Remainders

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Remainder

What is a remainder:

The 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!

Remainder of a fraction

In an improper fraction where the numerator is greater than the denominator, there are 33 ways to find the remainder:

  1. The first method – Understanding approach
  2. The second method – Mathematical approach
  3. The third method – Converting an improper fraction to a mixed number

Remainder of a decimal fraction

To find the remainder of a decimal fraction, proceed as follows:
Everything that appears to the left of the decimal point is called the whole number.
Everything that appears to the right of the decimal point is called the remainder.

Remainder of a mixed number

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.

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Test yourself on part of an amount!

einstein

Determine the numerical value of the shaded area:

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Remainder

What is the remainder:

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!

Remainder of a fraction

In an improper fraction where the numerator is greater than the denominator, there are 33 ways to find the remainder:

The first method – an understanding approach

When we have an improper fraction where the numerator is greater than the denominator, like in the fraction
535 \over 3
we need to understand how many times 33 fits into 55 completely, and what remains is the remainder.
What is the smallest number close to 55 that is divisible by 33 without a remainder? The answer is 33.
33 divided by 33 is 11, so the whole number is 11.
We can say in other words that 33 fits 11 time into 55 completely, so the whole number is 11.
Now let's move to the remainder -
If we "fit" 33 once, we get 33, but the numerator is 55. Therefore, we are left with 22.
Note – 
5βˆ’3=25-3=2
Therefore, the remainder isΒ 
232 \over 3
Because after fitting 33 once, 22 out of 33 is left, which meansΒ 232 \over 3

The second way – a mathematical way

Let's understand the method through an example:
8/38/3
We ask, what is the largest number closest to 88 that is divisible by 33 without a remainder.
The answer is 66.
We divide 66 by 33 and get the whole number.
Now we subtract from 88 the result of the multiplication of:
Β The whole number we got*3
And we write the answer in the numerator with the denominator 33.
The fraction we get is our remainder.
6:3=26:3=2
22 is the whole number.
8βˆ’(3β‹…2)=28-(3\cdot2)=2
The result 22 will be the numerator and the denominator will be 33 as in the original exercise.
The remainder isΒ 

232 \over 3

The third method – converting an improper fraction to a mixed number

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 11.
When the fraction is less than 11 – the entire fraction is the remainder.

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Remainder of a decimal 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:

Mathematical concept of division showing the whole number and remainder. Visual representation to explain quotient and remainder in long division. Fundamental arithmetic concept.

Let's see an example:
What is the remainder in the decimal:
75.875.8
The answer is remainder 88.
It can also be written as: 0.80.8

Note –
If you see the decimal number 75.0875.08 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 75.0875.08
the remainder is 0.080.08 and not 88!!
Important notes -

  • When we have a decimal number with 00 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 00 or that there is no remainder.

Remainder of a mixed number

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 5235 \frac{2}{3}
will of course be 232 \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 33 lipsticks – 11 lipstick for each.
They paid together and received a total change of 20β‚ͺ20 β‚ͺ.
33 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 206=623\frac{20}{6} = 6 \frac{2}{3}
From here, we understand that each one received 66 shekels and 232 \over 3.
The 232 \over 3 is the remainder that each one received.

Do you know what the answer is?

Examples with solutions for Part of an Amount

Exercise #1

Determine the numerical value of the shaded area:

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the total number of divisions in the grid.
  • Step 2: Count the number of shaded divisions.
  • Step 3: Calculate the fraction of the shaded area relative to the total.

Now, let's work through each step:
Step 1: The grid is divided into 10 equal vertical columns.
Step 2: Of these columns, 1 column is shaded.
Step 3: Since there are 10 columns in total, the shaded area represents 110\frac{1}{10} of the total area.

Finally, the fraction 110\frac{1}{10} can be expressed as the decimal 0.10.1.

Therefore, the numerical value of the shaded area is 0.10.1.

Answer

0.1

Exercise #2

Determine the numerical value of the shaded area:

Step-by-Step Solution

To solve this problem, let's follow the outlined plan:

  • Step 1: Count the number of shaded sections.
  • Step 2: Count the total number of sections in the rectangle.
  • Step 3: Express the number of shaded sections as a fraction of the total sections.
  • Step 4: Convert this fraction to a decimal to find the numerical value.

Now, let's apply these steps:
Step 1: The given diagram shows that there are 4 vertical stripes shaded.
Step 2: The total number of vertical stripes (including both shaded and unshaded) is 10.
Step 3: The fraction of shaded area is 410\frac{4}{10}.
Step 4: Convert 410\frac{4}{10} to a decimal. This equals 0.40.4.

Therefore, the numerical value of the shaded area is 0.4.

Answer

0.4

Exercise #3

Determine the numerical value of the shaded area:

Step-by-Step Solution

To solve this problem, we'll follow a few simple steps to calculate the shaded area by counting strips and converting to a decimal:

  • Step 1: Identify the total number of vertical strips in the entire rectangle. From the diagram, there are 10 strips in total.
  • Step 2: Count the number of shaded vertical strips. According to the diagram, 5 strips are shaded.
  • Step 3: Write the fraction of the shaded area relative to the total area. The fraction is 510\frac{5}{10}.
  • Step 4: Simplify the fraction, which is already simplified, and then convert it to a decimal. 510=0.5\frac{5}{10} = 0.5.

Therefore, the solution to the problem is 0.5 0.5 .

Answer

0.5

Exercise #4

Determine the number of hundredths in the following number:

0.96

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Define the place value of each digit in the decimal number.
  • Step 2: Identify the specific digit in the hundredths place.
  • Step 3: Determine the number of hundredths in 0.96.

Now, let's work through each step:

Step 1: Consider the decimal number 0.960.96. In decimal representation, the digit immediately after the decimal point represents tenths, and the digit following that represents hundredths.

Step 2: In the number 0.960.96, the digit 99 is in the tenths place, and the digit 66 is in the hundredths place.

Step 3: Therefore, the number of hundredths in 0.960.96 is 66.

Thus, the solution to the problem is that there are 6 hundredths in the number 0.960.96.

Answer

6

Exercise #5

Determine the number of ones in the following number:

0.73

Video Solution

Step-by-Step Solution

To solve this problem, let's carefully examine the decimal number 0.73 0.73 digit by digit:

  • The first digit after the decimal point is 7 7 .
  • The second digit after the decimal point is 3 3 .

We observe that there are no digits in the sequence of 0.73 0.73 that are the number '1'. Therefore, there are no '1's in the decimal number 0.73 0.73 .

Thus, the number of ones in the number 0.73 0.73 is 0.

The correct choice, given the options, is choice id 1: 0.

Answer

0

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