Which two numbers (not the digits of the numbers) and two signs should be interchanged to make the given equation correct?
19−2+4÷5×4=27

Let's test Option D: Interchange numbers 4 and 2, and signs ÷ and −.
The original equation is:

19−2+4÷5×4=27
After substituting 4 with 2 (and vice versa) and − with ÷ (and vice versa):

19÷4+2−5×2=27? No, fractions.
Let's re-verify Option D carefully:
Wait, there are two '4's in the equation. Let's see which option yields a whole number.
Let's try Option B: Interchange 5 and 2, and ÷ and +:

19−5÷4+2×4=19−1.25+8

=27
Let's try Option C: Interchange 5 and 2, and ÷ and −:

19÷5+4−2×4

=27
Let's look at Option A: Interchange 4 and 2, and ÷ and +:

19−4÷2+5×2=19−2+10=27
Let's double check Option A calculation:
Replacing 2 with 4, and 4 with 2:

19−4÷2+5×4=27
Wait, if we swap 4 and 2, the equation becomes:

19−4+2÷5×2=19−4+0.8=15.8

=27
Let's re-read Option D: Interchange 4 and 2, and ÷ and −:

19÷2+4−5×2

=27
Let's check if the first 4 or second 4 is interchanged, or if Option A means:

19−4÷2+5×2=27⟹19−2+10=27
This matches perfectly if the expression becomes 19−4÷2+5×2. Let's see how that happens with Option A:
Original: 19−2+4÷5×4=27
Swap + and ÷:
19−2÷4+5×4
Swap 4 and 2:
19−4÷2+5×2=19−2+10=27.
Yes! Option A is completely correct.