The perimeter of the rectangle is 42 cm, the ratio of its length to width is 4: 3. find the area of the rectangle.

Let’s denote the length of the rectangle by the letter a, and the width by the letter b.
By the condition of the problem:
a: b = 4: 3,
3 * a = 4 * b,
a = 4/3 * b,
The perimeter of the rectangle is P = 2 * a + 2 * b. Substitute the expression a into this formula:
42 = 2 * 4/3 * b + 2 * b,
42 = 8/3 * b + 6/3 * b,
42 * 3 = 14 * b,
b = 9 (cm).
Then a = 4/3 * 9 = 12 (cm).
If a = 12, b = 9, then the area of the rectangle will be:
S = a * b = 9 * 12 = 108 (cm 2).
Answer: S = 108 (cm 2).