How to extract a whole number from a fraction. What is a number fraction

How to separate the whole part from an improper fraction?

You highlight how many times the denominator fits in the numerator, then subtract the denominator from the numerator, the denominator remains unchanged.
try to calculate it on a calculator))
Divide the numeral by the denominator and write the number to the left of the decimal point.
if you need to select the fractional part:
You multiply the selected integer part by the denominator and subtract the resulting number from the numerator. That is:
79/3
1. select the whole part: 26
2. multiply the selected integer part by the denominator: 26*3
3. subtract the resulting number from the numerator 79-(26*3)
Select the whole part from improper fractions and arrange the resulting mixed numbers in descending order: 13/5, 53/10, 52/9, 23/5, 3/2, 49/2, 35/9, 35/11, 12/5 , 31/9, 5/4, 33/5, 31/7, 7/4, 35/8, 51/8, 6/5, 57/10. Given letters B, A, A, B, L, V, K, R, I, E, E, S, A, L, S, O, J, K. Decipher the name English writer late 19th century the beginning of the 20th century and the name of one of his works (a: 5+5+5; b; 6+12)
Source: mathematics
divide the numerator by the denominator, the number before the decimal point is the whole part, then multiply the whole part by the denominator and subtract it from the original numerator. This figure will be the numerator.
for example: 88/16=5.5
16*5=80
88-80=8
5 8/16=5 1/2
thank you everyone
Divide the numerator by the denominator and write the resulting number as an integer and the remainder as the numerator and the denominator remains the same.
It seems correct about 3/2. You just need to divide the numerator by the denominator with the remainder. Then the quotient is the whole part, the remainder is the numerator, and the divisor is the denominator (i.e., it remains as it was). For example
48/13. Divide 48 by 13 to get 3 and the remainder is 9. So 48/13=3 whole 9/13
25/22, 22/22 is one whole, and 3/22 remains, and then 1 whole and 3/22
Damn, I learned how to do this first. Only then did the Internet appear, I learned how to use it correctly and it was not long before I found this site)
1) To convert an improper fraction into a mixed fraction, you need to: divide the numerator by the denominator with a remainder using a column, the partial quotient is the whole part, the remainder is the numerator and the denominator is the same.
2) To turn a mixed fraction into an improper one, you need to: multiply the whole part by the denominator and add the numerator, the resulting number goes into the numerator, but the denominator remains the same.
233 divide by number and know bersh first number and multiply
for example 1000/9.... you easily divide 1000 by 9... you get 111, which is an integer and the remainder goes to the numerator and the denominator remains the same 9....
for example, 23/3 - divide the numerator by the denominator using a calculator (if you have one nearby), take the first number, multiply by the denominator and get the whole part of this fraction. From the numerator you subtract the number that was obtained when multiplied by the denominator, and you get a proper fraction. In your answer, write the whole part and the proper fraction next to it.
If there is no calculator nearby, then you divide a little intuitively and then do the same.
The best fractions are those whose denominator is 2, 5 or 10 :)
Divide the numerator by the denominator - you get the whole part and the remainder (fraction)
Magic
In order to convert a number, you need to divide the numerator by the denominator with the remainder, i.e. find out how many “integer” times it contains. And this incomplete quotient will be whole part. Then the remainder (if there is one) is given by the numerator, and the divisor is the denominator of the fractional part (to make it clearer, you need to multiply the denominator by the integer that you received earlier, and then subtract from the NUMERATOR what you now received)
For example: 136/28 = 4 whole 24/28, this is a reducible fraction = 4 whole 6/7
I divided 136 by 28 and got 4. Then, to find out the numerator, I multiplied 28 by 4 to get 112, and subtracted 112 from 136. To reduce, you need to divide both the numerator and the denominator by the same number (in this case it is 4)
Good luck!
To isolate the whole part from an improper fraction, you need to divide the resulting numerator by the denominator
write the number as an integer part, and the remainder as the numerator, and the denominator is the same.

