>>Math: Adding numbers with different signs
33. Addition of numbers with different signs
If the air temperature was equal to 9 °C, and then it changed to - 6 °C (i.e., decreased by 6 °C), then it became equal to 9 + (- 6) degrees (Fig. 83).
To add the numbers 9 and - 6 using , you need to move point A (9) to the left by 6 unit segments (Fig. 84). We get point B (3).
This means 9+(- 6) = 3. The number 3 has the same sign as the term 9, and its module equal to the difference between the moduli of terms 9 and -6.
Indeed, |3| =3 and |9| - |- 6| = = 9 - 6 = 3.
If the same air temperature of 9 °C changed by -12 °C (i.e., decreased by 12 °C), then it became equal to 9 + (-12) degrees (Fig. 85). Adding the numbers 9 and -12 using the coordinate line (Fig. 86), we get 9 + (-12) = -3. The number -3 has the same sign as the term -12, and its module is equal to the difference between the modules of the terms -12 and 9.
Indeed, | - 3| = 3 and | -12| - | -9| =12 - 9 = 3.
To add two numbers with different signs, you need to:
1) subtract the smaller one from the larger module of the terms;
2) put in front of the resulting number the sign of the term whose modulus is greater.
Usually, the sign of the sum is first determined and written, and then the difference in modules is found.
For example:
1) 6,1+(- 4,2)= +(6,1 - 4,2)= 1,9,
or shorter 6.1+(- 4.2) = 6.1 - 4.2 = 1.9;
When adding positive and negative numbers you can use micro calculator. To enter a negative number into a microcalculator, you need to enter the modulus of this number, then press the “change sign” key |/-/|. For example, to enter the number -56.81, you need to press the keys sequentially: | 5 |, | 6 |, | ¦ |, | 8 |, | 1 |, |/-/|. Operations on numbers of any sign are performed on a microcalculator in the same way as on positive numbers.
For example, the sum -6.1 + 3.8 is calculated using program
? The numbers a and b have different signs. What sign will the sum of these numbers have if the larger module is negative?
if the smaller modulus is negative?
if the larger modulus is a positive number?
if the smaller modulus is a positive number?
Formulate a rule for adding numbers with different signs. How to enter a negative number into a microcalculator?
TO 1045. The number 6 was changed to -10. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is it equal to sum 6 and -10?
1046. The number 10 was changed to -6. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of 10 and -6?
1047. The number -10 was changed to 3. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of -10 and 3?
1048. The number -10 was changed to 15. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of -10 and 15?
1049. In the first half of the day the temperature changed by - 4 °C, and in the second half - by + 12 °C. By how many degrees did the temperature change during the day?
1050. Perform addition:
1051. Add:
a) to the sum of -6 and -12 the number 20;
b) to the number 2.6 the sum is -1.8 and 5.2;
c) to the sum -10 and -1.3 the sum of 5 and 8.7;
d) to the sum of 11 and -6.5 the sum of -3.2 and -6.
1052. Which number is 8; 7.1; -7.1; -7; -0.5 is the root equations- 6 + x = -13.1?
1053. Guess the root of the equation and check:
a) x + (-3) = -11; c) m + (-12) = 2;
b) - 5 + y=15; d) 3 + n = -10.
1054. Find the meaning of the expression:
1055. Follow the steps using a microcalculator:
a) - 3.2579 + (-12.308); d) -3.8564+ (-0.8397) +7.84;
b) 7.8547+ (- 9.239); e) -0.083 + (-6.378) + 3.9834;
c) -0.00154 + 0.0837; e) -0.0085+ 0.00354+ (- 0.00921).
P 1056. Find the value of the sum:
1057. Find the meaning of the expression:
1058. How many integers are located between the numbers:
a) 0 and 24; b) -12 and -3; c) -20 and 7?
1059. Imagine the number -10 as the sum of two negative terms so that:
a) both terms were integers;
b) both terms were decimal fractions;
c) one of the terms was a regular ordinary fraction.
