Open Middle®
Directions: Using the digits 0 to 9 at most one time each and choosing either multiplication/division or addition/subtraction, place a digit in each box to make a true statement.
Hint
Have you considered fractions greater than one?
Answer
There are many answers including:
6/9 ÷ 4/3 = 1/2 + 0/7
6/9 x 3/4 = 1/2 + 0/7
3/4 ÷ 6/9 = 1/2 + 5/8
3/4 x 9/6 = 1/2 + 5/8
3/4 ÷ 6/9 = 2/1 – 7/8
3/4 x 9/6 = 2/1 – 7/8
6/9 x 0/7 = 4/8 – 1/2
6/9 ÷ 4/3 = 1/2 + 0/7
6/9 x 3/4 = 1/2 + 0/7
3/4 ÷ 6/9 = 1/2 + 5/8
3/4 x 9/6 = 1/2 + 5/8
3/4 ÷ 6/9 = 2/1 – 7/8
3/4 x 9/6 = 2/1 – 7/8
6/9 x 0/7 = 4/8 – 1/2
Source: Brian Errey
There are 1,168 solutions for _/_ * _/_ = _/_ + _/_. Since you can swap addends or factors (dividing the number of solutions by 4) and since you can swap the numerators or the denominators of the factors (dividing the number of solutions again by 2) there only really are 1,168/8 = 146 unique solutions, ranging from 2/4 * 3/9 = 0/5 + 1/6 = 0.1667 (like 5 others of the 146) to 6/3 * 9/2 = 8/4 + 7/1 = 9/4 * 8/2 = 7/1 + 6/3 = 9.0 (only these two add up to 9):
1/3 * 4/2 = 0/5 + 6/9 = 0.6667
1/3 * 4/2 = 0/7 + 6/9 = 0.6667
1/3 * 4/2 = 0/8 + 6/9 = 0.6667
1/3 * 7/2 = 4/8 + 6/9 = 1.1667
1/3 * 9/2 = 0/5 + 6/4 = 1.5
1/3 * 9/2 = 0/7 + 6/4 = 1.5
1/3 * 9/2 = 0/8 + 6/4 = 1.5
1/4 * 8/3 = 0/2 + 6/9 = 0.6667
1/4 * 8/3 = 0/5 + 6/9 = 0.6667
1/4 * 8/3 = 0/7 + 6/9 = 0.6667
1/4 * 9/2 = 3/6 + 5/8 = 1.125
1/4 * 9/3 = 0/2 + 6/8 = 0.75
1/4 * 9/3 = 0/5 + 6/8 = 0.75
1/4 * 9/3 = 0/7 + 6/8 = 0.75
1/6 * 4/2 = 0/5 + 3/9 = 0.3333
1/6 * 4/2 = 0/7 + 3/9 = 0.3333
1/6 * 4/2 = 0/8 + 3/9 = 0.3333
1/6 * 4/3 = 0/5 + 2/9 = 0.2222
1/6 * 4/3 = 0/7 + 2/9 = 0.2222
1/6 * 4/3 = 0/8 + 2/9 = 0.2222
1/6 * 8/3 = 0/2 + 4/9 = 0.4444
1/6 * 8/3 = 0/5 + 4/9 = 0.4444
1/6 * 8/3 = 0/7 + 4/9 = 0.4444
1/6 * 8/4 = 0/2 + 3/9 = 0.3333
1/6 * 8/4 = 0/5 + 3/9 = 0.3333
1/6 * 8/4 = 0/7 + 3/9 = 0.3333
1/6 * 9/2 = 0/5 + 3/4 = 0.75
1/6 * 9/2 = 0/7 + 3/4 = 0.75
1/6 * 9/2 = 0/8 + 3/4 = 0.75
