Seven times a two-digit positive number is equal to four times the number obtained by reversing the order of the digits. How many such two-digit numbers are there?

Let the number be 10a + b where a, b are digits, a ≥ 1. Then 7(10a + b) = 4(10b + a), giving 70a + 7b = 40b + 4a, so 66a = 33b, i.e., b = 2a. Possible: a=1,b=2 (12); a=2,b=4 (24); a=3,b=6 (36); a=4,b=8 (48). Check 12: 7×12=84, reversed=21, 4×21=84 ✓. Check 24: 7×24=168, reversed=42, 4×42=168 ✓. Check 36: 7×36=252, reversed=63, 4×63=252 ✓. Check 48: 7×48=336, reversed=84, 4×84=336 ✓. So 4 such numbers exist. Answer: 4.