Remainders

A remainder is the part left over when we divide a number by another number and it does not divide evenly.
In a fraction, we will see that the remaining part also needs to be divided equally, and this will be our remainder – exactly that equal part that is divided among everyone!

In an improper fraction where the numerator is greater than the denominator, there are $3$ ways to find the remainder:

When we have an improper fraction where the numerator is greater than the denominator, like in the fraction
$5 \over 3$
we need to understand how many times $3$ fits into $5$ completely, and what remains is the remainder.
What is the smallest number close to $5$ that is divisible by $3$ without a remainder? The answer is $3$.
$3$ divided by $3$ is $1$, so the whole number is $1$.
We can say in other words that $3$ fits $1$ time into $5$ completely, so the whole number is $1$.
Now let's move to the remainder -
If we "fit" $3$ once, we get $3$, but the numerator is $5$. Therefore, we are left with $2$.
Note – 
$5-3=2$
Therefore, the remainder is 
$2 \over 3$
Because after fitting $3$ once, $2$ out of $3$ is left, which means $2 \over 3$

Let's understand the method through an example:
$8/3$
We ask, what is the largest number closest to $8$ that is divisible by $3$ without a remainder.
The answer is $6$.
We divide $6$ by $3$ and get the whole number.
Now we subtract from $8$ the result of the multiplication of:
 The whole number we got*3
And we write the answer in the numerator with the denominator $3$.
The fraction we get is our remainder.
$6:3=2$
$2$ is the whole number.
$8-(3\cdot2)=2$
The result $2$ will be the numerator and the denominator will be $3$ as in the original exercise.
The remainder is 

$2 \over 3$

We ask how many times the denominator goes into the numerator?
This will be our whole number.
What remains will be the numerator in the fraction found in the mixed number.

Note –
Sometimes there are fractions where the numerator is larger than the denominator, but the denominator fits exactly into the numerator a whole number of times without a remainder, and therefore there is no remainder.

When the numerator is equal to the denominator - there is no remainder and the whole is $1$.
When the fraction is less than $1$ – the entire fraction is the remainder.

Click here to learn more about the remainder of a fraction

To find the remainder of a decimal fraction, proceed as follows:
Everything to the left of the decimal point is called the whole part.
Everything to the right of the decimal point is called the remainder.
In other words:

Let's see an example:
What is the remainder in the decimal:
$75.8$
The answer is remainder $8$.
It can also be written as: $0.8$

Note –
If you see the decimal number $75.08$ and are asked what the remainder is,
you need to remember that everything to the right of the decimal point is the remainder, so in the decimal number $75.08$
the remainder is $0.08$ and not $8$!!
Important notes -

• When we have a decimal number with $0$ in its whole numbers, meaning it has no whole numbers, the entire number is actually the remainder.
• When there are no digits after the decimal point, we can determine that the remainder is $0$ or that there is no remainder.

Click here to learn more about the remainder of a decimal fraction

In a mixed number composed of a whole number and a fraction -
the remainder is always the non-whole part!
This means that the remainder is always the fractional part of the mixed number.

For example –
The remainder in the fraction $5 \frac{2}{3}$
will of course be $2 \over 3$ because it is the fractional part of the mixed number.
Word problem exercise –
Gal, Shani, and Bar went to Super-Pharm to buy $3$ lipsticks – $1$ lipstick for each.
They paid together and received a total change of $20 ₪$.
$3$ The girls decided to split the change equally.
How much will each receive and what is the remainder?

Solution:
We need to understand how much each one received, so we divide $\frac{20}{6} = 6 \frac{2}{3}$
From here, we understand that each one received $6$ shekels and $2 \over 3$.
The $2 \over 3$ is the remainder that each one received.

Click here to learn more about the remainder of a mixed fraction