Remainder of a fraction

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Remainder of a fraction

In a mixed number of a whole number and a fraction -
the fraction is the remainder.

In a fraction greater than 11 where the numerator is greater than the denominator -
The remainder consists of a denominator and numerator, which is the part left after finding how many whole numbers are in the fraction.

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Write the fraction as a mixed number:

\( \frac{10}{7}= \)

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Remainder of a fraction

What is a remainder?

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 44 pizza triangles among 33 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.
Given that after we gave each child one triangle, there was one spare triangle that we divide into 33 parts between each child.

How do we identify the remainder of a fraction?

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 take a look at some examples to help us better understand this concept:
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:
4254 \frac{2}{5}
We can immediately identify that there are whole numbers and a remainder of -
252\over 5

In a fraction where the numerator is larger than the denominator -
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:
929 \over 2
We need to understand how many times 22 goes into 99 as a whole number and what remains is our remainder.
What's the closest number to 99 that is divisible by 22 without a remainder? The answer is 8.
8 divided by 22 is 44, so there are 44 whole numbers.
In other words, we can say that 22 goes into 99 44 times as a whole number, so the whole number is 44.
Are we done? Not at all.
If we "put in" 44 times 22, we get 88, but the numerator is 99. Therefore, we're left with 11.
Note -
98=19-8=1
So the remainder is
12 1\over 2
because after we put in 44 times 22, we have 11 left out of 22, meaning one half.

Let's look at another example.
What is the remainder in the fraction:
434 \over 3
Let's ask, how many times does 33 go into 44 as a whole number?
The answer is 11 time
Then we have a remainder of
131 \over 3

Another example with a mathematical solution:
If it was complicated to understand the remainder concept verbally, try to understand it through a calculation exercise.
What is the remainder in the fraction -
14314 \over 3
Let's ask, what is the closest number to 1414 that is divisible by 33 without a remainder.
The answer is 1212.
Let's divide 1212 by 33 to get the whole number.
Now let's subtract from 1414 the result of multiplying:
the whole number we got 33\cdot
and write the answer in the numerator with denominator 33.
The fraction we get is our remainder.
12:3=412:3=4
44 is the whole number.
14(34)=214-(3\cdot4)=2
The result 22 will be the numerator and the denominator will be 33 like in the original exercise.
The remainder is
232 \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
848 \over 4
The numerator is indeed larger than the denominator but 44 goes into 88 twice without a remainder, so there is no remainder.
84=2\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 11.
Like for example in the fraction:
22=1\frac{2}{2} = 1

Bonus tip –
What is the remainder in a fraction less than 11, for example in the fraction
353\over5
The answer is the entire fraction, meaning,
the remainder is
353\over5
since the whole number is 00.

And now let's practice!
Write what is the remainder in each of the following numbers and explain.
3133 \frac{1}{3}

Solution:
The remainder is
131 \over 3
It can be clearly seen that there are 33 whole numbers and one-third remainder.

What is the remainder in the fraction-
636 \over 3

Solution:
No remainder. 33 goes into 66 exactly twice.
What is the remainder in the fraction:
747 \over 4

44 goes into 77 one time 11 with a remainder

343 \over 4
Therefore this is our remainder.
74=134\frac{7}{4} = 1\frac{3}{4}

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Examples with solutions for Fractions as Divisors

Exercise #1

Write the fraction as a mixed number:

107= \frac{10}{7}=

Video Solution

Step-by-Step Solution

To solve the problem, we will convert the given improper fraction 107\frac{10}{7} to a mixed number by dividing the numerator by the denominator.

  • Step 1: Divide the numerator (10) by the denominator (7). This gives a quotient and a remainder.

  • Step 2: Calculating 10÷710 \div 7 gives a quotient of 1 because 7 goes into 10 once.

  • Step 3: Multiply the quotient by the divisor (1×7=7 1 \times 7 = 7 ).

  • Step 4: Subtract the product obtained in step 3 from the original numerator to find the remainder: 107=310 - 7 = 3.

  • Step 5: Compose the mixed number using the quotient as the whole number and the remainder over the divisor as the fraction part: 37\frac{3}{7}.

