Open Middle®
Directions: Using the integers 1 to 10 at most one time each, fill in the boxes so that the sum is equal to 1.
Hint
What three unit fractions add up to exactly 1 whole? What numbers would you use for the denominators? Why?
Answer
There are many solutions. Here are some possible solutions:
1/6 + 4/8 + 3/9
3/6 + 2/8 + 1/4
1/6 + 4/8 + 3/9
3/6 + 2/8 + 1/4
Source: Joshua Nelson
Our class came up with two different solutions:
1.) 1/4 + 2/8 + 5/10
2.) 3/6 + 1/4 + 2/8
Sorry, I forget it.is “2.)”
all 48 solutions
1/4 + 2/8 + 3/6= 1
1/4 + 2/8 + 5/10= 1
1/4 + 3/6 + 2/8= 1
1/4 + 5/10 + 2/8= 1
1/5 + 2/4 + 3/10= 1
1/5 + 3/10 + 2/4= 1
1/5 + 3/10 + 4/8= 1
1/5 + 4/8 + 3/10= 1
1/6 + 2/4 + 3/9= 1
1/6 + 3/9 + 2/4= 1
1/6 + 3/9 + 4/8= 1
1/6 + 3/9 + 5/10= 1
1/6 + 4/8 + 3/9= 1
1/6 + 5/10 + 3/9= 1
1/10 + 2/5 + 3/6= 1
1/10 + 2/5 + 4/8= 1
1/10 + 3/6 + 2/5= 1
1/10 + 4/8 + 2/5= 1
2/4 + 1/5 + 3/10= 1
2/4 + 1/6 + 3/9= 1
2/4 + 3/10 + 1/5= 1
2/5 + 1/10 + 3/6= 1
2/5 + 1/10 + 4/8= 1
2/5 + 3/6 + 1/10= 1
2/5 + 4/8 + 1/10= 1
2/8 + 1/4 + 3/6= 1
2/8 + 1/4 + 5/10= 1
2/8 + 3/6 + 1/4= 1
2/8 + 5/10 + 1/4= 1
3/6 + 1/4 + 2/8= 1
3/6 + 1/10 + 2/5= 1
3/6 + 2/5 + 1/10= 1
3/6 + 2/8 + 1/4= 1
3/9 + 1/6 + 2/4= 1
3/9 + 1/6 + 4/8= 1
3/9 + 1/6 + 5/10= 1
3/10 + 1/5 + 2/4= 1
3/10 + 1/5 + 4/8= 1
3/10 + 2/4 + 1/5= 1
3/10 + 4/8 + 1/5= 1
4/8 + 1/5 + 3/10= 1
4/8 + 1/6 + 3/9= 1
4/8 + 1/10 + 2/5= 1
4/8 + 2/5 + 1/10= 1
4/8 + 3/10 + 1/5= 1
5/10 + 1/4 + 2/8= 1
5/10 + 1/6 + 3/9= 1
5/10 + 2/8 + 1/4= 1
All 48 solutions reduce to 9 unique solutions. (ignore the order)
a/b+c/d+e/f=1
let a<c<e, done.
1/4 + 2/8 + 3/6= 1
1/4 + 2/8 + 5/10= 1
1/5 + 2/4 + 3/10= 1
1/5 + 3/10 + 4/8= 1
1/6 + 2/4 + 3/9= 1
1/6 + 3/9 + 4/8= 1
1/6 + 3/9 + 5/10= 1
1/10 + 2/5 + 3/6= 1
1/10 + 2/5 + 4/8= 1