In this question, there are numerous information's of immense important already given. Based on those given information's the answer to the question can be easily deduced.
Ratio of the perimeter of two rectangles = 4:7
Perimeter of the larger rectangle = 42 inches
Let us assume the common ratio = x
Then
7x = 42
x = 42/7
= 6
Then
Perimeter of the smaller rectangle = 4 * 6 inches
= 24 inches.
So the perimeter of the smaller rectangle is 24 inches. I hope the procedure and the answer is clear enough for you to understand. Based on this procedure, you can attempt similar problems without requiring any outside help.