Video tutorial “Isolation of an integer part from an improper fraction. Improper fraction representation of a mixed number

It is customary to write without the $ "+" $ sign in the form $ n \ frac (a) (b) $.

Example 1

For example, the sum $ 4 + \ frac (3) (5) $ is written $ 4 \ frac (3) (5) $. Such notation is called a mixed fraction, and the number that corresponds to it is called a mixed number.

Definition 1

Mixed number is a number that is equal to the sum of a natural number $ n $ and a regular fraction $ \ frac (a) (b) $, and is written as $ n \ frac (a) (b) $. In this case, the number $ n $ is called $ n \ frac (a) (b) $, and the number $ \ frac (a) (b) $ is called the fractional part of the number /

For mixed numbers, the equalities $ n \ frac (a) (b) = n + \ frac (a) (b) $ and $ n + \ frac (a) (b) = n \ frac (a) (b) $ hold.

Example 2

For example, the number $ 7 \ frac (4) (9) $ is a mixed number, where the natural number $ 7 $ is its integer part, $ \ frac (4) (9) $ is its fractional part. Examples of mixed numbers: $ 17 \ frac (1) (2) $, $ 456 \ frac (111) (500) $, $ 23000 \ frac (4) (5) $.

There are numbers in mixed notation that contain an incorrect fraction in the fractional part. For example, $ 3 \ frac (54) (5) $, $ 56 \ frac (9) (2) $. The recording of these numbers can be represented as the sum of their integer and fractional parts. For example, $ 3 \ frac (54) (5) = 3 + \ frac (54) (5) $ and $ 56 \ frac (9) (2) = 56 + \ frac (9) (2) $. Such numbers are not suitable for the definition of a mixed number, because the fractional part of the mixed numbers must be a regular fraction.

The number $ 0 \ frac (2) (7) $ is also not a mixed number, since $ 0 $ is not a natural number.

Converting a mixed number to an improper fraction

Algorithm for converting a mixed number to an improper fraction:

Write the mixed number $ n \ frac (a) (b) $ as the sum of the integer and fractional parts of this number, i.e. as $ n + \ frac (a) (b) $.

Replace the whole part of the original mixed number with a fraction with the denominator $ 1 $.

Add the fractions $ \ frac (n) (1) $ and $ \ frac (a) (b) $ to get the desired improper fraction equal to the original mixed number.

Example 3

Expand mixed number $ 7 \ frac (3) (5) $ as improper fraction.

Solution.

Let's use the algorithm for converting a mixed number into an improper fraction.

Mixed number $ 7 \ frac (3) (5) = 7 + \ frac (3) (5) $.

Let's write the number $ 7 $ as $ \ frac (7) (1) $.

Add up the fractions $ \ frac (7) (1) + \ frac (3) (5) = \ frac (35) (5) + \ frac (3) (5) = \ frac (38) (5) $.

Let's write a short record of this solution:

Answer:$ 7 \ frac (3) (5) = \ frac (38) (5) $

The whole algorithm for converting a mixed number $ n \ frac (a) (b) $ into an improper fraction is reduced to \ textit (a formula for converting a mixed number into an improper fraction):

Example 4

Write the mixed number $ 14 \ frac (3) (5) $ as an improper fraction.

Solution.

Let's use the formula $ n \ frac (a) (b) = \ frac (n \ cdot b + a) (b) $ to convert the mixed number to an improper fraction. In this example, $ n = 14 $, $ a = 3 $, $ b = 5 $.

We get $ 14 \ frac (3) (5) = \ frac (14 \ cdot 5 + 3) (5) = \ frac (73) (5) $.

Answer:$ 14 \ frac (3) (5) = \ frac (73) (5) $

Isolating the whole part from an improper fraction

When obtaining a numerical solution, it is not customary to leave an answer in the form of an incorrect fraction. An improper fraction is converted to an equal natural number (if the numerator is completely divisible by the denominator), or the whole part is extracted from the improper fraction (if the numerator is not completely divisible by the denominator).

