Addition and subtraction of mixed numbers: features and examples. Subtraction of mixed fractions

Mixed fractions can be subtracted just like simple fractions. To subtract mixed numbers of fractions, you need to know several subtraction rules. Let's explore these rules with examples.

Subtraction of mixed fractions with the same denominator.

Consider an example with the condition that the reduced integer and fractional parts are greater than the subtracted whole and fractional parts, respectively. Under these conditions, the deduction takes place separately. Subtract the whole part from the whole part, and the fractional part from the fractional part.

Let's consider an example:

Perform mixed fraction subtraction \ (5 \ frac (3) (7) \) and \ (1 \ frac (1) (7) \).

\ (5 \ frac (3) (7) -1 \ frac (1) (7) = (5-1) + (\ frac (3) (7) - \ frac (1) (7)) = 4 \ frac (2) (7) \)

The correctness of the subtraction is checked by addition. Let's check the subtraction:

\ (4 \ frac (2) (7) +1 \ frac (1) (7) = (4 + 1) + (\ frac (2) (7) + \ frac (1) (7)) = 5 \ frac (3) (7) \)

Consider an example with the condition when the fractional part of the reduced is less, respectively, the fractional part of the subtracted. In this case, we borrow one from the whole in the diminished one.

Let's consider an example:

Perform mixed fraction subtraction \ (6 \ frac (1) (4) \) and \ (3 \ frac (3) (4) \).

The reduced \ (6 \ frac (1) (4) \) has a fractional part less than the fractional part of the subtracted \ (3 \ frac (3) (4) \). That is, \ (\ frac (1) (4)< \frac{1}{3}\), поэтому сразу отнять мы не сможем. Займем у целой части у 6 единицу, а потом выполним вычитание. Единицу мы запишем как \(\frac{4}{4} = 1\)

\ (\ begin (align) & 6 \ frac (1) (4) -3 \ frac (3) (4) = (6 + \ frac (1) (4)) - 3 \ frac (3) (4) = (5 + \ color (red) (1) + \ frac (1) (4)) - 3 \ frac (3) (4) = (5 + \ color (red) (\ frac (4) (4)) + \ frac (1) (4)) - 3 \ frac (3) (4) = (5 + \ frac (5) (4)) - 3 \ frac (3) (4) = \\\\ & = 5 \ frac (5) (4) -3 \ frac (3) (4) = 2 \ frac (2) (4) = 2 \ frac (1) (4) \\\\ \ end (align) \)

Next example:

\ (7 \ frac (8) (19) -3 = 4 \ frac (8) (19) \)

Subtracting a mixed fraction from an integer.

Example: \ (3-1 \ frac (2) (5) \)

The decreasing 3 does not have a fractional part, so we cannot immediately subtract it. Let us borrow one from the integer part of 3, and then perform the subtraction. We will write the unit as \ (3 = 2 + 1 = 2 + \ frac (5) (5) = 2 \ frac (5) (5) \)

\ (3-1 \ frac (2) (5) = (2 + \ color (red) (1)) - 1 \ frac (2) (5) = (2 + \ color (red) (\ frac (5 ) (5))) - 1 \ frac (2) (5) = 2 \ frac (5) (5) -1 \ frac (2) (5) = 1 \ frac (3) (5) \)

Subtraction of mixed fractions with different denominators.

Consider an example with the condition if the fractional parts of the reduced and subtracted with different denominators. You need to bring to a common denominator, and then perform the subtraction.

Subtract two mixed fractions with different denominators \ (2 \ frac (2) (3) \) and \ (1 \ frac (1) (4) \).

The common denominator is 12.

\ (2 \ frac (2) (3) -1 \ frac (1) (4) = 2 \ frac (2 \ times \ color (red) (4)) (3 \ times \ color (red) (4) ) -1 \ frac (1 \ times \ color (red) (3)) (4 \ times \ color (red) (3)) = 2 \ frac (8) (12) -1 \ frac (3) (12 ) = 1 \ frac (5) (12) \)

Questions on the topic:
How to subtract mixed fractions? How to solve mixed fractions?
Answer: you need to decide what type the expression belongs to and, by the type of expression, apply the solution algorithm. Subtract the whole from the whole part, subtract the fractional part from the fractional part.

