One Hundred, and One

1. Take the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ in that order. Placing only plus and minus signs between them, and running neighbours together into larger numbers where you wish, make the result equal to $100$.

2. Using all ten digits $0$ through $9$, each exactly once, write an expression equal to $1$.

Solution

Part 1. A tidy answer using only three signs is $$123 - 45 - 67 + 89 = 100.$$ Reading from the left, $123 - 45 = 78$, then $78 - 67 = 11$, then $11 + 89 = 100$. There are others, for instance $1 + 2 + 3 - 4 + 5 + 6 + 78 + 9 = 100$, but the first is the neatest.

Part 2. Make two halves and add them: $$\frac{148}{296} + \frac{35}{70} = \frac{1}{2} + \frac{1}{2} = 1.$$ Each fraction is exactly one half, since $296 = 2 \times 148$ and $70 = 2 \times 35$, and between the two of them the ten digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ each appear once.