Definition 2

Isolating the whole part from an improper fraction is called replacing a fraction with a mixed number equal to it.

To isolate the whole part from an improper fraction, you need to represent the improper fraction $ \ frac (a) (b) $ as a mixed number $ q \ frac (r) (b) $, where $ q $ is an incomplete quotient, $ r $ is remainder of dividing $ a $ by $ b $. Thus, the integer part is equal to the incomplete quotient of $ a $ divided by $ b $, and the remainder is equal to the numerator of the fractional part.

Let us prove this statement. To do this, it suffices to show that $ q \ frac (r) (b) = \ frac (a) (b) $.

Let's convert the mixed number $ q \ frac (r) (b) $ into an improper fraction using the formula:

Because $ q $ is an incomplete quotient, $ r $ is the remainder of dividing $ a $ by $ b $, then the equality $ a = b \ cdot q + r $ is valid. Thus, $ \ frac (q \ cdot b + r) (b) = \ frac (a) (b) $, whence $ q \ frac (r) (b) = \ frac (a) (b) $, which was required to be shown.

Thus, we formulate \ textit (the rule for separating the integer part from an improper fraction) $ \ frac (a) (b) $:

Divide $ a $ by $ b $ with the remainder, while determining the incomplete quotient $ q $ and the remainder $ r $.

Write down the mixed number $ q \ frac (r) (b) $, equal to the original fraction $ \ frac (a) (b) $.

Example 5

Select the integer part from the fraction $ \ frac (107) (4) $.

Solution.

Let's do long division:

Picture 1.

So, as a result of dividing the numerator $ a = 107 $ by the denominator $ b = 4 $, we get the incomplete quotient $ q = 26 $ and the remainder $ r = 3 $.

We get that the improper fraction $ \ frac (107) (4) $ is equal to the mixed number $ q \ frac (r) (b) = 26 \ frac (3) (4) $.

Answer: $ \ frac ((\ rm 107)) ((\ rm 4)) (\ rm = 26) \ frac ((\ rm 3)) ((\ rm 4)) $.

Adding a mixed number and a natural number

Rule of addition of mixed and natural numbers:

To add a mixed and natural number, you need to add this natural number to the integer part of the mixed number, the fractional part remains unchanged:

where $ a \ frac (b) (c) $ is a mixed number,

$ n $ is a natural number.

Example 6

Add mixed $ 23 \ frac (4) (7) $ and $ 3 $.

Solution.

Answer:$ 23 \ frac (4) (7) + 3 = 26 \ frac (4) (7). $

Adding two mixed numbers

When adding two mixed numbers, their whole parts and fractional parts are added.

Example 7

Add mixed numbers $ 3 \ frac (1) (5) $ and $ 7 \ frac (4) (7) $.

Solution.

Let's use the formula:

\ \

Answer:$ 10 \ frac (27) (35). $

How to select the whole part from an improper fraction? To select a whole part from an incorrect fraction, you need to: Divide the numerator by the denominator with the remainder; The incomplete quotient will be the whole part; The remainder (if any) gives the numerator, and the divisor is the denominator of the fractional part. Run No. 1057, 1058, 1059, 1060.1062, 1063.1064.7.

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Mixed numbers

"Summary of a lesson in mathematics" - Follow the model. a) 4/7 + 2/7 = (4 + 2) / 7 = 6/7 b, c, d (at the board) e) 7 / 9-2 / 9 = (7-2) / 9 = 5 / 9 f, g, h (at the board). 12 kg of cucumbers were harvested in the garden. 2/3 of all cucumbers were pickled. 6 / 7-3 / 7 = (6-3) / 7 = 3/7 2/11 + 5/11 = (2 + 5) / 22 = 7/22 9 / 10-8 / 10 = (9-8 ) / 10 = 2/10. Show the fraction 2/8 + 3/8. Formulate a rule for subtraction. Learning new material:

"Comparison of decimal fractions" - The purpose of the lesson. Compare the numbers: Verbal counting. 9.85 and 6.97; 75.7 & 75.700; 0.427 and 0.809; 5.3 & 5.03; 81.21 & 81.201; 76.005 and 76.05; 3.25 & 3.502; Read the fractions: 41.1; 77.81; 21.005; 0.0203. 41.1; 77.81; 21.005; 0.0203. Equalize the number of decimal places. Lesson plan. Decimal places. Consolidation lesson in grade 5.