1060. What is the distance (in unit segments) between the points of the coordinate line with coordinates:
a) 0 and a; b) -a and a; c) -a and 0; d) a and -Za?
M 1061. Radii of geographical parallels earth's surface, on which the cities of Athens and Moscow are located, are respectively 5040 km and 3580 km (Fig. 87). How much shorter is the Moscow parallel than the Athens parallel?
1062. Write an equation to solve the problem: “A field with an area of 2.4 hectares was divided into two sections. Find square each site, if it is known that one of the sites:
a) 0.8 hectares more than another;
b) 0.2 hectares less than another;
c) 3 times more than another;
d) 1.5 times less than another;
e) constitutes another;
e) is 0.2 of the other;
g) constitutes 60% of the other;
h) is 140% of the other.”
1063. Solve the problem:
1) On the first day, the travelers traveled 240 km, on the second day 140 km, on the third day they traveled 3 times more than on the second, and on the fourth day they rested. How many kilometers did they travel on the fifth day, if for 5 days they drove an average of 230 km per day?
2) Father’s monthly income is 280 rubles. My daughter's scholarship is 4 times less. How much does a mother earn per month if there are 4 people in the family? youngest son- a schoolboy and each person receives an average of 135 rubles?
1064. Follow these steps:
1) (2,35 + 4,65) 5,3:(40-2,9);
2) (7,63-5,13) 0,4:(3,17 + 6,83).
1066. Present each of the numbers as a sum of two equal terms:
1067. Find the value of a + b if:
a) a= -1.6, b = 3.2; b) a=- 2.6, b = 1.9; V) 
1068. There were 8 apartments on one floor of a residential building. 2 apartments had a living area of 22.8 m2, 3 apartments - 16.2 m2, 2 apartments - 34 m2. What living area did the eighth apartment have if on this floor on average each apartment had 24.7 m2 of living space?
1069. The freight train consisted of 42 cars. There were 1.2 times more covered cars than platforms, and the number of tanks was equal to the number of platforms. How many cars of each type were on the train?
1070. Find the meaning of the expression 
N.Ya.Vilenkin, A.S. Chesnokov, S.I. Shvartsburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school
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numbers with different signs
To ensure that the student, in less time than before, masters a large amount of knowledge, thorough and effective - this is one of the main tasks of modern pedagogy. In this regard, there is a need to begin studying new things by repeating old, already studied, known material on a given topic. In order for the repetition to proceed quickly and in order to have the most obvious connection between the new and the old, it is necessary to organize the recording of the studied material in a special way when explaining.
As an example, I will tell you how I teach students to add and subtract numbers with different signs using a coordinate line. Before studying the topic directly and during lessons in the 5th and 6th grades, I pay a lot of attention to the structure of the coordinate line. Before starting to study the topic “Addition and subtraction of numbers with different signs,” it is necessary that each student firmly knows and is able to answer the following questions:
1) How is the coordinate line constructed?
2) How are the numbers located on it?
3) What is the distance from the number 0 to any number?
Students should understand that moving along a straight line to the right leads to an increase in the number, i.e. the addition action is performed, and to the left - to its decrease, i.e. the action of subtracting numbers is performed. To prevent boredom from working with the coordinate line, there are many non-standard game problems. For example, this one.
A straight line has been drawn along the highway. The length of one unit segment is 2 m. Everyone moves only along a straight line. On number 3 are Gena and Cheburashka. They walked in different directions at the same time and stopped at the same time. Gena walked twice as far as Cheburashka and ended up on number 11. What number did Cheburashka end up on? How many meters did Cheburashka walk? Which of them walked slower and by how much?(Non-standard mathematics at school. - M., Laida, 1993, No. 62).
When I am firmly convinced that all students can cope with movements along a straight line, and this is very important, I move directly to teaching adding and subtracting numbers at the same time.
