Define the Initial Area

Let the original length of the garden be $L$ and original breadth be $B$. The initial area is $A_1 = L \times B$.

$A_1 = L \times B$

Determine New Dimensions

The length is increased by $40\%$, making the new length $L' = 1.40L$. The breadth is decreased by $20\%$, making the new breadth $B' = 0.80B$.

$L' = 1.40L, \quad B' = 0.80B$

Calculate the New Area

The new area $A_2$ is the product of the new length and new breadth: $A_2 = 1.40L \times 0.80B = 1.12(L \times B) = 1.12 A_1$.

$A_2 = 1.12 A_1$

Calculate Percentage Increase

The percentage increase in area is $\frac{A_2 - A_1}{A_1} \times 100\% = (1.12 - 1) \times 100\% = 12\%$. Alternatively, using the successive percentage formula: $40 - 20 + \frac{40 \times (-20)}{100} = 20 - 8 = 12\%$

$\text{Net} \, \text{Change} = 12\%$