“Number rounding rules” - 1.8. 48. Well done! 3. 3. Learn to apply the rounding rule using examples. Try to compare. Round whole numbers to tens. 1. Recall the rule for rounding numbers. Is it convenient to work with such a number? One hundred thousandths. 3. We write down the result. 5312.>. 2. Derive the rule for rounding decimal fractions to a given digit.

"Addition of mixed numbers" - 25. Example 4. Find the value of the difference 3 4 \ 9-1 5 \ 6. 3 4 \ 9 = 3 818; 1 5 \ 6 = 1 15 \ 18. 3 4 \ 9 = 3 8 \ 18 = 3 + 8 \ 18 = 2 + 1 + 8 \ 18 = 2 + 8 \ 18 + 18 \ 18 = 2 + + 26 \ 18 = 2 26 \ 18. Lesson synopsis in grade 6

Remember!

An improper fraction has the numerator equal to or greater than the denominator. That's why improper fraction or equal to one or greater than one.

Any incorrect fraction is always more correct.

How to select a whole part

You can select the whole part of an incorrect fraction. Let's see how this can be done.

To select a whole part from an incorrect fraction, you need to:

1. divide the numerator by the denominator with the remainder;
2. the resulting incomplete quotient is written in the whole part of the fraction;
3. the remainder is written in the numerator of the fraction;
4. the divisor is written into the denominator of the fraction.

Remember!

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

We got a mixed number from an improper fraction, but you can also perform the opposite action, that is, represent a mixed number as an improper fraction.

To represent a mixed number as an improper fraction, you need to:

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 paragraph 2 into the numerator of the fraction, and leave the denominator of the fractional part the same.

Example. Let's represent the mixed number as an improper fraction.

§ 1 Isolation of the whole part from an improper fraction

In this lesson, you will learn how to convert an improper fraction to a mixed number by highlighting the whole part, and vice versa, get an improper fraction from a mixed number.

First, let's remember what a mixed number and an improper fraction are.

Mixed number is a special form of notation of a number that contains whole and fractional parts.

An irregular fraction is a fraction whose numerator is greater than or equal to the denominator.

Consider the problem:

Let's divide 8 candies for three guys. How much will each get?

To find out how many sweets each child will receive, you need

But it is not customary to write the wrong fraction in the answer. It is previously replaced either by a natural number equal to it (when the numerator is divided entirely by the denominator), or the so-called separation of the whole part from the improper fraction is carried out (when the numerator is not completely divisible by the denominator).

Separating the whole part from an improper fraction is replacing a fraction with its equal mixed number.

To select a whole part from an incorrect fraction, you need to divide the numerator by the denominator with a remainder. In this case, the incomplete quotient will be the whole part, the remainder will be the numerator, and the divisor will be the denominator.

Let's get back to the problem.

So, we divide 8 by 3 with a remainder, we get 2 in the incomplete quotient and 2 in the remainder.

§ 2 Representation of a mixed number as an improper fraction

Let's do the following task:

We divide 49 by 13, we get 3 in the incomplete quotient (this will be the integer part) and in the remainder 10 (we will write this in the numerator of the fractional part).

The skill of representing mixed numbers as improper fractions is useful for performing various actions with mixed numbers. It's time to figure out how such a translation is carried out.

To represent the mixed number as an improper fraction, you need to multiply the denominator of the fraction by the whole part and add the numerator to the resulting product. As a result, we get a number that will be the numerator of the new fraction, and the denominator remains unchanged.

