7.4. Do there exist such two two-digit numbers that if 20 is added to the first one and 15 is subtracted from the second one, the resulting numbers will remain two-digit, and their product will be equal to the product of the original numbers?
Answer: they exist. Hint: An example can be given $a=16, b=27$. Indeed, $(16+20) \cdot(27-15)=16 \cdot 27=432$ (The analysis of the solution and all possible pairs are considered in problem 8.4).