Remainder of a fraction

A remainder is a part of a non-whole number.
It usually occurs when we divide one number by another and it doesn't divide evenly.
For example, if we want to divide $4$ pizza triangles among $3$ children.

How do we divide them?

Each child will get one pizza triangle and one third of a triangle.
The third of a triangle is the remainder.
Because after we gave each child one triangle, there was one spare triangle that we divide into $3$ parts between each child.

A fraction has several forms that we may encounter, and it's important to understand what is the remainder in each fraction.
In every fraction, the remainder is what's left from the whole number.
Let's understand through examples:
In a fraction in the form of a whole number and remainder (meaning a mixed number)-
It's easiest for us to identify what the remainder is.
For example, in this mixed number:
$4 \frac{2}{5}$
We can immediately identify that there are whole numbers and a remainder of -
$2\over 5$

In a fraction where the numerator is larger than the denominator -
However, in a fraction of this form, when the numerator is larger than the denominator, we cannot immediately identify the remainder. For example, in the fraction:
$9 \over 2$
We need to understand how many times $2$ goes into $9$ as a whole number and what remains is our remainder.
What's the closest number to $9$ that is divisible by $2$ without a remainder? The answer is 8.
8 divided by $2$ is $4$, so there are $4$ whole numbers.
In other words, we can say that $2$ goes into $9$ $4$ times as a whole number, so the whole number is $4$.
Are we done? Not at all.
If we "put in" $4$ times $2$, we get $8$, but the numerator is $9$. Therefore, we're left with $1$.
Note -
$9-8=1$
So the remainder is
$1\over 2$
because after we put in $4$ times $2$, we have $1$ left out of $2$, meaning one half.

Let's look at another example.
What is the remainder in the fraction:
$4 \over 3$
Let's ask, how many times does $3$ go into $4$ as a whole number?
The answer is $1$ time
And then we have a remainder of
$1 \over 3$

Another example with a mathematical solution:
If it was complicated to understand the remainder concept in words, try to understand it through a calculation exercise.
What is the remainder in the fraction -
$14 \over 3$
Let's ask, what is the closest number to $14$ that is divisible by $3$ without a remainder.
The answer is $12$.
Let's divide $12$ by $3$ to get the whole number.
Now let's subtract from $14$ the result of multiplying:
the whole number we got $3\cdot$
and write the answer in the numerator with denominator $3$.
The fraction we get is our remainder.
$12:3=4$
$4$ is the whole number.
$14-(3\cdot4)=2$
The result $2$ will be the numerator and the denominator will be $3$ like in the original exercise.
The remainder is
$2 \over 3$

Does every fraction where the numerator is greater than the denominator have a remainder?
Absolutely not!

Sometimes there are fractions where the numerator is larger than the denominator, but the denominator divides evenly into the numerator without a remainder, so there is no remainder.
Let's look at an example -
In the fraction
$8 \over 4$
The numerator is indeed larger than the denominator but $4$ goes into $8$ twice without a remainder, so there is no remainder.
$\frac{8}{4} = 2$

What happens when the numerator is equal to the denominator?
When the numerator equals the denominator there is no remainder and the whole number is $1$.
Like for example in the fraction:
$\frac{2}{2} = 1$

Bonus Note –
What is the remainder in a fraction less than $1$, for example in the fraction
$3\over5$
The answer is the entire fraction, meaning,
the remainder is
$3\over5$
since the whole number is $0$.

And now let's practice!
Write what is the remainder in each of the following numbers and explain.
$3 \frac{1}{3}$

Solution:
The remainder is
$1 \over 3$
It can be clearly seen that there are $3$ whole numbers and one-third remainder.

What is the remainder in the fraction-
$6 \over 3$

Solution:
No remainder. $3$ goes into $6$ exactly twice.
What is the remainder in the fraction:
$7 \over 4$

$4$ goes into $7$ one time $1$ with a remainder

$3 \over 4$
Therefore this is our remainder.
$\frac{7}{4} = 1\frac{3}{4}$