How to subtract a fraction from an integer? How to subtract a fraction from an integer?
Answer: you need to take a unit from an integer and write this unit as a fraction

\ (4 = 3 + 1 = 3 + \ frac (7) (7) = 3 \ frac (7) (7) \),

and then subtract the whole from the whole, subtract the fractional part from the fractional part. Example:

\ (4-2 \ frac (3) (7) = (3 + \ color (red) (1)) - 2 \ frac (3) (7) = (3 + \ color (red) (\ frac (7 ) (7))) - 2 \ frac (3) (7) = 3 \ frac (7) (7) -2 \ frac (3) (7) = 1 \ frac (4) (7) \)

Example # 1:
Subtract the correct fraction from one: a) \ (1- \ frac (8) (33) \) b) \ (1- \ frac (6) (7) \)

Solution:
a) We represent the unit as a fraction with the denominator 33. We get \ (1 = \ frac (33) (33) \)

\ (1- \ frac (8) (33) = \ frac (33) (33) - \ frac (8) (33) = \ frac (25) (33) \)

b) We represent the unit as a fraction with the denominator 7. We get \ (1 = \ frac (7) (7) \)

\ (1- \ frac (6) (7) = \ frac (7) (7) - \ frac (6) (7) = \ frac (7-6) (7) = \ frac (1) (7) \)

Example # 2:
Subtract mixed fraction from an integer: a) \ (21-10 \ frac (4) (5) \) b) \ (2-1 \ frac (1) (3) \)

Solution:
a) We borrow from an integer 21 units and write it like this \ (21 = 20 + 1 = 20 + \ frac (5) (5) = 20 \ frac (5) (5) \)

\ (21-10 \ frac (4) (5) = (20 + 1) -10 \ frac (4) (5) = (20 + \ frac (5) (5)) - 10 \ frac (4) ( 5) = 20 \ frac (5) (5) -10 \ frac (4) (5) = 10 \ frac (1) (5) \\\\\)

b) Let us borrow a unit from the integer 2 and write it like this \ (2 = 1 + 1 = 1 + \ frac (3) (3) = 1 \ frac (3) (3) \)

\ (2-1 \ frac (1) (3) = (1 + 1) -1 \ frac (1) (3) = (1 + \ frac (3) (3)) - 1 \ frac (1) ( 3) = 1 \ frac (3) (3) -1 \ frac (1) (3) = \ frac (2) (3) \\\\\)

Example # 3:
Subtract an integer from a mixed fraction: a) \ (15 \ frac (6) (17) -4 \) b) \ (23 \ frac (1) (2) -12 \)

a) \ (15 \ frac (6) (17) -4 = 11 \ frac (6) (17) \)

b) \ (23 \ frac (1) (2) -12 = 11 \ frac (1) (2) \)

Example No. 4:
Subtract the correct fraction from the mixed fraction: a) \ (1 \ frac (4) (5) - \ frac (4) (5) \)

\ (1 \ frac (4) (5) - \ frac (4) (5) = 1 \\\\\)

Example # 5:
Calculate \ (5 \ frac (5) (16) -3 \ frac (3) (8) \)

\ (\ begin (align) & 5 \ frac (5) (16) -3 \ frac (3) (8) = 5 \ frac (5) (16) -3 \ frac (3 \ times \ color (red) ( 2)) (8 \ times \ color (red) (2)) = 5 \ frac (5) (16) -3 \ frac (6) (16) = (5 + \ frac (5) (16)) - 3 \ frac (6) (16) = (4 + \ color (red) (1) + \ frac (5) (16)) - 3 \ frac (6) (16) = \\\\ & = (4 + \ color (red) (\ frac (16) (16)) + \ frac (5) (16)) - 3 \ frac (6) (16) = (4 + \ color (red) (\ frac (21 ) (16))) - 3 \ frac (3) (8) = 4 \ frac (21) (16) -3 \ frac (6) (16) = 1 \ frac (15) (16) \\\\ \ end (align) \)

Solving complex examples correctly is an impossible task for those who do not understand elementary rules and laws in mathematics. Addition and subtraction mixed numbers can rightfully be attributed to complex examples... However, with correct parsing the numbers themselves, you can easily carry out any action.

