Open Middle®
Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true.
Hint
What assumptions are we making about the fractions we are able to use?
Do fractions have to be in lowest terms?
Do fractions have to be in lowest terms?
Answer
There are 100 unique solutions. Three examples are:
1: 1 3/2 + 7 6/8 = 9 5/4
2: 1 3/9 + 4 7/2 = 8 5/6
3: 1 4/2 + 5 7/3 = 9 8/6
1: 1 3/2 + 7 6/8 = 9 5/4
2: 1 3/9 + 4 7/2 = 8 5/6
3: 1 4/2 + 5 7/3 = 9 8/6
Source: Ellen Metzger
Solution found by fifth grade students: 1-5/4 + 7-2/8 = 9-3/6
Solution found by 5th grade students- 4 1/2 + 5 7/3 = 9 8/6
Note: About half of the solutions in the solutions section aren’t really unique, but just have the whole part of the mixed numbers swapped. But in that case it’s also missing 6 solutions, making it 106 solutions (or 53, really):
1 4/2 + 5 3/9 = 7 8/6, the equivalent to solution #56: 5 4/2 + 1 3/9 = 7 8/6
4 5/6 + 1 9/2 = 8 7/3, the equivalent to solution #11: 1 5/6 + 4 9/2 = 8 7/3
1 7/6 + 8 2/4 = 9 5/3, the equivalent to solution #95: 8 7/6 + 1 2/4 = 9 5/3
7 1/3 + 2 4/8 = 9 5/6, the equivalent to solution #22: 2 1/3 + 7 4/8 = 9 5/6
4 3/9 + 1 7/2 = 8 5/6, the equivalent to solution #2: 1 3/9 + 4 7/2 = 8 5/6
1 2/4 + 7 3/9 = 8 5/6, the equivalent to solution #74: 7 2/4 + 1 3/9 = 8 5/6
There is no area of formal mathematics in which:
• a mixed number is defined to allow improper fractions, or
• improper mixed numbers are a standard final form.
All official conventions require the fractional part to be proper.
I found a solution not it the 100 unique solutions:
3 1/2 + 6 5/4 = 9 7/8
3 1/2 + 6 5/4 = 3.5 + 7.25 = 10.75
9 7/8 = 9.875
Not a solution
There are 362880 possible orderings of the numbers using improper fractions in mixed numbers, with 212 of those being valid.
There are 10080 possible orderings of the numbers using only proper fractions in mixed numbers, with only 8 being valid answers.
Ignoring trivial rearrangements, There are 90720 possible orderings of the numbers using improper fractions in mixed numbers, with 53 of those being valid. (0.08%)
Ignoring trivial rearrangements, There are 2520 possible orderings of the numbers using only proper fractions in mixed numbers, with only 2 being valid answers. (0.058%)
Ignoring trivial rearrangements, The only valid answers using proper fractions in mixed numers are:
– 2 1/3 + 7 4/8 = 9 5/6
– 2 1/6 + 5 3/9 = 7 4/8
Notice to educators: This is not an appropriate homework exercise.