Directions: Using the digits 1 to 9 at most one time each, fill in the boxes to create a fraction as close to one as possible.

### Hint

When you change the denominator, what happens to the size of the parts?

Name a fractions that you know is pretty close to 1. How do you know it is close to 1?

Which is closer to 1, 2/3 or 5/6?

### Answer

79/81

Source: Peter Morris

79/81

I believe the answer is 79/81. That is just 2/81 away from 1. Where as the reciprocal (81/79) is 2/79 away from 1, which is farther.

The way I came up with the answer is knowing that I wanted the numerator and denominator to be as close to each other as possible. ideally they would be equal to each other in this the fraction will be exactly one. But that is not possible given the constraints of using each digit no more than once. I couldn’t have the same digit in the tens place, so I would probably want a 9 and a 1 in the ones places, with the tens places just being one apart from each other. So, the closest I could get the numerator and denominator to each other would be two apart (because the digit zero is also not a choice) (each pair of consecutive two digit numbers either has the same number in the tens place or has a zero in the ones place of the second number). Then for the tens places, since 9 was already being used, I picked the next two largest digits, 7, and 8. Hence my answer of 79 / 81. Why the next largest digits? This goes back to a very basic understanding of fractions, where the denominator is the total number of pieces that something is being divided into (with a larger number meaning that the SIZE of those pieces is smaller), and the numerator is the NUMBER of those pieces that you take / have / shade in. So you want the distance away from 1 to be a small number of small pieces!!

Yay, this is so much fun! I enjoy puzzles and I enjoy explaining my thinking. 🙂 It makes me pretty confident that my answer is right. But, has anyone come up with an answer closer to mine? I suppose I should still think about an answer of 76/98 (which is 12/98 away from zero – a larger number of pieces away from zero, but at least those pieces are even smaller). Or 87/91 (which is 4/91 away from zero). So the question there would be, which is smaller: 12/98 or 4/91, or my original answer of 2/81. I stick with my original answer! 🙂