In this article we will talk about mixed numbers. First, let's define mixed numbers and give examples. Next, let's look at the connection between mixed numbers and improper fractions. After that we will show you how to translate mixed number into an improper fraction. Finally, let's study the reverse process, which is called separating the whole part from an improper fraction.

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Mixed numbers, definition, examples

Mathematicians agreed that the sum n+a/b, where n is a natural number, a/b is a proper fraction, can be written without the addition sign in the form. For example, the sum 28+5/7 can be briefly written as . Such a record was called mixed, and the number that corresponds to this mixed record was called a mixed number.

This is how we come to the definition of a mixed number.

Definition.

Mixed number is a number equal to the sum of the natural number n and the proper ordinary fraction a/b, and written in the form . In this case, the number n is called whole part of the number, and the number a/b is called fractional part of a number.

By definition, a mixed number is equal to the sum of its integer and fractional parts, that is, the equality is true, which can be written like this: .

Let's give examples of mixed numbers. The number is a mixed number natural number 5 is the integer part of the number, and is the fractional part of the number. Other examples of mixed numbers are .

Sometimes you can find numbers in mixed notation, but having an improper fraction as a fraction, for example, or. These numbers are understood as the sum of their integer and fractional parts, for example, And . But such numbers do not fit the definition of a mixed number, since the fractional part of mixed numbers must be a proper fraction.

The number is also not a mixed number, since 0 is not a natural number.

The relationship between mixed numbers and improper fractions

Follow connection between mixed numbers and improper fractions best with examples.

Let there be a cake and another 3/4 of the same cake on the tray. That is, according to the meaning of addition, there are 1+3/4 cakes on the tray. Having written down the last amount as a mixed number, we state that there is a cake on the tray. Now cut the whole cake into 4 equal parts. As a result, there will be 7/4 of the cake on the tray. It is clear that the “quantity” of the cake has not changed, so .

From the example considered, the following connection is clearly visible: Any mixed number can be represented as an improper fraction.

Now let there be 7/4 of the cake on the tray. Having folded a whole cake from four parts, there will be 1 + 3/4 on the tray, that is, a cake. From this it is clear that .

From this example it is clear that An improper fraction can be represented as a mixed number. (In the special case, when the numerator of an improper fraction is divided evenly by the denominator, the improper fraction can be represented as a natural number, for example, since 8:4 = 2).

Converting a mixed number to an improper fraction

To perform various operations with mixed numbers, the skill of representing mixed numbers as improper fractions is useful. In the previous paragraph, we found out that any mixed number can be converted into an improper fraction. It's time to figure out how such a translation is carried out.

Let us write an algorithm showing how to convert a mixed number to an improper fraction:

Let's look at an example of converting a mixed number to an improper fraction.

Example.

Express a mixed number as an improper fraction.

Solution.

Let's perform all the necessary steps of the algorithm.

A mixed number is equal to the sum of its integer and fractional parts: .

Having written the number 5 as 5/1, the last sum will take the form .

To finish converting the original mixed number into an improper fraction, all that remains is to add fractions with different denominators: .

A short summary of the entire solution is: .

Answer:

So, to convert a mixed number to an improper fraction, you need to perform the following chain of actions: . Finally received , which we will use further.

Example.

Write the mixed number as an improper fraction.

Solution.

Let's use the formula to convert a mixed number to an improper fraction. In this example n=15 , a=2 , b=5 . Thus, .

Answer:

Separating the whole part from an improper fraction

It is not customary to write an improper fraction in the answer. The improper fraction is first replaced either by an equal natural number (when the numerator is divisible by the denominator), or the so-called separation of the whole part from the improper fraction is carried out (when the numerator is not divisible by the denominator).

Definition.

Separating the whole part from an improper fraction- This is the replacement of a fraction with an equal mixed number.