1/6 * 9/3 = 0/2 + 4/8 = 0.5
1/6 * 9/3 = 0/5 + 2/4 = 0.5
1/6 * 9/3 = 0/5 + 4/8 = 0.5
1/6 * 9/3 = 0/7 + 2/4 = 0.5
1/6 * 9/3 = 0/7 + 4/8 = 0.5
1/6 * 9/3 = 0/8 + 2/4 = 0.5
1/6 * 9/4 = 0/2 + 3/8 = 0.375
1/6 * 9/4 = 0/5 + 3/8 = 0.375
1/6 * 9/4 = 0/7 + 3/8 = 0.375
2/4 * 3/1 = 0/5 + 9/6 = 1.5
2/4 * 3/1 = 0/7 + 9/6 = 1.5
2/4 * 3/1 = 0/8 + 9/6 = 1.5
2/4 * 6/1 = 0/5 + 9/3 = 3.0
2/4 * 6/1 = 0/7 + 9/3 = 3.0
2/4 * 6/1 = 0/8 + 9/3 = 3.0
2/4 * 7/1 = 5/6 + 8/3 = 3.5
2/6 * 7/1 = 3/9 + 8/4 = 2.3333
2/8 * 7/1 = 3/6 + 5/4 = 1.75
2/8 * 9/1 = 3/6 + 7/4 = 2.25
2/9 * 3/1 = 0/5 + 4/6 = 0.6667
2/9 * 3/1 = 0/7 + 4/6 = 0.6667
2/9 * 3/1 = 0/8 + 4/6 = 0.6667
2/9 * 3/4 = 0/5 + 1/6 = 0.1667
2/9 * 3/4 = 0/7 + 1/6 = 0.1667
2/9 * 3/4 = 0/8 + 1/6 = 0.1667
2/9 * 6/1 = 0/5 + 4/3 = 1.3333
2/9 * 6/1 = 0/7 + 4/3 = 1.3333
2/9 * 6/1 = 0/8 + 4/3 = 1.3333
2/9 * 6/4 = 0/5 + 1/3 = 0.3333
2/9 * 6/4 = 0/7 + 1/3 = 0.3333
2/9 * 6/4 = 0/8 + 1/3 = 0.3333
3/4 * 6/1 = 0/5 + 9/2 = 4.5
3/4 * 6/1 = 0/7 + 9/2 = 4.5
3/4 * 6/1 = 0/8 + 9/2 = 4.5
3/4 * 7/1 = 6/8 + 9/2 = 5.25
3/6 * 5/1 = 2/8 + 9/4 = 2.5
3/6 * 9/1 = 8/4 + 5/2 = 4.5
3/6 * 9/4 = 1/2 + 5/8 = 1.125
3/8 * 4/1 = 0/2 + 9/6 = 1.5
3/8 * 4/1 = 0/5 + 9/6 = 1.5
3/8 * 4/1 = 0/7 + 9/6 = 1.5
3/8 * 6/1 = 0/2 + 9/4 = 2.25
3/8 * 6/1 = 0/5 + 9/4 = 2.25
3/8 * 6/1 = 0/7 + 9/4 = 2.25
3/9 * 4/1 = 0/2 + 8/6 = 1.3333
3/9 * 4/1 = 0/5 + 8/6 = 1.3333
3/9 * 4/1 = 0/7 + 8/6 = 1.3333
3/9 * 4/8 = 0/2 + 1/6 = 0.1667
3/9 * 4/8 = 0/5 + 1/6 = 0.1667
3/9 * 4/8 = 0/7 + 1/6 = 0.1667
3/9 * 5/1 = 2/4 + 7/6 = 1.6667
3/9 * 5/1 = 4/8 + 7/6 = 1.6667
3/9 * 5/4 = 1/6 + 2/8 = 0.4167
3/9 * 6/1 = 0/2 + 8/4 = 2.0
3/9 * 6/1 = 0/5 + 4/2 = 2.0
3/9 * 6/1 = 0/5 + 8/4 = 2.0
3/9 * 6/1 = 0/7 + 4/2 = 2.0
3/9 * 6/1 = 0/7 + 8/4 = 2.0
3/9 * 6/1 = 0/8 + 4/2 = 2.0
3/9 * 6/1 = 2/8 + 7/4 = 2.0
3/9 * 6/4 = 0/5 + 1/2 = 0.5
3/9 * 6/4 = 0/7 + 1/2 = 0.5
3/9 * 6/4 = 0/8 + 1/2 = 0.5
3/9 * 6/8 = 0/2 + 1/4 = 0.25
3/9 * 6/8 = 0/5 + 1/4 = 0.25
3/9 * 6/8 = 0/7 + 1/4 = 0.25
3/9 * 7/1 = 2/6 + 8/4 = 2.3333
