If four distinct positive numbers, a, b, c and d, in the order given are in proportion, then which of the following options is NOT correct?

The correct answer is: "c, d, b and a, in the order given are in proportion." — this statement is NOT correct.

Understanding the starting condition:

We are told that four distinct positive numbers a, b, c, d (in that order) are in proportion. This means:

\frac{a}{b}=\frac{c}{d}

This is the fundamental relationship, and it also implies (by cross-multiplication):

a\times d=b\times c

We will use these two facts to check each option.

Option 1: b, a, d, c — Is this in proportion?

Check if \frac{b}{a}=\frac{d}{c}

Starting from \frac{a}{b}=\frac{c}{d}, we can take the reciprocal of both sides:

\frac{b}{a}=\frac{d}{c} ✓

This is TRUE — it is a valid proportion (invertendo property).

Option 2: c, d, b, a — Is this in proportion?

Check if \frac{c}{d}=\frac{b}{a}

We know from the original condition that \frac{c}{d}=\frac{a}{b}

So the question becomes: is \frac{a}{b}=\frac{b}{a} ?

This would require a^2=b^2, i.e., a = b. But the problem states all four numbers are distinct. Therefore, this equality does NOT hold in general.

This statement is NOT CORRECT. ✗

Option 3: a, c, b, d — Is this in proportion?

Check if \frac{a}{c}=\frac{b}{d}

Cross-multiplying: a\times d=b\times c

This is exactly the cross-product we derived from the original proportion! ✓

This is TRUE — it is a valid proportion (alternendo property).

Option 4: d, c, b, a — Is this in proportion?

Check if \frac{d}{c}=\frac{b}{a}

From Option 1, we already proved \frac{b}{a}=\frac{d}{c}

This is the same relation, just written in reverse order. ✓

This is TRUE — it is a valid proportion (inverse alternendo property).

Conclusion:
The only option that is NOT a valid proportion is Option 2: c, d, b and a. Claiming \frac{c}{d}=\frac{b}{a} would contradict the fact that a, b, c, d are all distinct, because it would force a = b.