Too close to call
Weekly Problem 24 - 2012
Can you put these very close fractions into order?
Can you put these very close fractions into order?
Problem
Arrange these fractions in order, from smallest to largest:
$$\frac{111110}{111111} \qquad \frac{222221}{222223} \qquad \frac{333331}{333334}$$
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $\frac{111110}{111111} < \frac{333331}{333334} < \frac{222221}{222223}$
$$\frac{111110}{111111} = 1 - \frac{1}{111111} = 1 - \frac{6}{666666}$$ $$\frac{222221}{222223} = 1 - \frac{2}{222223} = 1 - \frac{6}{666669}$$ $$\frac{333331}{333334} = 1 - \frac{3}{333334} = 1 - \frac{6}{666668}$$ Now, $$\frac{6}{666669} < \frac{6}{666668} < \frac{6}{666666}$$ and so
$$\frac{111110}{111111} < \frac{333331}{333334} < \frac{222221}{222223}$$