The sides of a rectangle are in a 3: 4 ratio. what is the length of the shorter sides of the rectangle if it's perimeter is 42cm

Comment
The perimeter of any rectangle is  P = 2L + 2W. It is customary to make the length (L)  the longest side.  That being said, the object is to set up a proportion and substitute that into the Perimeter formula.

Givens
P = 42 cm
W / L = 3/4

Set up the two equations from the givens.
P = 2L + 2W = 42
W = 3/4 L

Step One
Substitute the second equation into the first.
2L + 2(3/4)L = 42  Make an improper fraction of second term on the left.
2L + 6/4 L = 42     Reduce the fraction 
2L + 3/2 L = 42     Multiply through by 2
4L + 3L  = 84       Combine like terms.
7L = 84                Divide by 7
L = 84/7
L = 12

Step 2
Find the Width.
3/4 = W/12          Cross multiply
4W = 36              Divide by 4
W = 36/4
W = 9

Check
P = 2L + 2W
2L + 2W = 42
2*12 + 2*9 =? 42
24 + 18 =? 42
42 = 42