Open Middle®
Directions: Using the digits 1 to 9, at most one time each, place a digit in each box 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.