What it is?

A mixed number is a combination of an integer part and a fractional part. For example, there are 2 and 3, of which 2 is a prime number, but 3 is already mixed, where 3 is whole part, and - fractional. The presented varieties are added and subtracted in different ways, but do not entail difficulties in independent decision examples.

Complete parsing of the example

For a full presentation of the essence of mixed meaning, an example should be given of a task that will help to display the meaning of the narration of the conceived. So, Vasya rode a circle around the school on a bicycle in 1 minute and 30 seconds, and then walked another circle in 3 minutes and 30 seconds. How much time did Vasya spend on the whole walk around the school?

This example is aimed at adding mixed numbers, which in this case do not even have to be converted into seconds beforehand. It turns out that the addition is carried out by separately adding minutes and seconds. As a result, we get the following result:

Addition of minutes - 1 + 3 = 4.
Addition of seconds = 30 + 30 = 60 seconds = 1 minute.
Total value 4 minutes + 1 minute = 5 minutes.

If we proceed from the mathematical display, then the presented actions can be distinguished in one expression:

From the above, it becomes clear that the mixed numbers should be added separately in parts - first whole parts, and then fractional ones. If the fractional number gives another integer value, it is also added to the integer value obtained earlier. The fractional part is added to the resulting integer value - a mixed number is obtained.

Addition rules

To consolidate what has been learned, a rule for adding mixed numbers should be given. Here you should use the following sequence:

To begin with, separate the parts from the meaning - into whole and fractional.
Now add the whole pieces.
Next, add up the fractional ones.
If an integer part can be extracted from a fractional number - converted into a mixed value - then a similar breakdown is carried out.
The resulting integer part from the fractional value is added to the previously obtained integer value.
The fractional part is added to the whole part.

For clarification, a few examples should be given:

The addition of mixed numbers follows the same algorithm as subtraction, so the next step will be discussed in detail below.

Subtraction rules

As in the first case, there is a rule for subtracting mixed values, but it is fundamentally different from the previous sequence. So, here you should adhere to the sequence:

An example for subtraction is represented as: decremented - subtracted = difference.
In connection with the above equation, you should first compare the fractional parts of the presented numbers.
If the fractional part to be reduced has a larger fraction, it means that the subtraction is carried out according to the same criterion as in addition - first integers are subtracted, and then fractional values. Both results add up.
If the fractional value to be reduced is less, it means that they are previously converted to an incorrect fraction and a standard subtraction is performed.
The whole part and the fractional part are determined from the obtained difference.

For clarification, you should give the following examples:

From the presented article, it became clear how to carry out the addition and subtraction of mixed numbers. In the example described above, it can be seen that it is not always necessary to modify the numbers - to convert them from simple fractions to complex ones. It is often enough to simply add or subtract whole and fractional values separately, which for a person with more experience can be easily carried out in the mind.

The article discusses in detail examples, the solution of which is presented in full accordance with mathematical rules and fundamentals. Individual situations are analyzed, for each an example of the modifications that can be encountered in solving problems and complex examples is given.

In this lesson, you will learn the rules of addition and subtraction of mixed numbers, learn how to solve various problems on the topic "Addition and subtraction of mixed numbers." The addition and subtraction of mixed numbers is based on a property of these numbers. When adding, you can use the displacement and combining properties, and when subtracting numbers, you can use the properties of subtracting a number from a sum and subtracting a sum from a number.

First, let's remember what mixed numbers are. A mixed number is a number written in such a way that it has an integer part and a fractional part. For example, . Here 3 is the whole part, - fractional.

Suppose we were given such a task. Vasya ran the first of two laps of the distance in 1 minute 40 seconds, and the second lap - in 1 minute 20 seconds. How long did it take Vasya to run the entire distance and how much faster did he run the second lap than the first?

Solution

It's easy to see that we can add minutes to minutes, seconds to seconds. It turns out 2 minutes + 60 seconds, i.e. 3 minutes. But, on the other hand, 40 seconds are minutes, and 20 seconds are. And then, by analogy, in order to add up these mixed numbers, we can not translate them into irregular fractions, but immediately add whole minutes with each other, and separately - fractional ones. This gives 2 minutes and, that is, another whole minute. Total 3 minutes.