Each student is given a reference note. By analyzing the provisions of the notes and relying on existing geometric visual pictures of the coordinate line, students gain new knowledge. (The outline is shown in the figure). Studying a topic begins by writing down in a notebook the questions that will be discussed.
1 . How to perform addition using a coordinate line? How to find an unknown term? Let's look at the relevant part of the outline??. Let us remember that a add b- it means to increase a on b and movement along the coordinate line occurs to the right. We recall how the components of addition and the laws of addition are named and calculated, as well as the properties of zero during addition. Are these parts?? And?? notes. Therefore, the following questions written in the notebook are:
1). Addition is movement to the right.
SL. + SL. = C; SL. = C - SL.
2). Addition laws:
1) displacement law: a+ b= b+ a;
2) combination law: (a+ b) + c= a+ (b+ c) = (a+ c) + b
3). Properties of zero during addition: a+ 0= a; 0+ a= a; a+ (- a) = 0.
4). Subtraction is a movement to the left.
U. - V. = R.; U. = V. + R.; V. = U. - R.
5). Addition can be replaced by subtraction, and subtraction can be replaced by addition.
4 + 3 = - 1 3 - 4 = -1
4 + 3 = 3 + (- 4) = 3 - 4 = - 1
according to the commutative law of addition
6). This is how the parentheses open:
+ (a+ b+ c) = + a+ b+ c
"gentleman"
- (a + b + c) = - a - b - c
"robber"
2 . Laws of addition.
3 . List the properties of zero during addition.
4 . How to subtract numbers using a coordinate line? Rules for finding unknown subtrahends and minuends.
5 . How do you go from addition to subtraction and from subtraction to addition?
6 . How to open parentheses preceded by: a) a plus sign; b) minus sign?
The theoretical material is quite voluminous, but since each part of it is connected and, as it were, “flows” from one another, memorization occurs successfully. Working with notes doesn't end there. Each part of the outline is associated with the text of the textbook, which is read in class. If after this the student believes that the part being analyzed is completely clear to him, then he lightly paints over the text of the summary in the appropriate frame, as if saying: “I understand this.” If there is something unclear, then the frame is not painted over until everything becomes clear. The white part of the notes is the signal “Figure it out!”
The teacher's goal, which should be achieved by the end of the lesson, is this: students, leaving the lesson, should remember that addition is movement along a coordinate line to the right, and subtraction is to the left. All students learned to open brackets. The remaining time of the lesson is devoted to opening the brackets. We open brackets orally and in writing in tasks like:
|
); - 20 + (- 7 + (- 5)). |
||||
Homework assignment. Answer the questions written in the notebook by reading the textbook paragraphs indicated in the notes.
In the next lesson we will practice the algorithm for adding and subtracting numbers. Each student has a card on their desk with instructions:
1) Write down an example.
2) Open the brackets, if any.
3) Draw a coordinate line.
4) Mark the first number on it without scale.
5) If the number is followed by a “+” sign, then move to the right, and if there is a “-” sign, then move to the left by as many unit segments as the second term contains. Draw it diagrammatically and put a sign next to the number you are looking for?
6) Ask the question “Where is zero?”
7) Determine the sign of the number that has question mark, which is a solution, like this: if? is to the right of 0, then the answer has a + sign, but what if? is to the left of 0, then the answer has a sign - . Write the found sign in the answer after the = sign.
8) Mark three segments on the drawing.
9) Find the length of the segment from zero to sign?
Example 1.- 35 + (- 9) = - 35 - 9 = - 44.
1. I copy the example and open the parentheses.
2. I draw a picture and reason like this:
a) I mark - 35 and move to the left by 9 unit segments; I put a sign next to the desired number?;
b) I ask myself: “Where is zero?” I answer: “Zero is to the right - 35 by 35 unit segments, which means the sign of the answer is -, so? to the left of zero";
c) looking for the distance from 0 to the sign?. To do this, I calculate 35 + 9 = 44 and assign the resulting number in response to the - sign.