It remains to find out how you can isolate the whole part from an improper fraction.

It's very simple: the improper fraction a/b is equal to a mixed number of the form, where q is the partial quotient, and r is the remainder of a divided by b. That is, the integer part is equal to the incomplete quotient of dividing a by b, and the remainder is equal to the numerator of the fractional part.

Let's prove this statement.

To do this, it is enough to show that . Let's convert the mixed into an improper fraction as we did in the previous paragraph: . Since q is an incomplete quotient, and r is the remainder of dividing a by b, then the equality a=b·q+r is true (if necessary, see

Mixed numbers. Selecting a whole part

Among ordinary fractions There are two different types.
Proper and improper fractions
Let's look at fractions.

Please note that in the first two fractions (3/7 and 5/7) the numerators are smaller than the denominators. Such fractions are called proper.

A proper fraction has a numerator less than its denominator. Therefore, a proper fraction is always less than one.

Let's look at the two remaining fractions.
The fraction 7/7 has a numerator equal to the denominator (such fractions are equal to units), and the fraction 11/7 has a numerator greater than the denominator. Such fractions are called improper.

An improper fraction has a numerator equal to or greater than its denominator. Therefore, an improper fraction is either equal to one or greater than one.

Any improper fraction is always greater than a proper fraction.

How to select an entire part
An improper fraction can have a whole part. Let's look at how this can be done.

To isolate the whole part from an improper fraction, you need to:
1. divide the numerator by the denominator with the remainder;
2. We write the resulting incomplete quotient into the whole part of the fraction;
3. write the remainder into the numerator of the fraction;
4. We write the divisor into the denominator of the fraction.

Example. Let's select the whole part from the improper fraction 11/2.
. Divide the numerator by the denominator in a column.

. Now let's write down the answer.

The resulting number above, containing an integer and a fractional part, is called a mixed number.

We got a mixed number from an improper fraction, but we can also do the opposite, that is, represent the mixed number as an improper fraction.
To represent a mixed number as an improper fraction:
1. multiply its integer part by the denominator of the fractional part;
2. add the numerator of the fractional part to the resulting product;
3. write the resulting amount from point 2 into the numerator of the fraction, and leave the denominator of the fractional part the same.
Example. Let's represent a mixed number as an improper fraction.
. Multiply the integer part by the denominator.

3 . 5 = 15
. Add the numerator.

15 + 2 = 17
. We write the resulting amount into the numerator of the new fraction, and leave the denominator the same.

Any mixed number can be represented as the sum of an integer and a fractional part.

Any natural number can be written as a fraction with any natural denominator.

The quotient of dividing the numerator by the denominator of such a fraction will be equal to the given natural number.
Examples.

Do you want to feel like a sapper? Then this lesson is for you! Because now we will study fractions - these are such simple and harmless mathematical objects that, in their ability to “blow the mind”, surpass the rest of the algebra course.

The main danger of fractions is that they occur in real life. This is how they differ, for example, from polynomials and logarithms, which you can study and easily forget after the exam. Therefore, the material presented in this lesson can, without exaggeration, be called explosive.

A number fraction (or just a fraction) is a pair of integers written separated by a slash or a horizontal bar.

Fractions written through a horizontal line:

The same fractions written with a slash:
5/7; 9/(−30); 64/11; (−1)/4; 12/1.

Fractions are usually written through a horizontal line - it’s easier to work with them this way, and they look better. The number written on top is called the numerator of the fraction, and the number written below is called the denominator.

Any integer can be represented as a fraction with a denominator of 1. For example, 12 = 12/1 is the fraction from the example above.

In general, you can put any whole number into the numerator and denominator of a fraction. The only limitation is that the denominator must be different from zero. Remember the good old rule: “You can’t divide by zero!”

If the denominator still has a zero, the fraction is called an indefinite fraction. Such a record is meaningless and cannot be used in calculations.