It was possible to do all this and so. Note that the mixed number is the sum of its integer and fractional parts. And then we will use the displacement property:

What about subtraction? Same. For purely practical reasons, the first lap is the same in minutes as the second, and in seconds - 20 longer (or a third of a minute). Could be so:

Think you already understand the algorithm? From the whole we subtract (add to the whole) the whole, from the fractional - fractional. Let's look at a few more examples.

Let's fix these calculations with a rule. To add two mixed numbers, you need:

add their whole parts;
add their fractional parts;
if necessary, convert the sum of fractional parts to a mixed number;
add up the resulting numbers.

Let's move on to subtraction. Let's look at a few examples and then formulate a general algorithm.

Find errors in addition examples

Let's take a closer look at the first example: the mixed number was replaced by a fraction, and the number -, but these fractions are not equal. If we decide to convert fractions to incorrect ones, we get the following:

Now let's move on to the second example, in which actions are performed according to the algorithm we have considered. As you can see, all actions were performed correctly, however, it is customary to write mixed numbers so that their fractional part is a regular fraction. Therefore, we will represent the fraction as a mixed number, and then we will perform the addition.

If you go according to the plan, then you need to subtract from. We cannot do this. Then we will do as we do when subtracting natural numbers: borrow from the senior category. Only the whole part will play the role of the senior category here. After all, the unit is, so you can write instead. And then - according to plan:

Let's fix these calculations with a rule. To subtract one mixed number from another, you must:

compare the fractional parts of the reduced and subtracted;
if the fractional part of the reduced is greater, then subtract the whole part from the whole part, the fractional part from the fractional part, and add the results;
if the fractional part of the subtracted is greater, then we convert one unit from the whole part of the reduced one into a fraction so that the fraction becomes incorrect, and then we subtract the whole from the whole part, and the fractional from the fractional part, and add the results.

Find errors in subtraction examples

Let's look at the first example. According to the algorithm, we must first represent 12 as a mixed number, and then perform the subtraction:

Let's look at the second example. Here is an error when subtracting fractional parts: we need to subtract the fractional part of the subtracted from the fractional part of the reduced, and not vice versa. To do this, we have to take 1 unit and represent it as a fraction.

In this lesson, we got acquainted with mixed numbers, learned how to add and subtract them, and formulated algorithms for addition and subtraction. We learned that to add and subtract mixed numbers it is not at all necessary to translate them into improper fractions, but rather simply add or subtract whole parts and add or subtract fractional parts, and then write down the final answer.

In each case, we had one subtlety. For addition, we understood that sometimes the sum of fractional parts is obtained in the form of an incorrect fraction, therefore, if necessary, the resulting incorrect fraction must be reduced to the correct one, that is, to select the whole part. And during the subtraction, such subtlety appeared that it was not always possible to subtract the fractional part of the subtracted from the fractional part of the subtracted, so we had to "borrow" a unit from the whole part and convert it to a fractional one in order to get an incorrect fraction, from which it was already possible to subtract the fractional part ...

Bibliography

Maths. Grade 5. Zubareva I.I., Mordkovich A.G. 14th ed., Rev. and add. - M .: 2013.
Vilenkin N.Ya. and other Mathematics. 5 cl. - M: Mnemosina, 2013.
Erina T.M. Mathematics grade 5 Slave. notebook for uch. Vilenkina 2013 .-- M: Mnemosina, 2013.

Website of the festival of pedagogical ideas " Public lesson» ()
School Assistant website ()
Website schools.keldysh.ru ()

Homework

>> Math: Addition and Subtraction of Mixed Numbers - Grade 6

12. Addition and subtraction of mixed numbers

The displacement and combination properties of addition make it possible to reduce the addition of sour cream numbers to the addition of their whole parts and to the addition of their fractional parts.
Example 1. Find the value of the sum
Solution. Let us reduce the fractional parts of the numbers to the smallest total 8, then represent the mixed numbers as the sum of their integer and fractional parts:

Example 2. Let's find the value of the sum.
Solution. First, we bring the fractional parts of these numbers to the smallest common denominator 12, then we add the whole and fractional parts separately:

To add mixed numbers, you need:

1) bring the fractional parts of these numbers to the lowest common denominator;

2) separately perform the addition of whole parts and separately fractional parts.