4/3 * 9/2 = 0/5 + 6/1 = 6.0
4/3 * 9/2 = 0/7 + 6/1 = 6.0
4/3 * 9/2 = 0/8 + 6/1 = 6.0
4/6 * 9/2 = 0/5 + 3/1 = 3.0
4/6 * 9/2 = 0/7 + 3/1 = 3.0
4/6 * 9/2 = 0/8 + 3/1 = 3.0
4/6 * 9/3 = 0/5 + 2/1 = 2.0
4/6 * 9/3 = 0/7 + 2/1 = 2.0
4/6 * 9/3 = 0/8 + 2/1 = 2.0
4/8 * 6/1 = 0/2 + 9/3 = 3.0
4/8 * 6/1 = 0/5 + 9/3 = 3.0
4/8 * 6/1 = 0/7 + 9/3 = 3.0
4/8 * 7/3 = 1/2 + 6/9 = 1.1667
4/8 * 9/1 = 6/3 + 5/2 = 4.5
4/9 * 6/1 = 0/2 + 8/3 = 2.6667
4/9 * 6/1 = 0/5 + 8/3 = 2.6667
4/9 * 6/1 = 0/7 + 8/3 = 2.6667
4/9 * 6/8 = 0/2 + 1/3 = 0.3333
4/9 * 6/8 = 0/5 + 1/3 = 0.3333
4/9 * 6/8 = 0/7 + 1/3 = 0.3333
5/3 * 9/2 = 4/8 + 7/1 = 7.5
5/6 * 7/1 = 4/3 + 9/2 = 5.8333
5/6 * 7/2 = 1/4 + 8/3 = 2.9167
5/6 * 7/4 = 1/3 + 9/8 = 1.4583
5/6 * 9/3 = 1/2 + 8/4 = 2.5
5/6 * 9/3 = 4/8 + 2/1 = 2.5
5/8 * 6/1 = 3/2 + 9/4 = 3.75
6/3 * 7/2 = 8/4 + 5/1 = 7.0
6/3 * 9/2 = 8/4 + 7/1 = 9.0
6/8 * 7/1 = 3/4 + 9/2 = 5.25
6/8 * 7/3 = 1/2 + 5/4 = 1.75
6/8 * 9/3 = 1/2 + 7/4 = 2.25
6/9 * 7/1 = 4/2 + 8/3 = 4.6667
6/9 * 7/2 = 1/3 + 8/4 = 2.3333
7/4 * 8/2 = 6/3 + 5/1 = 7.0
7/4 * 8/3 = 1/6 + 9/2 = 4.6667
7/4 * 9/3 = 2/8 + 5/1 = 5.25
7/6 * 8/4 = 3/9 + 2/1 = 2.3333
7/6 * 9/4 = 1/8 + 5/2 = 2.625
7/6 * 9/4 = 5/8 + 2/1 = 2.625
8/4 * 9/2 = 6/3 + 7/1 = 9.0
8/4 * 9/3 = 0/2 + 6/1 = 6.0
8/4 * 9/3 = 0/5 + 6/1 = 6.0
8/4 * 9/3 = 0/7 + 6/1 = 6.0
8/6 * 9/3 = 0/2 + 4/1 = 4.0
8/6 * 9/3 = 0/5 + 4/1 = 4.0
8/6 * 9/3 = 0/7 + 4/1 = 4.0
8/6 * 9/4 = 0/2 + 3/1 = 3.0
8/6 * 9/4 = 0/5 + 3/1 = 3.0
8/6 * 9/4 = 0/7 + 3/1 = 3.0
The solutions for _/_ ÷ _/_ = _/_ + _/_ are the same: Just swap the numerator and denominator of the second fraction and you’ll have an equation equivalent to the first. So you’ll get 1,168 solutions again, with 8 of them being equivalent, for example
9.0 = (6/2 ÷ 3/9 = 8/4 + 7/1) = (6/3 ÷ 2/9 = 8/4 + 7/1) = (9/2 ÷ 3/6 = 8/4 + 7/1) = (9/3 ÷ 2/6 = 8/4 + 7/1) (plus another 4 solutions with swapped addends), making for 1,168/8 = 146 unique solutions with the same results as before.