The first step is to multiply the integer part 5 by the denominator 7 to get 35.

The second step is to add the numerator 4 to the resulting product 35, it will be 39.

Now let's write 39 in the numerator and leave 7 in the denominator.

Thus, in this lesson you learned how to convert an improper fraction to a mixed number, for this you need to divide the numerator by the denominator with the remainder. Then the incomplete quotient will be the integer part, the remainder will be the numerator, and the divisor will be the denominator of the fractional part of the mixed number.

Also, you got acquainted with the representation of a mixed number in the form of an improper fraction. In order to represent the mixed number as an improper fraction, you need to multiply the denominator of the fractional part of the mixed number by an integer part and add the numerator to the resulting product.

List of used literature:

1. Mathematics grade 5. Vilenkin N.Ya., Zhokhov V.I. et al. 31st ed., erased. - M: 2013.
2. Didactic materials in mathematics grade 5. Author - Popov M.A. - year 2013
3. We calculate without errors. Works with self-test in mathematics 5-6 grades. Author - Minaeva S.S. - year 2014
4. Didactic materials in mathematics grade 5. Authors: Dorofeev G.V., Kuznetsova L.V. - 2010
5. Control and independent work in mathematics, grade 5. Authors - Popov M.A. - year 2012
6. Maths. Grade 5: textbook. for general education students. institutions / I. I. Zubareva, A. G. Mordkovich. - 9th ed., Erased. - M .: Mnemosina, 2009

Lesson summary in grade 5

“Mixed numbers. Isolation of the whole part from an improper fraction "

During the classes

Organizing time. Greetings.

We will hold an oral account and break all records

Verbal counting.

Find the mistakes

Correct fractions.

b)

Let's write on the blackboard what we can't compare yet.

2. Perform division:

45: 9=5 ; 0: 67=0; 234: 1=234;

567: 567 = 1; 34: 17 = 2; a: a = 1;

3. Perform division with remainder:

6 = 2 (rest 2)

3 = 8 (rest 1)

48: 9 = 5 (rest 3)

Follow the steps:

We cannot solve the last example; let us write it out.

Explanation of the new material

What is shown in the picture? How many pieces were the cake divided into? How many parts did you take? Present as a fraction.

What's in this picture? It can be seen that the cake is on different trays. How many pieces are on the first tray? Second?

It can be denoted as such a number:

1 - whole part, - fractional part.

The sum of the integer and fractional parts is calledmixed number .

Determine from the picture what mixed number is equal to a fraction?

That is, we saw a connection between an improper fraction and a mixed number.

Let's draw conclusions: we can turn an improper fraction into a mixed number, i.e. as they say in mathematics, select the whole part from an irregular fraction.

The rule for separating the whole part from an improper fraction:

Divide the numerator by the denominator with remainder

The incomplete quotient will be the whole part

The remainder gives the numerator, and the divisor is the denominator of the fractional part

Work on the topic of the lesson.

Select a whole part from an improper fraction (along with the class):

Select the whole part from the irregular fraction (at the board)

Compare

Historical information.

In the old days in Russia, coins were used in denominations of less than one kopeck:

penny - Ph. andpolushka - Ph.

Other coins also had names:

3 k. - altyn, 5 k. - penny, 15 k. - five altyn,

10 k. - a dime, 20 k. Two-kopeck coins,

25 k. - a quarter, 50 k. - fifty dollars.

Independent work

How can you imagine

1 dime, 1 altyn, three polushki .

Reflection

What's your mood?

Write the fraction that best suits your knowledge:

2 (can not understand anything)

2 (it was interesting, but not clear)

3 (difficult, the topic is not interesting)

3 (it was difficult, but I will definitely make an effort to study the topic)

4 (some examples caused difficulties)

4 (everything is clear, but I cannot help)

5 (everything is clear, I can help others)

I hope your grade will only increase with each lesson! And to get a grade of 5, you need to work not only in the classroom, but also at home.

Homework.