Fractions – 9
THE CHALLENGE
Use the numbers 2 to 9 at most once each to fill in these boxes. There are a lot of answers that are essentially the same, so organize them with increasing numerators and denominators.
Notes
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THE CHALLENGE & EXPLORATION
Write the equation with variables to make it easier to talk about: A/B × C/D = 1. Multiplying both sides by B and D turns this into A × C = B × D.
As there is only one 5 and one 7, they cannot be put into this equation. Thinking in terms of primes helps a lot. We need to balance the number of 2’s and 3’s on both sides of this equation using the remaining six numbers: 2, 3, 4, 6, 8, and 9.
The 3’s are more limited, so there are only three possibilities.
- There are no multiples of 3. This only leaves three numbers, which is not enough.
- 3 is on one side and 6 is on the other side. We will need to use two of the three remaining numbers.
- 3 x 4 = 6 x 2; 3 x 8 = 6 x 4
- 9 is on one side and 3 and 6 are on the other side.
- 9 x 2 = 3 x 6
So, there are three solutions (plus three more if you flip the fractions):
- (3 x 4) / (2 x 6)
- (3 x 8) / (4 x 6)
- (2 x 9) / (3 x 6)