As for _/_ * _/_ = _/_ – _/_ (note that since the left side is non-negative, the right side has to be too, making the third fraction > fourth fraction), there are 1,398 solutions, ranging from 2/9 * 3/4 = 1/6 – 0/5 = 0.1667 (and 5 others) to 6/4 * 7/3 = 8/1 – 9/2 = 12.5 (unique).
Like before, _/_ ÷ a/b = _/_ – _/_ is equivalent to _/_ * b/a = _/_ – _/_, which means it has the same 1,398 solutions as above, just with the numerator and denominator in the second fraction swapped.
Unfortunately, it seems I can’t post a list of all the solutions due to some limit on this website. Tried posting them in chunks, but my comments don’t show up.
There are 1,168 solutions for _/_ * _/_ = _/_ + _/_. Since you can swap addends or factors (dividing the number of solutions by 4) and since you can swap the numerators or the denominators of the factors (dividing the number of solutions again by 2) there only really are 1,168/8 = 146 unique solutions, ranging from 2/4 * 3/9 = 0/5 + 1/6 = 0.1667 (like 5 others of the 146) to 6/3 * 9/2 = 8/4 + 7/1 = 9/4 * 8/2 = 7/1 + 6/3 = 9.0 (only these two add up to 9):
1/3 * 4/2 = 0/5 + 6/9 = 0.6667
1/3 * 4/2 = 0/7 + 6/9 = 0.6667
1/3 * 4/2 = 0/8 + 6/9 = 0.6667
1/3 * 7/2 = 4/8 + 6/9 = 1.1667
1/3 * 9/2 = 0/5 + 6/4 = 1.5
1/3 * 9/2 = 0/7 + 6/4 = 1.5
1/3 * 9/2 = 0/8 + 6/4 = 1.5
1/4 * 8/3 = 0/2 + 6/9 = 0.6667
1/4 * 8/3 = 0/5 + 6/9 = 0.6667
1/4 * 8/3 = 0/7 + 6/9 = 0.6667
1/4 * 9/2 = 3/6 + 5/8 = 1.125
1/4 * 9/3 = 0/2 + 6/8 = 0.75
1/4 * 9/3 = 0/5 + 6/8 = 0.75
1/4 * 9/3 = 0/7 + 6/8 = 0.75
1/6 * 4/2 = 0/5 + 3/9 = 0.3333
1/6 * 4/2 = 0/7 + 3/9 = 0.3333
1/6 * 4/2 = 0/8 + 3/9 = 0.3333
1/6 * 4/3 = 0/5 + 2/9 = 0.2222
1/6 * 4/3 = 0/7 + 2/9 = 0.2222
1/6 * 4/3 = 0/8 + 2/9 = 0.2222
1/6 * 8/3 = 0/2 + 4/9 = 0.4444
1/6 * 8/3 = 0/5 + 4/9 = 0.4444
1/6 * 8/3 = 0/7 + 4/9 = 0.4444
1/6 * 8/4 = 0/2 + 3/9 = 0.3333
1/6 * 8/4 = 0/5 + 3/9 = 0.3333
1/6 * 8/4 = 0/7 + 3/9 = 0.3333
1/6 * 9/2 = 0/5 + 3/4 = 0.75
1/6 * 9/2 = 0/7 + 3/4 = 0.75
1/6 * 9/2 = 0/8 + 3/4 = 0.75
1/6 * 9/3 = 0/2 + 4/8 = 0.5
