How do you convert an improper fraction like \frac{38}{3} into a mixed fraction?

The answer to convert \frac{38}{3} into a mixed number is:

\frac{38}{3} = 12\frac{2}{3}

How can you prove and check this solution?

## What Is An Improper And A Mixed Fraction?

An improper fraction is a fraction where the numerator is
**
larger
**
than the denominator. A mixed fraction is a fraction that contains a
**
whole number and a fraction
**
together.

An example of an improper fraction is:

\frac{38}{3}

The reason why this is an improper fraction is that the number on
**
top
**
of the
vinculum
(the numerator) is
*
larger
*
than the number beneath the vinculum (the denominator).

An example of a mixed fraction is:

12 \frac{2}{3}

The reason why this is a
**
mixed number
**
(or
**
mixed fraction
**
) is that the number contains a whole number
**
and
**
a fraction.

## How To Convert An Improper Fraction To A Mixed Number

To change an improper fraction into a mixed fraction, check if you can reduce the numerator and denominator of the improper fraction by the highest common factor if a common factor greater than 1 exists.

Once you have reduced the fraction, calculate how many times the denominator goes
*
wholly
*
into the numerator.

The number of times the denominator can go into the numerator is known as the
**
quotient
**
or
**
whole number
**
, and the
*
remainder
*
from that operation will be the
**
new numerator
**
.

To construct your
*
new mixed fraction
*
place the
**
whole number
**
outside to the
*
left
*
of the fraction, and the new numerator as the new fractions’ numerator, and leave the denominator as is (it will either be the reduced denominator or your original denominator).

\frac{\text{improper numerator}}{\text{denominator}} \ = \ \text{WHOLE NUMBER} \ \frac{\text{remainder}}{\text{denominator}}

Here’s how you can change the improper fraction \frac{38}{3} to a mixed fraction.

## Improper To Mixed Fraction Step-By-Step

To change an improper fraction to a mixed fraction use this simple step by step approach.

First, check if the improper fraction can be reduced further.

(Doing this simple check
*
first
*
will help reduce the number of steps required to get to the answer.)

Start by asking yourself the question:

*
Does the numerator and denominator share a common factor other than 1?
*

No, there are no common factors between 38 and 3, so no reducing is needed.

The next step requires you to perform some division:

**
How many times does the denominator go into the numerator?
**

There are two ways to answer this question: subtracting the denominator from the numerator until the numerator is
*
less than or equal to
*
the denominator OR just performing division.

### Subtraction Method

To find the answer using the subtraction method would look like this:

How many times does 3 go into 38 using subtraction?

Subtract the 3 from 38.

38 - 3 = 35

Start counting how many times you perform the subtraction of 3. This count will be the whole number of our mixed number result, and as this is the first time the whole number is now 1.

Keep repeating the subtraction by 3:

35 - 3 = 32

Whole number is now 2.

32 - 3 = 29

Whole number is now 3.

29 - 3 = 26

Whole number is now 4.

26 - 3 = 23

Whole number is now 5.

23 - 3 = 20

Whole number is now 6.

20 - 3 = 17

Whole number is now 7.

17 - 3 = 14

Whole number is now 8.

14 - 3 = 11

Whole number is now 9.

11 - 3 = 8

Whole number is now 10.

8 - 3 = 5

Whole number is now 11.

5 - 3 = 2

Whole number is now 12.

As the current result of 2 is greater than 3 you now stop subtracting as you’ve also found your new numerator.

Therefore, from this subtraction expedition, you have found that the number 3 goes into 38, 12 times with 2 remaining.

Therefore, wrapping this result into a mixed fraction would produce the result:

\frac{38}{3} \ = \ 12\frac{2}{3}

### Division Method

Here’s how the process will look using short division:

\begin{aligned} 1 \ 2 \ \ & r2 \\ 3\overline{\smash{)} \ 3 \ 8 \ } \ & \\ \end{aligned}

To perform this method start by dividing the divisor 3 by the first digit of the dividend 38 (being also 3):

\fcolorbox{red}{yellow}{3} \ \overline{\smash{\Large{)}} \ \fcolorbox{red}{yellow}{3} \ 8 \ }

How many times does 3 go into 3?

3 \div 3 = 1

Write this number above the vinculum :

\begin{aligned} \fcolorbox{red}{yellow}{1} \ \ \ & \\ 3 \overline{\smash{)} \ 3 \ 8 \ } & \end{aligned}

Next, move along to the next digit: how many times does 3 go into 8?

\begin{aligned} 1 \ \ \ \ \ \ & \\ \fcolorbox{red}{yellow}{3} \ \overline{\smash{\Large{)}} \ 3 \ \fcolorbox{red}{yellow}{8} \ } \end{aligned}

It goes twice with 2 remainder:

\begin{aligned} 1 \fcolorbox{red}{yellow}{2} \ & \fcolorbox{red}{yellow}{r2} \\ 3 \ \overline{\smash{)} \ 3 \ 8 \ \ \ } & \end{aligned}

Therefore, using this technique would mean the number above the vinculum (12) is the whole number and the remainder (2) is the new numerator.

As you can see both approaches achieve the same result, use whichever is easiest for you.

## Convert 38/3 To A Mixed Number: Summary

To change the improper fraction
^{
38
}
⁄
_{
3
}
to a mixed fraction, calculate the number of times the denominator (3) goes into the numerator (38), and from that result, place the whole number (12) to the left of the fraction, the remainder (2) as the new numerator to produce the mixed fraction:
12
^{
2
}
⁄
_{
3
}
.