Hi, Mike! Just curious as to how you would best explain the following question:

Which of the following is the decimal equivalent to 2/7? (Note: A bar indicates a digit pattern that is repeated.)

A. 0. 285714 (all digits after decimal repeat: 0.285714285714…)

B. 0.28571428 (no repeating digits)

C. 0.28571428 (only digits 28 repeat: 0.285714282828…)

D. 0.2857142857 (no repeating digits)

E. 0.2857142857 (only digits 2857 repeat: 0.28571428572857…)Thank you!

Some calculators will make this pretty easy, but let’s assume that you don’t have one of those, or that this isn’t a question where calculators are allowed. In that case, I’d explain it with long division. Forgive the tedium of reading the steps spelled out below, it would go pretty fast doing it on paper while talking through it.

7 goes into 2 zero times, but it goes into 20 **2** times with a remainder of 6.

7 goes into 6 zero times, but it goes into 60 **8** times with a remainder of 4.

7 goes into 4 zero times, but it goes into 40 **5** times with a remainder of 5.

7 goes into 5 zero times, but it goes into 50 **7** times with a remainder of 1.

7 goes into 1 zero times, but it goes into 10 **1** time with a remainder of 3.

7 goes into 3 zero times, but it goes into 30 **4** times with a remainder of 2.

NOW HERE’S THE IMPORTANT PART: From here on out, you’re doing stuff you’ve already done!

7 goes into 2 zero times, but it goes into 20 **2** times with a remainder of 6. YOU’VE ALREADY FIGURED OUT that after 2, your next digit will be 8 because you already know that:

7 goes into 6 zero times, but it goes into 60 **8** times with a remainder of 4. Again, you already know what will happen with that remainder of 4: it’ll make your next digit 5.

7 goes into 4 zero times, but it goes into 40 **5** times with a remainder of 5. Etc.

The point is—and you can keep going as long as it takes for it to become obvious—you’re in a loop you can’t escape. All digits after the decimal will repeat.

(Screenshot from this useful site I found.)