In this article, we will summarize all the basic rules of mathematics that will accompany you in every exercise - the commutative property of addition, the commutative property of multiplication, the distributive property, and all the others!
Shall we begin?
The commutative property can be found in two cases, with addition and with multiplication.
You can read general features of the commutative property at this link.
Thanks to it, we can change the place of the addends without altering the result.
The property is also valid in algebraic expressions.
Rule:
x\cdotnumber~any=number~any\cdot x
Click here to see a more detailed explanation about the commutative property of addition.
Thanks to it, we can change the place of the factors without altering the product.
The property is also valid in algebraic expressions.
Rule:
x\cdotnumber~any=number~any\cdot x
Click here to see a more detailed explanation about the commutative property of multiplication.
In the same way, the commutative property can also be found in two cases, with division and with multiplication.
You can read general features of the distributive property at this link.
It allows us to distribute - it separates an exercise with several numbers and multiplication operations into another simpler one that has numbers and addition or subtraction operations without changing the result.
The property is also valid in algebraic expressions.
Multiply the number outside the parentheses by the first number inside the parentheses and, to this product, add or subtract - according to the sign of the exercise - the product of the number outside the parentheses with the second number inside the parentheses.
Additionally
The distributive property allows us to make small changes to the numbers in the exercise to round them as much as possible, making the exercise easier.
For example:
In the exercise:
We can change the number to the expression
and rewrite the exercise:
Then continue with the distributive property:
You can read about the distributive property of multiplication at this link.
We will choose the expression that is inside the parentheses - we will take one element at a time and multiply it in the given order by each of the elements in the second expression, keeping the subtraction and addition signs.
Then we will do the same with the second element of the chosen expression.
You can read about the extended distributive property right here.
Thanks to it, we can round the number we want to divide, always taking into account that the rounded number can actually be divided by the other.
This is done without affecting the original number to preserve its value.
For example:
We will round up the number to . To preserve the value of we will write
We will get:
We will divide by and subtract the quotient of divided by
We will get:
Click here to see a more detailed explanation of the commutative property of division.
Solve:
\( 2-3+1 \)
Solve:
\( 3-4+2+1 \)
Solve:
\( -5+4+1-3 \)
Solve the exercise:
84:4=
Solve the following exercise
?=24:12
Solve:
We use the substitution property and add parentheses for the addition operation:
Now, we solve the exercise according to the order of operations:
0
Solve:
We will use the substitution property to arrange the exercise a bit more comfortably, we will add parentheses to the addition operation:
We first solve the addition, from left to right:
And finally, we subtract:
2
Solve:
According to the order of operations, addition and subtraction are on the same level and, therefore, must be resolved from left to right.
However, in the exercise we can use the substitution property to make solving simpler.
-5+4+1-3
4+1-5-3
5-5-3
0-3
-3
Solve the exercise:
84:4=
There are several ways to solve the following exercise,
We will present two of them.
In both ways, we begin by decomposing the number 84 into smaller units; 80 and 4.
Subsequently we are left with only the 80.
Continuing on with the first method, we will then further decompose 80 into smaller units;
We know that:
And therefore, we are able to reduce the exercise as follows:
Eventually we are left with
which is equal to 20
In the second method, we decompose 80 into the following smaller units:
We know that:
And therefore:
which is also equal to 20
Now, let's remember the 1 from the first step and add it in to our above answer:
Thus we are left with the following solution:
21
Solve the following exercise
?=24:12
We will use the distributive property of division and split the number 24 into a sum of 12 and 12, which makes the division operation easier and allows us to solve the exercise without a calculator.
Note - it's best to choose to split the number based on knowledge of multiples. In this case of the number 12 because we need to divide by 12.
Reminder - The distributive property of division actually allows us to split the larger term in a division problem into a sum or difference of smaller numbers, which makes the division operation easier and allows us to solve the exercise without a calculator
We will use the formula of the distributive property
(a+b):c=a:c+b:c
Therefore the answer is section a - 2.
2
Solve the following exercise
?=93:3
\( 100-(30-21)= \)
\( 11\times3+7= \)
\( 12:(2\times2)= \)
\( 12\times13+14= \)
Solve the following exercise
?=93:3
We will use the distributive property of division and split the number 93 into a sum of 90 and 3, which makes the division operation easier and allows us to solve the exercise without a calculator.
Note - it's best to choose to split the number based on knowledge of multiples. In this case, we use 3 because we need to divide by 3. Additionally, in this case, splitting by tens and ones is suitable and makes the division operation easier.
Reminder - The distributive property of division essentially allows us to split the larger term in the division problem into a sum or difference of smaller numbers, which makes the division operation easier and allows us to solve the exercise without a calculator
We will use the formula of the distributive property
(a+b):c=a:c+b:c
Therefore, the answer is option B - 31.
31
According to the order of operations, we first solve the exercise within parentheses:
Now we obtain:
In this exercise, it is not possible to use the substitution property, therefore we solve it as is from left to right according to the order of arithmetic operations.
That is, we first solve the multiplication exercise and then we add:
According to the order of operations, we first solve the exercise within parentheses:
Now we divide:
According to the order of operations, we start with the multiplication exercise and then with the addition.
Now we get the exercise:
\( 12\times5\times6= \)
\( 13+2+8= \)
\( 133+30= \)
\( 13+5+5= \) ?
\( 13+7+100= \) ?
According to the rules of the order of operations, we solve the exercise from left to right:
360
We will use the commutative property and first solve the addition exercise on the right:
Now we get:
23
In order to solve the question, we first use the distributive property for 133:
We then use the distributive property for 33:
We rearrange the exercise into a more practical form:
We solve the middle exercise:
Which results in the final exercise as seen below:
163
?
First, solve the right-hand exercise since adding the numbers together will give us a round number:
Now we have an easier exercise to solve:
23
?
First, we'll solve the left-hand side of the exercise since adding these numbers together gives us a round number:
This leaves us with a much easier exercise to solve:
120