1/6 * 9/3 = 0/5 + 2/4 = 0.5
1/6 * 9/3 = 0/5 + 4/8 = 0.5
1/6 * 9/3 = 0/7 + 2/4 = 0.5
1/6 * 9/3 = 0/7 + 4/8 = 0.5
1/6 * 9/3 = 0/8 + 2/4 = 0.5
1/6 * 9/4 = 0/2 + 3/8 = 0.375
1/6 * 9/4 = 0/5 + 3/8 = 0.375
1/6 * 9/4 = 0/7 + 3/8 = 0.375
2/4 * 3/1 = 0/5 + 9/6 = 1.5
2/4 * 3/1 = 0/7 + 9/6 = 1.5
2/4 * 3/1 = 0/8 + 9/6 = 1.5
2/4 * 6/1 = 0/5 + 9/3 = 3.0
2/4 * 6/1 = 0/7 + 9/3 = 3.0
2/4 * 6/1 = 0/8 + 9/3 = 3.0
2/4 * 7/1 = 5/6 + 8/3 = 3.5
2/6 * 7/1 = 3/9 + 8/4 = 2.3333
2/8 * 7/1 = 3/6 + 5/4 = 1.75
2/8 * 9/1 = 3/6 + 7/4 = 2.25
2/9 * 3/1 = 0/5 + 4/6 = 0.6667
2/9 * 3/1 = 0/7 + 4/6 = 0.6667
2/9 * 3/1 = 0/8 + 4/6 = 0.6667
2/9 * 3/4 = 0/5 + 1/6 = 0.1667
2/9 * 3/4 = 0/7 + 1/6 = 0.1667
2/9 * 3/4 = 0/8 + 1/6 = 0.1667
2/9 * 6/1 = 0/5 + 4/3 = 1.3333
2/9 * 6/1 = 0/7 + 4/3 = 1.3333
2/9 * 6/1 = 0/8 + 4/3 = 1.3333
2/9 * 6/4 = 0/5 + 1/3 = 0.3333
2/9 * 6/4 = 0/7 + 1/3 = 0.3333
2/9 * 6/4 = 0/8 + 1/3 = 0.3333
3/4 * 6/1 = 0/5 + 9/2 = 4.5
3/4 * 6/1 = 0/7 + 9/2 = 4.5
3/4 * 6/1 = 0/8 + 9/2 = 4.5
3/4 * 7/1 = 6/8 + 9/2 = 5.25
3/6 * 5/1 = 2/8 + 9/4 = 2.5
3/6 * 9/1 = 8/4 + 5/2 = 4.5
3/6 * 9/4 = 1/2 + 5/8 = 1.125
3/8 * 4/1 = 0/2 + 9/6 = 1.5
3/8 * 4/1 = 0/5 + 9/6 = 1.5
3/8 * 4/1 = 0/7 + 9/6 = 1.5
3/8 * 6/1 = 0/2 + 9/4 = 2.25
3/8 * 6/1 = 0/5 + 9/4 = 2.25
3/8 * 6/1 = 0/7 + 9/4 = 2.25
3/9 * 4/1 = 0/2 + 8/6 = 1.3333
3/9 * 4/1 = 0/5 + 8/6 = 1.3333
3/9 * 4/1 = 0/7 + 8/6 = 1.3333
3/9 * 4/8 = 0/2 + 1/6 = 0.1667
3/9 * 4/8 = 0/5 + 1/6 = 0.1667
3/9 * 4/8 = 0/7 + 1/6 = 0.1667
3/9 * 5/1 = 2/4 + 7/6 = 1.6667
3/9 * 5/1 = 4/8 + 7/6 = 1.6667
3/9 * 5/4 = 1/6 + 2/8 = 0.4167
3/9 * 6/1 = 0/2 + 8/4 = 2.0
3/9 * 6/1 = 0/5 + 4/2 = 2.0
3/9 * 6/1 = 0/5 + 8/4 = 2.0
3/9 * 6/1 = 0/7 + 4/2 = 2.0
3/9 * 6/1 = 0/7 + 8/4 = 2.0
3/9 * 6/1 = 0/8 + 4/2 = 2.0
3/9 * 6/1 = 2/8 + 7/4 = 2.0
3/9 * 6/4 = 0/5 + 1/2 = 0.5
3/9 * 6/4 = 0/7 + 1/2 = 0.5
3/9 * 6/4 = 0/8 + 1/2 = 0.5
3/9 * 6/8 = 0/2 + 1/4 = 0.25
3/9 * 6/8 = 0/5 + 1/4 = 0.25
3/9 * 6/8 = 0/7 + 1/4 = 0.25
3/9 * 7/1 = 2/6 + 8/4 = 2.3333
4/3 * 9/2 = 0/5 + 6/1 = 6.0
4/3 * 9/2 = 0/7 + 6/1 = 6.0
4/3 * 9/2 = 0/8 + 6/1 = 6.0
4/6 * 9/2 = 0/5 + 3/1 = 3.0
