In this article we separate three basic concepts which are generally present in both Mathematics and Physics and Chemistry in the Enem tests. Exercises involving them exclusively do not present any difficulty to be solved, therefore, they are less frequent in the exam. These concepts usually appear indirectly. See what they are:
1st: Signal game
The set of integers is made up of all positive, negative, and zero integers. Due to the presence of negative numbers, which add rules to addition and multiplication, the basic operations between them present some differences that need to be adapted. Watch:
→ Sign Games: Sum of Whole Numbers
When adding two whole numbers, watch their signs to choose between the alternatives:
1) Equal signs
Add the numbers and keep the sign for the result. For example:
a) (– 16) + (– 44) = – 60
b) (+ 7) + (+ 13) = 20
Note that it is possible to write the same numeric expressions above in reduced form:
a) – 16 – 44 = – 60
b) 7 + 13 = 20
in short: When you add two negative numbers, the result will be negative. By adding two positive numbers, the result will be positive.
2) Different signs
Subtract the numbers and keep the sign of whichever is greater in magnitude, that is, whichever is greater regardless of sign. For example:
a) (+ 16) + (– 44) = – 28
b) (– 7) + (+ 13) = 6
Note that –44 is less than +16 simply because it is negative. However, ignoring the signs, 44 is greater than 16. Therefore, 44 is the largest in module and, therefore, its sign prevails in the result. You can also write the same numeric expressions as above in reduced form:
a) 16 - 44 = - 28
b) – 7 + 13 = 6
in short: when adding two numbers whose signs are different, subtract the numbers and keep for the result the sign of the one that is greater in modulus.
The same rules apply for numeric expressions that involve more than two numbers to be added, so to solve them, just add their terms two by two. It is not necessary to talk about subtraction, because, from the set of whole numbers, subtraction is an addition between numbers with different signs.
For more information and examples about the sum, read the text Operations between integers.
→ Sign Games: Integer Multiplication
The rules for signs in integer multiplication are the same for division. Check out:
1) Equal signs
When the signs are equals in a multiplication, the result will always be positive. For example:
a) (+ 16)·(+ 4) = + 64
b) (– 8)·(– 8) = + 64
Note that when you multiply two negative numbers, the result will be positive because these two numbers have equal signs. We advise you to always use parentheses for multiplication.
2) Different signs
When the signs are many different in a multiplication, the result will always be negative. For example:
a) 16·(– 2) = – 32
b) (– 7)·(+ 3) = – 21
The same rules apply for division. For more information on integer multiplication and sign play, read the text: Whole number multiplication.
2nd: Equations
Since this text deals with basic concepts, we will discuss definitions and properties of first-degree equations. To solve quadratic equations, we suggest reading the text Bhaskara's formula.
To solve a equation, that is, to find the numerical value of the unknown, it is necessary to complete the following three steps:
1) Put all the terms that have an unknown in the first member;
2) Put all the terms that no have unknowns in the second member;
3) Perform the resulting calculations;
4) Isolate the unknown.
For example:
12x - 4 = 6x + 20
Steps 1 and 2: 12x - 6x = 20 + 4
Step 3: 6x = 24
Step 4: x = 24
6
x = 4
For more information on troubleshooting equations and some examples, read the texts:
1) 1st degree equation with one unknown
2) Problems involving the use of equations
3) Introduction to the 1st degree equation
3rd: Rule of three simple
THE rule of three it is thus known for relating four values referring to two quantities, so that three of them are known. It works only for proportional quantities, that is, for that quantity that varies proportionally to the variation of another quantity.
the greatness Travelled distance, for example, is proportional to the magnitude Speed. Over a period of time, the higher the speed, the longer the distance covered.
Example:
Let's say a man is used to commuting to work inside the city at an average speed of 40 km/h. Knowing that the home-work route is 20 km, how many kilometers would it reach if it were at 110 km/h?
Note that speed and distance covered are proportional. Obviously, within the same amount of time, this man will reach a much greater distance by walking at 110 km/h. To find this distance, we can set up the following table:
Now, just set up an equality, following the same position of the elements in the table, and use the rule "Product of extremes by means".
40 = 20
110x
40x = 20·110
40x = 2200
x = 2200
40
x = 55
For more information, discussions and examples regarding the simple and compound rule of three, see the texts:
The) Simple three rule
B) Percentage using rule of three
ç) rule of three compound
To deepen your knowledge about proportionality, which underlies the rule of three, read the texts:
The) Proportional numbers
B) Proportionality between quantities
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/tres-conceitos-basicos-matematica-para-enem.htm
