

[ EDIT ] The above shows how to calculate the potential range of the exchange rate in one case. The same needs to be repeated for the remaining $4$ cases, and if the $5$ resulting intervals do not all have a common point, then it can be concluded that multiple conversion rates must have been used.

[ EDIT by OP for remaining data couples ]

$$ 29.82\color{red}{49}\ldots \simeq \frac{225,283,000 - 500}{7,553,000 + 500} \lt x \lt \frac{225,283,000 + 500}{7,553,000 - 500} \simeq 29.82\color{red}{89}\ldots $$

$$ 29.82\color{red}{15}\ldots \simeq \frac{111,518,000 - 500}{3,739,000 + 500} \lt x \lt \frac{111,518,000 + 500}{3,739,000 - 500} \simeq 29.82\color{red}{97}\ldots $$

$$ 29.8\color{red}{21}\ldots \simeq \frac{62,938,000 - 500}{2,110,000 + 500} \lt x \lt \frac{62,938,000 + 500}{2,110,000 - 500} \simeq 29.8\color{red}{35}\ldots $$

$$ 29.8\color{red}{16}\ldots \simeq \frac{38,777,000 - 500}{1,300,000 + 500} \lt x \lt \frac{38,777 + 500}{1,300 - 500} \simeq 29.8\color{red}{40}\ldots $$

O0123
  • 143
dxiv
  • 76,497
  • So you're claiming that the document I got the year numbers above from actually used multiple different conversion rates in the same table? – O0123 Jun 16 '18 at 02:52
  • @VincentMiaEdieVerheyen Changed it to a CW, feel free to add the numbers and final conclusion to it. – dxiv Jun 16 '18 at 03:11
  • @VincentMiaEdieVerheyen Recheck the numbers, the second one on the LHS should rather be $,29.8215 \simeq \frac{111,518,000 - 500}{3,739,000 + 500},$. If the other numbers are correct, then all intervals include the first one, so there are common points. – dxiv Jun 16 '18 at 16:41