Math CentralQuandaries & Queries


Question from isabelle, a student:

how do you turn 6.333... into a fraction in simplest form?

We have two responses for you

Hi Isabelle.

A fraction in simplest form has a whole number on the top and another whole number on the bottom. The one on the bottom can't be a zero and in simplest form, the top and bottom don't have any common factors except 1.

Let me show you how to do this for the number 3.1111...

I need to convert 3.1111.... into a whole number divided by another whole number. I can see that 3.11111... = 3 + 0.11111.... The three is a whole number already, but how can I make the 0.1111... into a kind of a whole number divided by a whole number?

If I multiply it by 9, I get 0.999999.... which is the same as one, and that's a whole number. But I can't just multiply a thing by nine and say it is the same, can I?

What if I multiplied by 9 and then divided by 9? That would be okay, because I'm undoing the multiplication.

So 3.1111... = 3.1111... x 9 / 9.

What is 3.1111.... x 9? it is 9 x 3 + 9 x 0.1111..... and that is 27 + 1, which is 28. Now we know that 3.1111.... = 28 / 9. And that's a fraction. Nothing goes into both 28 and 9, so it is also in simplest form.

Now you know how to do these kinds of problems, Isabelle. Try it for 6.333...

Stephen La Rocque.



Let me try 4.777... I would approach it by letting y = 4.777... and subtracting x from 10y. This gives

10y - y = 47.777... - 4.777... = 43

and thus

9y = 43 and thus y = 43/9.

43 and 9 have no common factors so the fraction is in simplest form. You can check your answer by

43 ÷ 9 = 4.777...



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