The main property of a fraction

Fractions a /b and c /d are said to be equal if ad = bc.

From this definition it follows that the same fraction can be written in different ways. For example, 1/2 = 2/4, since 1 4 = 2 2. Of course, there are many fractions that are not equal to each other. For example, 1/3 ≠ 5/4, since 1 4 ≠ 3 5.

A reasonable question arises: how to find all fractions equal to a given one? We give the answer in the form of a definition:

The main property of a fraction is that the numerator and denominator can be multiplied by the same number other than zero. This will result in a fraction equal to the given one.

This is very important property- remember it. Using the basic property of a fraction, you can simplify and shorten many expressions. In the future, it will constantly “pop up” in the form of various properties and theorems.

Improper fractions. Selecting a whole part

If the numerator is less than the denominator, it is called a proper fraction. Otherwise (i.e., when the numerator is greater than or at least equal to the denominator), the fraction is called improper, and an integer part can be distinguished in it.

The whole part is written with a large number in front of the fraction and looks like this (marked in red):

To isolate the whole part of an improper fraction, you need to follow three simple steps:

Find how many times the denominator fits in the numerator. In other words, find the maximum integer that, when multiplied by the denominator, will still be less than the numerator (at most, equal). This number will be the integer part, so we write it in front;
Multiply the denominator by the integer part found in the previous step, and subtract the result from the numerator. The resulting “stub” is called the remainder of the division; it will always be positive (in extreme cases, zero). We write it in the numerator of the new fraction;
We rewrite the denominator without changes.

Well, is it difficult? At first glance, it may be difficult. But with a little practice, you will be able to do it almost orally. In the meantime, take a look at the examples:

Task. Select the whole part in the indicated fractions:

In all examples, the whole part is highlighted in red, and the remainder of the division is highlighted in green.

Pay attention to the last fraction, where the remainder of the division turns out to be zero. It turns out that the numerator is completely divided by the denominator. This is quite logical, because 24: 6 = 4 is a hard fact from the multiplication table.

If everything is done correctly, the numerator of the new fraction will definitely be less than the denominator, i.e. the fraction will become correct. I will also note that it is better to highlight the whole part at the very end of the problem, before writing down the answer. Otherwise, the calculations can be significantly complicated.

Going to an improper fraction

There is also a reverse operation, when we get rid of the whole part. This is called the improper fraction transition and is much more common because working with improper fractions is much easier.

The transition to an improper fraction is also performed in three steps:

Multiply the whole part by the denominator. The result can be quite big numbers, but this should not bother us;
Add the resulting number to the numerator of the original fraction. Write the result in the numerator of the improper fraction;
Rewrite the denominator - again, without changes.

Here are specific examples:

Task. Convert to improper fraction:

For clarity, the integer part is again highlighted in red, and the numerator of the original fraction is highlighted in green.

Consider the case when the numerator or denominator of the fraction contains a negative number. For example:

In principle, there is nothing criminal in this. However, working with such fractions can be inconvenient. Therefore, in mathematics it is customary to place minuses as fraction signs.

This is very easy to do if you remember the rules:

“Plus for minus gives minus.” Therefore, if the numerator contains a negative number, and the denominator contains a positive number (or vice versa), feel free to cross out the minus and put it in front of the entire fraction;
"Two negatives make an affirmative". When there is a minus in both the numerator and the denominator, we simply cross them out - no additional actions are required.

Of course, these rules can also be applied in the opposite direction, i.e. You can enter a minus sign under the fraction sign (most often in the numerator).

We deliberately do not consider the “plus on plus” case - with it, I think, everything is clear. Let's see how these rules work in practice:

Task. Take out the negatives of the four fractions written above.

Pay attention to the last fraction: there is already a minus sign in front of it. However, it is “burned” according to the rule “minus for minus gives a plus.”

Also, do not move minuses in fractions with the whole part highlighted. These fractions are first converted to improper fractions - and only then do calculations begin.