, 2024

We begin by determining the number of inner toothpicks used to
make a $20$ by $24$ grid of squares. A grid containing $20$ rows has $19$ horizontal lines of inner toothpicks, each of which contains $24$ toothpicks (since there are $24$ columns). Thus, the number of inner toothpicks positioned horizontally is $19\times24=456$. A grid containing $24$ columns has $23$ vertical lines of inner toothpicks, each of which contains $20$ toothpicks (since there are $20$ rows). Thus, the number of inner toothpicks positioned vertically is $23\times20=460$. In total, there are $456+460=916$ inner toothpicks. Next, we determine the total number of toothpicks used to make a $20$ by $24$ grid of squares. There are $21$ horizontal lines of toothpicks, each of which contains $24$ toothpicks. There are $25$ vertical lines of toothpicks, each of which contains $20$ toothpicks. Thus, there are a total of $21\times24+25\times20=1004$ toothpicks used to make a $20$ by $24$ grid. (Alternately, we could have determined that there are $88$ outer toothpicks, and so there are $916+88=1004$ toothpicks in total.) The percentage of inner toothpicks used is $\frac{916}{1004}\times100\%$, which is $91\%$ when rounded to the nearest
percent.