Open Middle®
Directions: Using the digits 1 to 9 at most one time each, place a digit in each box to make a product that’s as close to 4/11 as possible.
Hint
How can you tell how close your product is to 4/11? How can you tell which of two fractions is closest?
Answer
I used to think that the closest product to 4/11 was 5/7 x 1/2 (or equivalents like 1/2 x 5/7 or 5/7 x 2/4).
However Mr. Ries’ fifth graders found an even better answer. They got 6/8 x 2/4 which gives a product that is a closer to 4/11. There will be equivalent alternative answers such as 3/4 x 1/2 or 6/8 x 1/2.
Source: Robert Kaplinsky
Hi Robert,
My 5th graders from Missoula, MT wanted me to message you to let you know they found a closer answer to 4/11 than 5/14. 6/8 x 2/4 gives a product of 12/32 which is pretty close to 12/33 (an equivalent fraction to 4/11). Thank you for all of these awesome problems!! We love them.
This is fantastic. Your fifth graders are absolutely right! I’ll be sure to update the problem’s answer and give them credit. Tell them they got a closer answer than the guy who made the problem!
I think your original answer of 5/14 is actually closer to 4/11 than 3/8 and its equivalents.
4/11 – 5/14 = 1/154
3/8 – 4/11 = 1/ 88
1/88 > 1/154, so 5/14 is closer to 4/11….
Just wondering-do your students convert to decimals OR are they using other reasoning?
Hi! I actually think 1/3 is exactly equivalent!
1/3 is definitely equivalent to 4/12 but not quite equivalent to 4/11.
These problems always bring up the best discussions in class. Thank you!
Our fifth grade team thinks we found an even closer solution, which delighted everyone!
4/7 x 5/8
I think Mr. Ries’ class have it hong because I multiplied all of them and for 5/7 x 1/2 I got 19360/41648 and 19712/41648 and the difference is 352 and Mr. Ries’ class answer is 16368/41648 and 15872/41648 and the difference is 496 so I think your solution is better.
2/3 x 4/7 =8/21
8/12 is close to 8/22, which is equivalent to 4/11
The solution suggested by Mr. Ries is not the best, nor better than the original solution given.
6/8 * 2/4 (and products of fractions with an equivalent result) give 12/32 = 3/8 = 0.375 as a result, which is close to 4/11, but not closer than the original solution, 5/7 * 1/2 = 5/14.
|5/14 – 4/11| = 0.00649350…, while
|3/8 – 4/11| = 0.0113636…, which is greater than 0.00649350, meaning the distance between 3/8 and 4/11 is greater than the distance between 5/14 and 4/11.
Here are the actual solutions closest to 4/11:
1/2 * 5/7 = 0.35714285714285715 = 4/11 + -0.006493506493506496
1/7 * 5/2 = 0.35714285714285715 = 4/11 + -0.006493506493506496
2/4 * 5/7 = 0.35714285714285715 = 4/11 + -0.006493506493506496
2/7 * 5/4 = 0.35714285714285715 = 4/11 + -0.006493506493506496
3/6 * 5/7 = 0.35714285714285715 = 4/11 + -0.006493506493506496
3/7 * 5/6 = 0.35714285714285715 = 4/11 + -0.006493506493506496
4/7 * 5/8 = 0.35714285714285715 = 4/11 + -0.006493506493506496
4/8 * 5/7 = 0.35714285714285715 = 4/11 + -0.006493506493506496
5/2 * 1/7 = 0.35714285714285715 = 4/11 + -0.006493506493506496
5/4 * 2/7 = 0.35714285714285715 = 4/11 + -0.006493506493506496
5/6 * 3/7 = 0.35714285714285715 = 4/11 + -0.006493506493506496
5/7 * 1/2 = 0.35714285714285715 = 4/11 + -0.006493506493506496
5/7 * 2/4 = 0.35714285714285715 = 4/11 + -0.006493506493506496
5/7 * 3/6 = 0.35714285714285715 = 4/11 + -0.006493506493506496
5/7 * 4/8 = 0.35714285714285715 = 4/11 + -0.006493506493506496
5/8 * 4/7 = 0.35714285714285715 = 4/11 + -0.006493506493506496
They are all equal to 5/14.
Note: Three fourts of them just have the factors swapped or the numerator or denominator swapped, so really, there are only these unique solutions:
1/2 * 5/7 = 0.35714285714285715 = 4/11 + -0.006493506493506496
2/4 * 5/7 = 0.35714285714285715 = 4/11 + -0.006493506493506496
3/6 * 5/7 = 0.35714285714285715 = 4/11 + -0.006493506493506496
4/7 * 5/8 = 0.35714285714285715 = 4/11 + -0.006493506493506496
real answer: 1/2 x 4/5 = 4/10
REAL CLOSER TO 4/11
IM RIGHT