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?
Age
11 to 14
Challenge level
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.
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
$$\frac{111110}{111111} < \frac{333331}{333334} < \frac{222221}{222223}$$