Thus, the mixed number representation of 107\frac{10}{7} is 137\mathbf{1\frac{3}{7}}.

Answer

137 1\frac{3}{7}

Exercise #2

Write the fraction as a mixed number:

128= \frac{12}{8}=

Video Solution

Step-by-Step Solution

To solve this problem, we need to convert the improper fraction 128 \frac{12}{8} into a mixed number.

Here's how we'll do it:

  • The first step is to divide the numerator by the denominator: 12÷8 12 \div 8 .
  • This division gives us a quotient of 1 and a remainder of 4.
  • The quotient, 1, becomes the whole number part of our mixed number.
  • The remainder is used as the new numerator over the original denominator to form the fractional part: 48\frac{4}{8}.
  • The mixed number is thus 148 1\frac{4}{8} .
  • Finally, since 48\frac{4}{8} can be simplified, we reduce it to 12\frac{1}{2}.

Thus, the mixed number representation is correctly simplified as 112 1\frac{1}{2} .

However, when selecting from the given choices, the correct choice based on the options provided is 148 1\frac{4}{8} (Choice 4), which matches the unsimplified form.

Therefore, the solution to the problem is 148 1\frac{4}{8} .

Answer

148 1\frac{4}{8}

Exercise #3

Write the fraction as a mixed number:

139= \frac{13}{9}=

Video Solution

Step-by-Step Solution

To convert the improper fraction 139\frac{13}{9} into a mixed number, we follow these steps:

  • Step 1: Perform the division of the numerator by the denominator. Divide 13 by 9.
  • Step 2: Determine the whole number part by using the quotient of the division.
  • Step 3: Find the remainder to establish the fractional part.
  • Step 4: Write the mixed number using the whole number from Step 2 and the fractional part formed by the remainder and original denominator.

Let's carry out these steps in detail:

Divide 13 by 9:

13÷9=1 13 \div 9 = 1 with a remainder of 4 4 .

This division tells us that 9 fits into 13 a total of 1 time, with a remainder of 4.

The whole number part of our mixed number is therefore 1, and the remainder 4 forms the numerator of our fractional part over the original denominator, which is 9.

So, the fractional part is 49\frac{4}{9}.

Therefore, the improper fraction 139\frac{13}{9} as a mixed number is 149\mathbf{1\frac{4}{9}}.

Answer

149 1\frac{4}{9}

Exercise #4

Write the fraction as a mixed number:

1610= \frac{16}{10}=

Video Solution

Step-by-Step Solution

To solve the problem of converting the fraction 1610 \frac{16}{10} to a mixed number, we proceed with the following steps:

  • Step 1: Identify the numerator (16) and the denominator (10).
  • Step 2: Divide the numerator by the denominator to find the whole number part.
    Dividing 16 by 10 gives us a quotient of 1 (whole number) and a remainder of 6.
  • Step 3: Express the result as a mixed number.
    The whole number part is 1, and the remainder is the numerator of the fractional part over the original denominator. This is 610\frac{6}{10}.
  • Step 4: Write the final mixed number as: 1610 1\frac{6}{10} .

Therefore, the mixed number form of the fraction 1610 \frac{16}{10} is 1610 1\frac{6}{10} .

Answer

1610 1\frac{6}{10}

Exercise #5

Write the fraction as a mixed number:

1711= \frac{17}{11}=

Video Solution

Step-by-Step Solution

To convert the improper fraction 1711 \frac{17}{11} to a mixed number, we proceed as follows:

  • Step 1: Perform the division 17÷11 17 \div 11 . We find: - The quotient (whole number) is 1 since 11 goes into 17 once.
    - The remainder is 6 because 17(11×1)=6 17 - (11 \times 1) = 6 .

  • Step 2: Express the remainder as a fraction over the original denominator. Hence, the fractional part is 611 \frac{6}{11} .

  • Step 3: Combine the quotient and the remainder fraction to form the mixed number: 1611 1\frac{6}{11} .

Therefore, the mixed number equivalent of the fraction 1711 \frac{17}{11} is 1611 1\frac{6}{11} .

Answer

1611 1\frac{6}{11}

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