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### Numbers Containing Whole Number, Decimal, and Fraction at Once

```Date: 10/19/2004 at 15:32:57
From: nesha
Subject: solving a problem that has both a decimal and a fraction

When dealing with a problem with a whole number, decimal, and a
fraction all in one number, what kind of formula would I use?
Example: 3.366 2/3.  The problem asks to rename the following decimals
as proper fractions in lowest terms.

I know how to find a fraction from a decimal, and I know how to find
a decimal from a fraction.  The formulas are different, so when the
problem has both I don't know what kind of formula to use.

```

```
Date: 10/19/2004 at 18:47:49
From: Doctor Ian
Subject: Re: solving a problem that has both a decimal and a fraction

Hi Nesha,

never, ever going to come across something like

3.366 2/3

in real life.  There is absolutely no point in writing a number this
way.

However, when in doubt, start from definitions.  When we write a
decimal number, like 3.366, it's really a sum:

3    6      6
3 + -- + --- + ----
10   100   1000

So what would 3.366 2/3 mean?  It would mean

3    6     6 2/3
3 + -- + --- + -------
10   100    1000

which is to say,

3    6    20/3
3 + -- + --- + ----
10   100   1000

or

3    6     20
3 + -- + --- + ----
10   100   3000

Does that make sense?  Now you need to convert everything to a common

9000    900    180     20   10100
---- + ---- + ---- + ---- = -----
3000   3000   3000   3000    3000

/   /           /   /
2 * 2         * 5 * 5       * 101
= ---------------------------------
2 * 2 * 2 * 3 * 5 * 5 * 5
/   /           /   /

101
= ---------
2 * 3 * 5

101
= ---
30

To check, divide 101 by 30:

3.3 6 6
____________
30 ) 1 0 1.0 0 0
9 0
----
1 1 0
9 0
-----
2 0 0
1 8 0
-----
2 0 0
1 8 0
-----
2 0

Now, we could keep going; or we could stop here, and think about what
to do with the numerator.

If you think back to when you were first learning division, you worked
on problems like

5
___
3 ) 17             17 divided by 3 = 5 remainder 2
15
--
2

Later on, you learned to interpret the remainder as a fraction, with
the denominator equal to the divisor:

5
___
3 ) 17             17 divided by 3 = 5 2/3
15
--
2

It's the same thing here:

3.3 6 6
____________
30 ) 1 0 1.0 0 0         101 divided by 30 = 3.366 20/30
9 0
----
1 1 0
9 0
-----
2 0 0
1 8 0
-----
2 0 0
1 8 0
-----
2 0

where 20/30 is the same as 2/3, and it means 2/3 of the last PLACE
VALUE for which you obtained part of the quotient.  In the case of 17
divided by 3, the last place is the units place, so it's

5 + (2/3 of 1)

In the case of 101 divided by 30, the last place is the thousands
place, so it's

3.366 + (20/30 of 1/1000)

which is what we would mean by

3.366 2/3

if we ever actually wrote it that way.  Which we don't.  :^D

Does this make sense?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 10/20/2004 at 10:28:51
From: nesha
Subject: Thank you (solving a problem that has both a decimal and a
fraction )

Thank you so much for the answer to my question.  My math book did not
explain in as much detail as you did.  Now I understand how they were
coming up with their answers, whereas before I was completely lost.
```
Associated Topics:
Middle School Fractions

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