Open Middle®
Directions: Using the digits 1 to 9 at most one time each, fill in the boxes to create two equivalent expressions.
Hint
Which part of the expression does the fraction affect?
Answer
There are many possibilities, here are some possibilities:
(1/2)(6x + 8) + 4x + 5 = 7x + 9
(1/2)(8x + 6) + 5x + 4 = 9x + 7
(1/2)(6x + 8) + 4x + 5 = 7x + 9
(1/2)(8x + 6) + 5x + 4 = 9x + 7
Source: Will Case
I’ve found QUITE a few possibilities for this one!
1/3(9x +6) +5x +2 = 8x + 4
1/2(6x +4) +5x + 7 = 8x + 9
2/3(9x +6) + 1x + 4 = 7x + 8
and of course, in all three of those situations, I can double the possibilities by swapping the numbers that are coefficients with the numbers that are constants, so for example this would also work:
1/3(6x + 9) + 2x + 5 = 4x + 8
1/3(9x +6) +5x +2 = 8x + 4
1/2(6x +4) +5x + 7 = 8x + 9
2/3(9x +6) + 1x + 4 = 7x + 8
this would also work:
1/3(6x + 9) + 2x + 5 = 4x + 8
Here are all the 24 solutions:
1/2 * (4x + 6) + 7x + 5 = 9x + 8
1/2 * (6x + 4) + 5x + 7 = 8x + 9
1/2 * (6x + 8) + 4x + 5 = 7x + 9
1/2 * (8x + 6) + 5x + 4 = 9x + 7
1/3 * (6x + 9) + 2x + 5 = 4x + 8
1/3 * (9x + 6) + 5x + 2 = 8x + 4
2/3 * (6x + 9) + 4x + 1 = 8x + 7
2/3 * (9x + 6) + 1x + 4 = 7x + 8
2/6 * (3x + 9) + 7x + 1 = 8x + 4
2/6 * (9x + 3) + 1x + 7 = 4x + 8
3/6 * (2x + 4) + 8x + 5 = 9x + 7
3/6 * (4x + 2) + 5x + 8 = 7x + 9
3/6 * (4x + 8) + 7x + 1 = 9x + 5
3/6 * (8x + 4) + 1x + 7 = 5x + 9
4/6 * (3x + 9) + 5x + 2 = 7x + 8
4/6 * (9x + 3) + 2x + 5 = 8x + 7
6/3 * (1x + 2) + 5x + 4 = 7x + 8
6/3 * (1x + 2) + 7x + 4 = 9x + 8
6/3 * (2x + 1) + 4x + 5 = 8x + 7
6/3 * (2x + 1) + 4x + 7 = 8x + 9
8/4 * (2x + 3) + 5x + 1 = 9x + 7
8/4 * (3x + 2) + 1x + 5 = 7x + 9
9/6 * (2x + 4) + 5x + 1 = 8x + 7
9/6 * (4x + 2) + 1x + 5 = 7x + 8
As Mr. Kit correctly said, half of them are the same as one of the others, except that the constants and coefficients are swapped.