Find an upper-bound for the denominator of a given expression

Question

I have the following variables:

• USD
• EUR
• CNY
• JPY
• GBP

Each one of them is non-negative with at most 2 digits after the decimal point.

I need to compute the maximum possible denominator in the rational representation of:

USD * 0.4173 + EUR * 0.3093 + CNY * 0.1092 + JPY * 0.0833 + GBP * 0.0809

I am inclined to say that the answer is 1000000, because I can:

1. Replace each variable x with the integer x * 100
2. Replace each constant X with the integer X * 10000
3. Use 100 * 10000 as the denominator

In other words, the expression above is equal to:

$\frac{USD\times100\times4173+EUR\times100\times3093+CNY\times100\times1092+JPY\times100\times833+GBP\times100\times809}{100\times10000}$

Is this reasoning correct?

P.S.: I am not asked to state the lowest possible maximum value, but an upper bound (so technically, the answer holds even if there's a lower answer which holds).

Thank you!

Answer 1

Your reasoning is indeed correct! With integer denominator and all the values at the numerator scaled by the appropriate factor it's like multiplying every addend by 1, so indeed the value of the sum doesn't change when rescaled by the same factor for every addend

Answer 2

The reasoning is correct, moreover...

I am not asked to state the lowest possible maximum value, but an upper bound

$100 \cdot 10000\,$ is the lowest/best upper bound in general. It is attained e.g. for $USD$ $= EUR$ $= CNY$ $= JPY = \color{red}{1}$ and $GBP = \color{red}{2}\,$, in which case the last fraction on the following line is irreducible:

$$\color{red}{1 \,\cdot\,} 0.4173 \,+\, \color{red}{\,1\, \cdot} 0.3093 \,+\, \color{red}{\,1\, \cdot} 0.1092 \,+\, \color{red}{1 \,\cdot\,} 0.0833 \,+\, \color{red}{2 \,\cdot\,} 0.0809 \,=\, 0.010809 \,=\, \dfrac{10809}{1000000}$$