For this math tutor, the busy exam season is winding down.  Now tutoring shifts to summer maintenance, adults in night courses, and general interest….

We all know that 6/11 is a decimal. Being a fraction of two integers, it must either repeat or terminate.

It’s actually the denominator (the bottom number) that determines whether the decimal will repeat or terminate. A fraction in lowest terms will terminate if the denominator’s factors are also factors of 10. (I’ll talk more about this in upcoming posts.) 11, of course, has only the factors 1 and 11. Since 11 is not a factor of 10, no reduced fraction with denominator 11 will terminate. Instead, each such fraction must repeat.

If you try 6/11 on a calculator (by entering 6÷11), you’ll get 0.54545454….

Entering 8/11 gives 0.72727272….

Entering 4/11 yields 0.36363636….

It seems that the repeating pair of digits always sum to 9. If you try 10/11, you’ll get 0.909090…..

Clearly, the lead digit is always one less than the numerator (the top number of the fraction). At the same time, the lead digit and the second one add to nine. Therefore, we can predict the decimal for a single digit over eleven, as follows:

Example: Predict the decimal for 2/11.

Solution: We know that the first digit of the decimal will be one less than 2, which will be 1. We know that the first digit, plus the second, will add to 9. Therefore, the second digit must be 8. Our prediction is that
2/11=0.181818…

Checking with the calculator, we see the prediction is correct:)

With this method, you needn’t wonder about the decimal for a single digit over 11. I’ll be talking more about fractions and decimals in posts this summer. Hopefully, we can look forward to some relaxing time together. Cheers.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.