We are given two rectangles, P and Q, with the following information:

• Rectangle P has a perimeter of 20 inches.

  • One side is $h$, and the other is $j + 4$.

• Rectangle Q has a perimeter of 30 inches.

  • One side is $3h$, and the other is $j + 1$.

Perimeter Formulas:

The perimeter $P$ of a rectangle is given by: $P = 2 \times (\text{length} + \text{width})$

Setting up the equations:

1. Rectangle P's Perimeter: $20 = 2 \times (h + (j + 4))$ Simplifying: $20 = 2 \times (h + j + 4) \quad \Rightarrow \quad 20 = 2h + 2j + 8 \quad \Rightarrow \quad 12 = 2h + 2j \quad \Rightarrow \quad 6 = h + j$ This gives us the first equation: $h + j = 6 \quad \text{(Equation 1)}$

2. Rectangle Q's Perimeter: $30 = 2 \times (3h + (j + 1))$ Simplifying: $30 = 2 \times (3h + j + 1) \quad \Rightarrow \quad 30 = 6h + 2j + 2 \quad \Rightarrow \quad 28 = 6h + 2j$ This gives us the second equation: $6h + 2j = 28 \quad \text{(Equation 2)}$

Solving the system of equations:

We now have the system of two equations:

1. $h + j = 6$
2. $6h + 2j = 28$

From Equation 1, solve for $j$: $j = 6 - h$

Substitute this into Equation 2: $6h + 2(6 - h) = 28 \quad \Rightarrow \quad 6h + 12 - 2h = 28 \quad \Rightarrow \quad 4h + 12 = 28 \quad \Rightarrow \quad 4h = 16 \quad \Rightarrow \quad h = 4$

Now substitute $h = 4$ back into Equation 1: $4 + j = 6 \quad \Rightarrow \quad j = 2$

Final Answer:

The values of $j$ and $h$ are: $j = 2, \quad h = 4$

Would you like further details or have any questions?

Here are 5 follow-up questions for deeper understanding:

1. How would the solution change if the perimeter of rectangle P was different?
2. What if rectangle Q had sides proportional to a different constant than $3h$?
3. Can you solve for the area of each rectangle using the values of $j$ and $h$?
4. What other shapes besides rectangles have similar perimeter formulas?
5. How would you solve for perimeter if one of the dimensions was a variable expression?

Tip: When solving for variables in two equations, always simplify and substitute systematically to avoid errors.