# converting decimals to fractions - help!

#### aaronsf

Hi everyone, I've got a very simple question whose answer I can't seem to understand. It's basic math, hopefully it's in the right forum as I didn't consider it complicated enough for the Algebra section. I've got the following problem:

$$\displaystyle \frac{0.93 - 0.90} {0.90}$$

I'm told to find the result as a decimal, to four decimal places, the convert it to a fraction. Ok. Got the first part, the answer is 0.0333. Then I convert it to a fraction. Got that, too: $$\displaystyle \frac{333} {10000}$$

The only problem is, the answer in my book says that the correct fraction is $$\displaystyle \frac{1} {30}$$. At face value $$\displaystyle \frac{333} {10000}$$ and $$\displaystyle \frac{1} {30}$$ are not equivalent fractions. And yet they both equal 0.0333! So how is $$\displaystyle \frac{333} {10000}$$ the wrong answer?

I decided to go about it another way---by using equivalent fractions in the original equation instead of decimals, i.e., (93/100 - 90/100) / (90/100); this time I did indeed come up with $$\displaystyle \frac{1} {30}$$!

But I'm still not getting why it is that way. $$\displaystyle \frac{1} {30}$$is exactly equal to 0.0333.... But when you try to then turn that 0.0333 back into a fraction, the resulting fraction is $$\displaystyle \frac{333} {10000}$$, which is NOT equal to $$\displaystyle \frac{1} {30}$$. And yet... it is equal to the decimal form of $$\displaystyle \frac{1} {30}$$.

How can both of those fractions equal 0.0333, and yet... they don't equal each other?

I assume the answer has something to do with the nature of the repeating 3 at the end of the number, but I can't figure out exactly what role it plays in it. There's got to be a practical difference. Suppose I had a piece of string that were 88 "units" long, and I wanted to cut off 1/30 of it. Well I'd then cut off 2.933333 units... but if I were to cut off 333/10000 units of string, then I'd cut off 2.9304 units of the string. So one string would indeed be longer than the other. And yet somehow they would actually be the same?

If anyone can help me understand this, I'd really appreciate it.

Thanks!

#### romsek

MHF Helper
if what you are saying is true your book is terrible.

let's take a closer look at this

$\dfrac{0.93-0.90}{0.90}= \dfrac{0.03}{0.90}$

now forget about to doing it to 4 digits for a moment. Let's calculate this precisely.

$\dfrac{0.03}{0.90} = \dfrac{\dfrac{3}{100}}{\dfrac{9}{10}} = \dfrac{3}{100}\dfrac{10}{9}=\dfrac{1}{30}$

Now, let's find the decimal representation of $\dfrac{1}{30}$ by doing the division

we find that

$\dfrac {1}{30} = 0.03333333 \dots = 0.0\bar{3}$

this is known as a repeating decimal. The $3$'s go on forever.

so, does $0.0333 = \dfrac{333}{10000} \overset{?}{=} \dfrac {1}{30} = 0.0\bar{3}$

NO! It does not.

1 person

#### aaronsf

I'm sorry but I'm not sure I understand why you're saying the book is horrible. My book is saying that the correct answers are 1/30 (for the fraction version) and 0.0333 (for the decimal version).

It seems you are confirming that 1/30 is the correct answer for the fraction.

My problem is, I was arriving at 333/10000 for the fraction.

So it seems I'm wrong; how does that make the book horrible?

#### romsek

MHF Helper
I'm sorry but I'm not sure I understand why you're saying the book is horrible. My book is saying that the correct answers are 1/30 (for the fraction version) and 0.0333 (for the decimal version).

It seems you are confirming that 1/30 is the correct answer for the fraction.

My problem is, I was arriving at 333/10000 for the fraction.

So it seems I'm wrong; how does that make the book horrible?
I took your post to mean that your book was saying that $0.0333 = \dfrac {1}{30}$ which it clearly does not

1 person

#### aaronsf

I took your post to mean that your book was saying that $0.0333 = \dfrac {1}{30}$ which it clearly does not

Ah. I see. The book doesn't say that it equals 0.0333, as I mentioned in my OP, it just asks to find the decimal version to the fourth decimal. So instead of it repeating endlessly, it just stops at 0.0333. So the book's answers are "1/30; 0.0333."

What I didn't get, was how to make sense of the fact that 333/10000 and 1/30 both equal 0.0333, even though they aren't equivalent fractions, and if 333/10000 would be an incorrect answer for the fractional part of the question.

#### romsek

MHF Helper
Ah. I see. The book doesn't say that it equals 0.0333, as I mentioned in my OP, it just asks to find the decimal version to the fourth decimal. So instead of it repeating endlessly, it just stops at 0.0333. So the book's answers are "1/30; 0.0333."

What I didn't get, was how to make sense of the fact that 333/10000 and 1/30 both equal 0.0333, even though they aren't equivalent fractions, and if 333/10000 would be an incorrect answer for the fractional part of the question.
$\dfrac 1{30} \neq 0.0333$

$\dfrac 1{30} = 0.033333333333333333333333333333333333 \dots$ with 3's going on forever

1 person