Example 2.- 35 + 9.
Example 3. 9 - 35.
We solve these examples using similar reasoning to Example 1. There cannot be other cases of arrangement of numbers, and each picture corresponds to one of the rules given in the textbook and requiring memorization. It has been verified (and repeatedly) that this method of addition is more rational. In addition, it allows you to add numbers even when the student thinks that he does not remember a single rule. This method works when working with fractions, you just need to bring them to common denominator and then draw a picture. For example,
Everyone uses the “instruction” card as long as there is a need for it.
Such work replaces the tedious and monotonous action of counting according to the rules of a living and actively working thought. There are many advantages: no need to cram and feverishly figure out which rule to apply; The structure of the coordinate line is easy to remember, and this is both in algebra and in geometry when calculating the value of a segment when a point on a line lies between two other points. This technique is effective both in classes with in-depth study of mathematics, and in classes with age norms, and even in correction classes.
Almost the entire mathematics course is based on operations with positive and negative numbers. After all, as soon as we begin to study the coordinate line, numbers with plus and minus signs begin to appear to us everywhere, in every new topic. There is nothing easier than adding ordinary positive numbers together; it is not difficult to subtract one from the other. Even arithmetic with two negative numbers is rarely a problem.
However, many people get confused about adding and subtracting numbers with different signs. Let us recall the rules by which these actions occur.
Adding numbers with different signs
If to solve a problem we need to add a negative number “-b” to some number “a”, then we need to act as follows.
- Let's take the modules of both numbers - |a| and |b| - and compare these absolute values with each other.
- Let us note which of the modules is larger and which is smaller, and subtract from greater value less.
- Let us put in front of the resulting number the sign of the number whose modulus is greater.
This will be the answer. We can put it more simply: if in the expression a + (-b) the modulus of the number “b” is greater than the modulus of “a,” then we subtract “a” from “b” and put a “minus” in front of the result. If the module “a” is greater, then “b” is subtracted from “a” - and the solution is obtained with a “plus” sign.
It also happens that the modules turn out to be equal. If so, then we can stop at this point - we are talking about opposite numbers, and their sum will always be equal to zero.
Subtracting numbers with different signs
We've dealt with addition, now let's look at the rule for subtraction. It is also quite simple - and in addition, it completely repeats a similar rule for subtracting two negative numbers.
In order to subtract from a certain number “a” - arbitrary, that is, with any sign - a negative number “c”, you need to add to our arbitrary number “a” the number opposite to “c”. For example:
- If “a” is a positive number, and “c” is negative, and you need to subtract “c” from “a”, then we write it like this: a – (-c) = a + c.
- If “a” is a negative number, and “c” is positive, and “c” needs to be subtracted from “a”, then we write it as follows: (- a)– c = - a+ (-c).
Thus, when subtracting numbers with different signs, we end up returning to the rules of addition, and when adding numbers with different signs, we return to the rules of subtraction. Memorizing these rules allows you to solve problems quickly and easily.
developing knowledge about the rule for adding numbers with different signs, the ability to apply it in the simplest cases;
development of skills to compare, identify patterns, generalize;
fostering a responsible attitude towards educational work.
Equipment: multimedia projector, screen.
Lesson type: lesson of learning new material.
PROGRESS OF THE LESSON
1. Organizational moment.
Stand up straight
They sat down quietly.
The bell has now rung,
Let's start our lesson.
Guys! Today guests came to our lesson. Let's turn to them and smile at each other. So, we begin our lesson.
Slide 2- Epigraph of the lesson: “He who does not notice anything does not study anything.
He who doesn’t study anything is always whining and bored.”
Roman Sef ( children's writer)
Slad 3 - I suggest playing the game “On the contrary”. Rules of the game: you need to divide the words into two groups: win, lie, warmth, gave, truth, good, loss, took, evil, cold, positive, negative.
