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Find the following sum using the commutative and associative property of addition:
$2012 + 176 + 244 + 308$
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Hint: In this problem, first we will find the required sum by using commutative property. Then, we will find the required sum by using associative property. Commutative property of addition states that numbers can be added in any order and we will get the same answer. Associative property of addition states that numbers can be added by regrouping and we will get the same answer.
Complete step by step solution: For every real number $a$ and $b$, we can write $a + b = b + a$. This is the commutative property of addition for two numbers. In the given problem, we need to find the sum of four numbers by using commutative property. For this, first we will make one group of first two numbers and one group of the last two numbers. That is, we write the given sum as $\left( {2012 + 176} \right) + \left( {244 + 308} \right)$.
Now we are going to use commutative property for two numbers. That is, $a + b = b + a$. Therefore,
$\left( {2012 + 176} \right) + \left( {244 + 308} \right)$
$ = \left( {176 + 2012} \right) + \left( {308 + 244} \right)$
$ = 2188 + 552$
$ = 552 + 2188\quad \left[ {\because a + b = b + a} \right]$
$ = 2740$
Therefore, the required sum is $2740$ by using commutative property.
Now we will find the required sum by using associative property. For every real number $a,b$ and $c$, we can write $\left( {a + b} \right) + c = a + \left( {b + c} \right)$. This is the associative property of addition for three numbers. In the given problem, we need to find the sum of four numbers by using associative property. For this, we will consider one group of the first three numbers in which we will make one group of the first two numbers. That is, we write the given sum as $\left[ {\left( {2012 + 176} \right) + 244} \right] + 308$.
Now we are going to use associative property for three numbers. That is, $\left( {a + b} \right) + c = a + \left( {b + c} \right)$.
$\left[ {\left( {2012 + 176} \right) + 244} \right] + 308$
$ = \left[ {2012 + \left( {176 + 244} \right)} \right] + 308\quad \left[ {\because \left( {a + b} \right) + c = a + \left( {b + c} \right)} \right]$
$ = 2012 + \left[ {\left( {176 + 244} \right) + 308} \right]$
$ = 2012 + 176 + \left( {244 + 308} \right)\quad \left[ {\because \left( {a + b} \right) + c = a + \left( {b + c} \right)} \right]$
$ = \left( {2012 + 176} \right) + \left( {244 + 308} \right)$
$ = 2188 + 552$
$ = 2740$
Therefore, the required sum is $2740$ by using associative property.
Note: Commutative property involves two or more numbers. Associative property can be used for three or more numbers. Commutative and associative properties can be used with addition and multiplication. We cannot use commutative property when we are talking about subtraction and division. For example, let us take two non-zero distinct real numbers $a = 4$ and $b = 5$. Then, $a - b = 4 - 5 = - 1$ and $b - a = 5 - 4 = 1$. Now we can say that $a - b \ne b - a$. Therefore, we can say that commutative property is not true. Similarly we cannot use associative property when we are talking about subtraction and division.
Complete step by step solution: For every real number $a$ and $b$, we can write $a + b = b + a$. This is the commutative property of addition for two numbers. In the given problem, we need to find the sum of four numbers by using commutative property. For this, first we will make one group of first two numbers and one group of the last two numbers. That is, we write the given sum as $\left( {2012 + 176} \right) + \left( {244 + 308} \right)$.
Now we are going to use commutative property for two numbers. That is, $a + b = b + a$. Therefore,
$\left( {2012 + 176} \right) + \left( {244 + 308} \right)$
$ = \left( {176 + 2012} \right) + \left( {308 + 244} \right)$
$ = 2188 + 552$
$ = 552 + 2188\quad \left[ {\because a + b = b + a} \right]$
$ = 2740$
Therefore, the required sum is $2740$ by using commutative property.
Now we will find the required sum by using associative property. For every real number $a,b$ and $c$, we can write $\left( {a + b} \right) + c = a + \left( {b + c} \right)$. This is the associative property of addition for three numbers. In the given problem, we need to find the sum of four numbers by using associative property. For this, we will consider one group of the first three numbers in which we will make one group of the first two numbers. That is, we write the given sum as $\left[ {\left( {2012 + 176} \right) + 244} \right] + 308$.
Now we are going to use associative property for three numbers. That is, $\left( {a + b} \right) + c = a + \left( {b + c} \right)$.
$\left[ {\left( {2012 + 176} \right) + 244} \right] + 308$
$ = \left[ {2012 + \left( {176 + 244} \right)} \right] + 308\quad \left[ {\because \left( {a + b} \right) + c = a + \left( {b + c} \right)} \right]$
$ = 2012 + \left[ {\left( {176 + 244} \right) + 308} \right]$
$ = 2012 + 176 + \left( {244 + 308} \right)\quad \left[ {\because \left( {a + b} \right) + c = a + \left( {b + c} \right)} \right]$
$ = \left( {2012 + 176} \right) + \left( {244 + 308} \right)$
$ = 2188 + 552$
$ = 2740$
Therefore, the required sum is $2740$ by using associative property.
Note: Commutative property involves two or more numbers. Associative property can be used for three or more numbers. Commutative and associative properties can be used with addition and multiplication. We cannot use commutative property when we are talking about subtraction and division. For example, let us take two non-zero distinct real numbers $a = 4$ and $b = 5$. Then, $a - b = 4 - 5 = - 1$ and $b - a = 5 - 4 = 1$. Now we can say that $a - b \ne b - a$. Therefore, we can say that commutative property is not true. Similarly we cannot use associative property when we are talking about subtraction and division.
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