If, when adding the fractional parts, you get an incorrect fraction, select the whole part from this fraction and add it to the resulting whole part.

When subtracting mixed numbers, use the properties of subtracting a sum from a number and subtracting a number from sums .

Example 3. Let's find the value of the difference.
Solution. Let us bring the fractional parts to the lowest common denominator 18 and represent these numbers as the sum of the integer and fractional parts:

They write shorter:

If the fractional part of the reduced is less than the fractional part of the subtracted, then one unit of the whole part of the reduced must be converted into a fraction with the same denominator.

Example 4. Find the value of the difference

Solution. Let's reduce the fractional parts of these numbers to the lowest common denominator 18:

Since the fractional part of the reduced is less than the fractional part of the subtracted, then the reduced is written as follows:

To subtract mixed numbers, you must: 1) bring the fractional parts of these numbers to the lowest common denominator; if the fractional part of the reduced one is less than the fractional part of the subtracted one, turn it into an irregular fraction, decreasing the whole part by one; 2) separately perform the subtraction of whole parts and separately fractional parts.

? Tell me how to fold the mixed the numbers and on what properties of addition is the addition of mixed numbers based. Tell us how to subtract mixed numbers and what properties the mixed number subtraction rule is based on.

TO 363. Perform addition:

364. Subtract:

365. Find the meaning of the expression:

366. Perform the action:

368. Find by the formula :

369. The school pool is filled through the first pipe in 4 hours, and through the second in 6 hours. What part of the pool will be left to fill after both pipes have been working together for an hour?

370. New car can dig a ditch in 8 hours, and the old one - in 12 hours. The new machine worked for 3 hours, and the old one for 5 hours. What part of the ditch is left to dig?

371. A piece of length m was cut from a tape 8 m long. Find the length the remainder.

372. One chess game lasted one hour, and the other one hour. How long did the third game last if 3 hours were spent on all three games?

373. When a piece was cut off from the rope, the remaining part had a length of 2 m. How long would the remaining part be if the rope was cut off by m less? m more?

374. Write down all numbers, the denominator of the fractional part of which is 12, greater and less.

375. A point is marked on the coordinate ray (Fig. 17). Mark the points on the ray coordinates which are equal:

376. Find the perimeter of triangle ABC, if AB = m, .

377. There are tons of cargo on one machine, and tons less on the other. How many tons of cargo are there on two vehicles?

378. There are kg of grapes in one box, which is less by kg than in the other box. How many kilograms of grapes are in two boxes?

379. Kg of paint was used to paint windows. It took kg less to paint the doors than to paint the floor. How many paints were consumed in total if kg went to paint the floor?

380. Three collective farm links have grown peas on squares ha. The first and second links grew peas on an area of hectares, and the second and third - on an area of hectares. Find the area of each lot.

381. On Monday, tons of beets were brought to the sugar factory, on Tuesday - 2 tons more than on Monday, and on Wednesday - on tons less than on Tuesday and Monday together. From 7 tons of beets, 1 ton of sugar is obtained. How much sugar will be obtained from the imported beets?

382. Three cans contain 10 liters of milk. In the first and second cans there were 1 liters, and in the second and third liters of milk. How many liters of milk were there in each can?

383. A motor ship runs along the river in 1 hour. The speed of the current is km / h. Find the speed of the ship against the current.

384 Speed of the boat along the river, km / h, and against the current, km / h. What is the speed of the current?

385. Fedya and Vasya walked towards each other. Every hour the distance between them decreased by km. Find Fedya's speed, if Vasya's speed

386. The first cyclist was catching up with the second, and the distance between them decreased every hour per km. What speed was the first cyclist traveling if the second was traveling at a speed of y km / h?

NS 388. Calculate orally:

389. Find the missing numbers:

390 Find the natural values of m for which the inequality is true:

391. By what percentage will the volume of a cube increase if the length of each of its edges is increased by 20%?

392. The postal plane took off from the airfield at 10.40 am, stayed in flight for 5 hours 15 minutes, and on the ground during landings 1 hour 37 minutes. When did the plane return to the airfield?