4/6 * 9/2 = 0/7 + 3/1 = 3.0
4/6 * 9/2 = 0/8 + 3/1 = 3.0
4/6 * 9/3 = 0/5 + 2/1 = 2.0
4/6 * 9/3 = 0/7 + 2/1 = 2.0
4/6 * 9/3 = 0/8 + 2/1 = 2.0
4/8 * 6/1 = 0/2 + 9/3 = 3.0
4/8 * 6/1 = 0/5 + 9/3 = 3.0
4/8 * 6/1 = 0/7 + 9/3 = 3.0
4/8 * 7/3 = 1/2 + 6/9 = 1.1667
4/8 * 9/1 = 6/3 + 5/2 = 4.5
4/9 * 6/1 = 0/2 + 8/3 = 2.6667
4/9 * 6/1 = 0/5 + 8/3 = 2.6667
4/9 * 6/1 = 0/7 + 8/3 = 2.6667
4/9 * 6/8 = 0/2 + 1/3 = 0.3333
4/9 * 6/8 = 0/5 + 1/3 = 0.3333
4/9 * 6/8 = 0/7 + 1/3 = 0.3333
5/3 * 9/2 = 4/8 + 7/1 = 7.5
5/6 * 7/1 = 4/3 + 9/2 = 5.8333
5/6 * 7/2 = 1/4 + 8/3 = 2.9167
5/6 * 7/4 = 1/3 + 9/8 = 1.4583
5/6 * 9/3 = 1/2 + 8/4 = 2.5
5/6 * 9/3 = 4/8 + 2/1 = 2.5
5/8 * 6/1 = 3/2 + 9/4 = 3.75
6/3 * 7/2 = 8/4 + 5/1 = 7.0
6/3 * 9/2 = 8/4 + 7/1 = 9.0
6/8 * 7/1 = 3/4 + 9/2 = 5.25
6/8 * 7/3 = 1/2 + 5/4 = 1.75
6/8 * 9/3 = 1/2 + 7/4 = 2.25
6/9 * 7/1 = 4/2 + 8/3 = 4.6667
6/9 * 7/2 = 1/3 + 8/4 = 2.3333
7/4 * 8/2 = 6/3 + 5/1 = 7.0
7/4 * 8/3 = 1/6 + 9/2 = 4.6667
7/4 * 9/3 = 2/8 + 5/1 = 5.25
7/6 * 8/4 = 3/9 + 2/1 = 2.3333
7/6 * 9/4 = 1/8 + 5/2 = 2.625
7/6 * 9/4 = 5/8 + 2/1 = 2.625
8/4 * 9/2 = 6/3 + 7/1 = 9.0
8/4 * 9/3 = 0/2 + 6/1 = 6.0
8/4 * 9/3 = 0/5 + 6/1 = 6.0
8/4 * 9/3 = 0/7 + 6/1 = 6.0
8/6 * 9/3 = 0/2 + 4/1 = 4.0
8/6 * 9/3 = 0/5 + 4/1 = 4.0
8/6 * 9/3 = 0/7 + 4/1 = 4.0
8/6 * 9/4 = 0/2 + 3/1 = 3.0
8/6 * 9/4 = 0/5 + 3/1 = 3.0
8/6 * 9/4 = 0/7 + 3/1 = 3.0
7/6 * 9/4 = 1/8 + 5/2 = 2.625
As for _/_ * _/_ = _/_ – _/_ (note that since the left side is non-negative, the right side has to be too, making the third fraction > fourth fraction), there are 1,398 solutions, ranging from 2/9 * 3/4 = 1/6 – 0/5 = 0.1667 (and 5 others) to 6/4 * 7/3 = 8/1 – 9/2 = 12.5 (unique).
Like before, _/_ ÷ a/b = _/_ – _/_ is equivalent to _/_ * b/a = _/_ – _/_, which means it has the same 1,398 solutions as above, just with the numerator and denominator in the second fraction swapped.
Unfortunately, it seems I can’t post a list of all the solutions due to some limit on this website. Tried posting them in chunks, but my comments don’t show up.