You appear to be referring to the fact that if a/b = c/d then ad = bc.
You know this is true because of the axioms of equality (If you do the same thing to both sides of an equation, then equality is maintained).
a/b = c/d
multiply by d
ad/b = c
multiply by b
ad = bc
If you wish to prove a specific proportion is true, you can test it with the equation ad = bc
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Multiply the cross products, and see if they are equal. If they are equal, the proportion is true. If they are unequal, the proportion is false.
They're equal
Proportion is the probability of a selected sample. probability is the true probability of all cases. If this is not what you are looking for then please specify.
The statement that the proportion of defective pieces is not less than 0 nor greater than 1.
A ratio is simply a comparison between 2 numbers. Example- 4:3 A proportion is 2 fractions equal to each other or "proportionate".