Question

1. leo’s garden, which is 6 m wide, has the same area as jen’s garden, which is 8 m wide. find the lengths of the two rectangular gardens if leo’s garden is 3 m longer than jen’s garden. first make a sketch.

Explanation:

Step1: Define variables

Let the length of Jen's garden be \( l_J \) and its width \( w_J = 6 \, \text{m} \). Let the length of Leo's garden be \( l_L \) and its width \( w_L = 8 \, \text{m} \). Since their areas are equal, \( A_J = A_L \), so \( l_J \times w_J = l_L \times w_L \). Also, \( l_L = l_J + 3 \).

Step2: Substitute values into area equation

Substitute \( w_J = 6 \), \( w_L = 8 \), and \( l_L = l_J + 3 \) into \( l_J \times 6 = l_L \times 8 \):
\( 6l_J = 8(l_J + 3) \)

Step3: Solve for \( l_J \)

Expand the right side: \( 6l_J = 8l_J + 24 \)
Subtract \( 8l_J \) from both sides: \( -2l_J = 24 \)
Divide by -2: \( l_J = -12 \)? Wait, that can't be. Wait, maybe I mixed up. Wait, the problem says "Leo’s garden, which is 6 m wide, has the same area as Jen’s garden, which is 8 m wide. Find the lengths of the two rectangular gardens if Leo’s garden is 3 m longer than Jen’s garden." Oh! I swapped the widths. Let's correct:

Let Jen's width \( w_J = 8 \, \text{m} \), length \( l_J \). Leo's width \( w_L = 6 \, \text{m} \), length \( l_L = l_J + 3 \). Area equal: \( l_J \times 8 = l_L \times 6 \).

Substitute \( l_L = l_J + 3 \):
\( 8l_J = 6(l_J + 3) \)

Expand: \( 8l_J = 6l_J + 18 \)

Subtract \( 6l_J \): \( 2l_J = 18 \)

Divide by 2: \( l_J = 9 \, \text{m} \)

Step4: Find \( l_L \)

\( l_L = l_J + 3 = 9 + 3 = 12 \, \text{m} \)

Answer:

Jen's garden length: \( 9 \, \text{m} \), Leo's garden length: \( 12 \, \text{m} \)