There are many contradictions in life. With their help we determine surrounding reality. For our lesson I need the last one: positive - negative.
What are we talking about in mathematics when we use these words? (About numbers.)
The great Pythagoras said: “Numbers rule the world.” I propose to talk about the most mysterious numbers in science - numbers with different signs. - Negative numbers appeared in science as the opposite of positive numbers. Their path to science was difficult because even many scientists did not support the idea of their existence.
What concepts and quantities do people measure with positive and negative numbers? (charges of elementary particles, temperature, losses, height and depth, etc.)
Slide 4- Words with opposite meanings are antonyms (table).
2. Setting the topic of the lesson.
Slide 5 (working with a table)– What numbers were studied in previous lessons?
– What tasks related to positive and negative numbers can you perform?
– Attention to the screen. (Slide 5)
– What numbers are presented in the table?
– Name the modules of numbers written horizontally.
– Please indicate greatest number, indicate the number with the largest modulus.
– Answer the same questions for numbers written vertically.
– Do the largest number and the number with the largest absolute value always coincide?
– Find the sum of positive numbers, the sum of negative numbers.
– Formulate the rule for adding positive numbers and the rule for adding negative numbers.
– What numbers are left to add?
– Do you know how to fold them?
– Do you know the rule for adding numbers with different signs?
– Formulate the topic of the lesson.
– What goal will you set for yourself? .Think about what we will do today? (Children's answers). Today we continue to learn about positive and negative numbers. The topic of our lesson is “Adding numbers with different signs.” Our goal is to learn how to add numbers with different signs without errors. Write down the date and topic of the lesson in your notebook.
3.Work on the topic of the lesson.
Slide 6.– Using these concepts, find the results of adding numbers with different signs on the screen.
– What numbers are the result of adding positive numbers and negative numbers?
– What numbers are the result of adding numbers with different signs?
– What determines the sign of the sum of numbers with different signs? (Slide 5)
– From the term with the largest modulus.
- It's like a tug of war. The strongest wins.
Slide 7- Let's play. Imagine that you are in a tug of war. . Teacher. Rivals usually meet in competitions. And today we will visit several tournaments with you. The first thing that awaits us is the final of the tug-of-war competition. Meet Ivan Minusov at number -7 and Petr Plyusov at number +5. Who do you think will win? Why? So, Ivan Minusov won, he really turned out to be stronger than his opponent, and was able to drag him to his negative side exactly two steps.
Slide 8.- . Now let's go to other competitions. The final of the shooting competition is before you. The best in this event were Minus Troikin with three balloons and Plus Chetverikov, who has four in stock balloon. And here guys, who do you think will be the winner?
Slide 9- The competitions showed that the strongest wins. So it is when adding numbers with different signs: -7 + 5 = -2 and -3 + 4 = +1. Guys, how do numbers with different signs add up? Students offer their own options.
The teacher formulates the rule and gives examples.
10 + 12 = +(12 – 10) = +2
4 + 3,6 = -(4 – 3,6) = -0,4
During the demonstration, students can comment on the solution appearing on the slide.
Slide 10- Teacher, let’s play another game “Battleship”. An enemy ship is approaching our coast, it must be knocked out and sunk. For this we have a gun. But to hit the target you need to make accurate calculations. Which ones you will see now. Are you ready? Then go ahead! Please do not be distracted, the examples change exactly after 3 seconds. Is everyone ready?
Students take turns coming to the board and calculating the examples that appear on the slide. – Name the stages of completing the task.
Slide 11- Work according to the textbook: p. 180 p. 33, read the rule for adding numbers with different signs. Comments on the rule.
– What is the difference between the rule proposed in the textbook and the algorithm you compiled? Consider the examples in the textbook with commentary.