M 393. Quadrangle with equal sides called the VIZ rhombus (Fig. 18). Consider if the rhombus is a regular polygon. What is the similarity between solving this problem and finding solutions to the double inequality 0< у<. 10 среди чисел 0,12; 15; 2,7; 10,5?

394. Prove the displacement and combination properties of addition for fractions with the same denominators on the basis of the same properties for natural numbers.

395. Perform the action:

396. Stamps of 3 grade, 5 grade and 10 grade were delivered to the kiosk for sale. The number of stamps of each type was the same. What is the cost of all stamps of 5 kopecks, if: a) the total value of all stamps is 21 rubles. 60 k., B) the cost of all stamps for 10 k. More cost all brands of 3 k. for 6 rubles. 30 k.?

397. Perform calculations using the calculator and round the result to thousandths:

3,281 0,57 + 4,356 0,278 -13,758:6,83.

398. Solve the problem:

1) To combat pests of gardens, a lime-sulfur broth is prepared, consisting of 6 parts of sulfur, 3 parts of quicklime and 50 parts of water (by weight). How much will it turn out kilograms decoction, if you take 8.8 kg more water than sulfur?

2) To prepare porcelain for 1 part of gypsum, take 2 parts of sand and 25 parts of clay (by weight). How many kilograms of porcelain will you get if you take 6.9 kg more clay than sand?

399. Do the following:

1) 7225:85 + 64 2345-248 838:619;
2) 54 3465-9025:95 + 360 272:712.

D 400. Take action:

a
401. Find the value of the difference:

402. Solve the equation:

404. One tractor driver plowed the fields, and the other the same field. How much of the field is left to plow?

406. Barrels of fuel are enough for work one engine for 7 hours and the other for 5 hours. What part of the fuel will remain from a full barrel after 2 hours of operation of the first engine and 3 hours of operation of the second engine?

406. For an expedition working in the taiga, a package with food was dropped from a helicopter, which fell to the ground after 3 seconds. From what height was this package dropped if in the first second it flew m, and in each next second it flew m more than in the previous one?

407. How long did it take to manufacture a part if it was processed on a lathe h, on a milling machine h and on a drilling machine h?

408. Find the value of the expression:

409. Two pedestrians came out of the two villages at the same time towards each other and met after 1.5 hours. The distance between the villages was 12.3 km. The speed of one pedestrian is 4.4 km / h. Find the speed of another pedestrian.

410. To make cherry jam for 3 parts of sugar, take 2 parts of berries (by weight). How many kilograms of sugar and how many kilograms of berries should you take to get 10 kg of jam, if it decreases 1.5 times during cooking?

411. Find the value of the expression:

a) (44.96 + 28.84: (13.7 -10.9)): 1.8;

b) 102.816: (3.2 6.3) + 3.84.

412. Solve the equation:

a) (x-4.7) 7.3 = 38.69; c) 23.5- (2, For + 1.2a) = 19.3;
b) (3.6-a) 5.8 = 14.5; d) 12.98- (3.8x- 1.3x) = 11.23.

A The branch of mathematics in which the properties of numbers and actions on them are studied is called number theory.

The beginning of the creation of number theory was laid by the ancient Greek scientists Pythagoras, Euclid, Eratosthenes and others.

Some problems in number theory are very simple - any sixth grader can understand them. But the solution of these problems is sometimes so difficult that it takes centuries, and there are still no answers to some questions. For example, ancient Greek mathematicians knew only one pair of friendly numbers - 220 and 284. And only in the 18th century. the famous mathematician, member of the St. Petersburg Academy of Sciences Leonard Euler found 65 more pairs of friendly numbers (one of them 17 296 and 18 416). However, a general way of finding pairs of friendly numbers is still not known.

Almost 250 years ago, Christian Goldbach, a member of the St. Petersburg Academy of Sciences, suggested that any odd number greater than 5 can be represented as the sum of three primes. For example: 21 = 3 + 7 + 11, 23 = 5 + 7 + 11, etc.

Only 200 years later, the remarkable Soviet mathematician, Academician Ivan Matveevich Vinogradov (1891-1983) managed to prove this assumption. But the statement "Any even number greater than 2 can be represented as the sum of two primes" (for example: 28 = 11 + 17, 56 = 19 + 37, 924 = 311 + 613, etc.) has not yet been proven ...