Let the original length and width of the garden be denoted by $L$ and $W$. The original area = $L \times W$.

The length of the garden is increased by 40%, so the new length = $L \times 1.40$. The width of the garden is decreased by 20%, so the new width = $W \times 0.80$.

The new area of the garden = New length $\times$ New width = $(L \times 1.40) \times (W \times 0.80)$.

Simplifying, the new area = $1.40 \times 0.80 \times L \times W = 1.12 \times L \times W$.

The percentage increase in the area = $\frac{\text{New area - Original area}}{\text{Original area}} \times 100$.

So, the percentage increase = $\frac{(1.12 \times L \times W - L \times W)}{(L \times W)} \times 100 = 0.12 \times 100 = 12\%$.

Therefore, the area of the new garden has increased by 12%.