Slide 12- Teacher - Now guys, let's conduct experiment. But not chemical, but mathematical! Let's take the numbers 6 and 8, plus and minus signs and mix everything well. Let's get four experimental examples. Do them in your notebook. (two students solve on the wings of the board, then the answers are checked). What conclusions can be drawn from this experiment?(The role of signs). Let's conduct 2 more experiments , but with your numbers (1 person at a time goes to the board). Let's come up with numbers for each other and check the results of the experiment (mutual check).
Slide 13 .- The rule is displayed on the screen in poetic form .
4. Reinforcing the topic of the lesson.
Slide 14 – Teacher - “All kinds of signs are needed, all kinds of signs are important!” Now, guys, we will divide you into two teams. Boys will be on Santa Claus's team, and girls will be on Sunny's team. Your task, without calculating the examples, is to determine which of them will have negative answers and which will have positive answers and write down the letters of these examples in a notebook. Boys are respectively negative, and girls are positive (cards from the application are issued). A self-test is being carried out.
Well done! Your sense of signs is excellent. This will help you complete the next task
Slide 15 - Physical education. -10, 0,15,18,-5,14,0,-8,-5, etc. (negative numbers - squat, positive numbers - pull up, jump)
Slide 16-Solve 9 examples yourself (task on cards in the app). 1 person at the board. Do a self-test. The answers are displayed on the screen, and students correct mistakes in their notebooks. Raise your hands if you have it right. (Marks are given only for good and excellent results)
Slide 17-Rules help us solve examples correctly. Let's repeat them. On the screen is an algorithm for adding numbers with different signs.
5.Organization of independent work.
Slide 18 -Fonline work through the game “Guess the word”(task on cards in the appendix).
Slide 19 - The score for the game should be “A”
Slide 20 -A now, attention. Homework. Homework should not cause you any difficulties.
Slide 21 - Laws of addition in physical phenomena. Come up with examples of adding numbers with different signs and ask them to each other. What new have you learned? Have we achieved our goal?
Slide 22 - That's the end of the lesson, let's sum it up now. Reflection. The teacher comments and grades the lesson.
Slide 23 - Thank you for your attention!
I wish you to have more positive and less negative in your life. I want to tell you guys, thank you for your active work. I think that you can easily apply the acquired knowledge in subsequent lessons. The lesson is over. Thank you all very much. Goodbye!
If the air temperature was 9°C, and then it changed to -6°C (i.e., decreased by 6°C), then it became equal to 9 + (-6) degrees (Fig. 83).
Rice. 83
To add the numbers 9 and -6 using the coordinate line, you need to move point A(9) to the left by 6 unit segments (Fig. 84). We get point B(3).
Rice. 84
This means 9 + (-6) = 3. The number 3 has the same sign as the term 9, and its module is equal to the difference between the modules of the terms 9 and -6.
Indeed, |3| = 3 and |9| - |-6| = 9 - 6 = 3.
If the same air temperature of 9°C changed by -12°C (i.e., decreased by 12°C), then it became equal to 9 + (-12) degrees (Fig. 85).
Rice. 85
Adding the numbers 9 and -12 using the coordinate line (Fig. 86), we get 9 + (-12) = -3. The number -3 has the same sign as the term -12, and its module is equal to the difference between the modules of the terms -12 and 9.
Rice. 86
Indeed, |-3| = 3 and |-12| - |-9| = 12 - 9 = 3.
Usually, the sign of the sum is first determined and written, and then the difference in modules is found.
For example:
You can use a calculator to add positive and negative numbers. To enter a negative number into a microcalculator, you need to enter the modulus of this number, then press the “change sign” key. For example, to enter the number -56.81, you need to press the keys sequentially: . Operations on numbers of any sign are performed on a microcalculator in the same way as on positive numbers. For example, the sum -6.1 + 3.8 is calculated using the program
In short, this program is written like this: 
.
Self-test questions
- The numbers a and b have different signs. What sign will the sum of these numbers have if the larger module is negative? if the smaller modulus is negative? if the larger modulus is a positive number? if the smaller modulus is a positive number?
- Formulate a rule for adding numbers with different signs.
