Interpretation of the regression coefficient of a proportion type IV

#1
If I want to use some proportion type independent variables in a logistic regression, then what will be the interpretation of the regression coefficients corresponding to those proportion type variables? Will that mean- "The change in log odds for per unit change in the proportions"?

But what will be meant by "per unit change" in this case? As the proportions lie within [0,1], I am getting a little confused with what a "per unit change" will mean in this scale. Does it mean 0.01 or 1%? (I am sorry for my noob thoughts!)

In my data the range of the proportions is 0 to 1, not multiplied by 100. Do I need to multiply them by 100? So that I can say "per unit change" means 1% change? I have seen that the coefficients do differ in scale if I multiply the proportions by 100. For example, a coefficient of -1.3 for proportions becomes -0.013 for percentages (when the proportions are multiplied by 100).
 
#2
"Per unit change" for a proportion type variable should always mean change by 1 (or 100%) unit in the independent variable. Just that for my example, Y goes down by 1.3 units when X (measured in proportions) increases by 1 unit also implies that, Y goes down by 0.013 (1.3/100=-0.013) units when X (measured in proportions) increases by 0.01 (1/100=0.01) units. That makes it consistent with the sense of proportions. Thank you all.