- How to enter a negative number into a microcalculator?
Do the exercises
1061. The number 6 was changed to -10. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of 6 and -10?
1062. The number 10 was changed to -6. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of 10 and -6?
1063. The number -10 was changed to 3. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of -10 and 3?
1064. The number -10 was changed to 15. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of -10 and 15?
1065. In the first half of the day the temperature changed by -4°C, and in the second - by +12°C. By how many degrees did the temperature change during the day?
1066. Perform addition:
- a) 26 + (-6);
- b) -70 + 50;
- c) -17 + 30;
- d) 80 + (-120);
- e) -6.3 + 7.8;
- e) -9 + 10.2;
- g) 1 + (-0.39);
- h) 0.3 + (-1.2);
1067. Add:
- a) to the sum of -6 and -12 the number 20;
- b) to the number 2.6 the sum is -1.8 and 5.2;
- c) to the sum -10 and -1.3 the sum of 5 and 8.7;
- d) to the sum of 11 and -6.5 the sum of -3.2 and -6.
1068. Which number is 8? 7.1; -7.1; -7; Is -0.5 the root of the equation -6 + x = -13.1?
1069. Guess the root of the equation and check:
- a) x + (-3) = -11;
- b) -5 + y = 15;
- c) t + (-12) = 2;
- d) 3 + n = -10.
1070. Find the meaning of the expression:
1071. Follow these steps using a microcalculator:
- a) -3.2579 + (-12.308);
- b) 7.8547 + (-9.239);
- c) -0.00154 + 0.0837;
- d) -3.8564 + (-0.8397) + 7.84;
- e) -0.083 + (-6.378) + 3.9834;
- e) -0.0085 + 0.00354 + (-0.00921).
1072. Find the value of the sum:
1073. Find the meaning of the expression:
1074. How many integers are located between the numbers:
- a) 0 and 24;
- b) -12 and -3;
- c) -20 and 7?
1075. Imagine the number -10 as the sum of two negative terms so that:
- a) both terms were integers;
- b) both terms were decimal fractions;
- c) one of the terms was a proper ordinary fraction.
1076. What is the distance (in unit segments) between points on a coordinate line with coordinates:
- a) 0 and a;
- b) -a and a;
- c) -a and 0;
- d) a and -Za?
1077. The radii of the geographical parallels of the earth's surface on which the cities of Athens and Moscow are located are respectively equal to 5040 km and 3580 km (Fig. 87). How much shorter is the Moscow parallel than the Athens parallel?
Rice. 87
1078. Write an equation to solve the problem: “A field of 2.4 hectares was divided into two sections. Find the area of each plot if it is known that one of the plots:
1079. Solve the problem:
- On the first day, the travelers traveled 240 km, on the second day 140 km, on the third day they traveled 3 times more than on the second, and on the fourth day they rested. How many kilometers did they travel on the fifth day, if for 5 days they drove an average of 230 km per day?
- A farmer with two sons placed the collected apples in 4 containers, an average of 135 kg each. The farmer collected 280 kg of apples, and the youngest son collected 4 times less. How many kilograms of apples did the eldest son collect?
1080. Follow these steps:
- (2,35 + 4,65) 5,3: (40 - 2,9);
- (7,63 - 5,13) 0,4: (3,17 + 6,83).
1081. Perform addition:
1082.
Imagine each of the numbers as the sum of two equal terms: 10; -8; -6.8; 
.
1083. Find the value of a + b if:
1084. There were 8 apartments on one floor of a residential building. There were 2 apartments with a living area of 22.8 m2, 3 apartments with 16.2 m2, and 2 apartments with 34 m2. What living area did the eighth apartment have if on this floor on average each apartment had 24.7 m2 of living space?
1085. The freight train consisted of 42 cars. There were 1.2 times more covered cars than platforms, and the number of tanks was equal to the number of platforms. How many cars of each type were on the train?
1086.
Find the meaning of the expression 