Compare Complex Expressions: Evaluating 1/9(4² - 6) ÷ 2 + 4 vs 4² - (6 ÷ 2 + 4)(1/7)

To solve a problem given in division or multiplication each of the terms that appear in its expression separately,

this is done within the framework of the order of operations, which states that multiplication precedes addition and subtraction, and that the preceding operations are performed before all others,

A. We will start with the terms that appear on the left in the given problem:

$\frac{1}{9}\cdot((4^2-3\cdot2):2+4)$First, we simplify the terms in the parentheses (the numerators) that multiply the fraction according to the order of operations, noting that the term in the parentheses includes within it an operation of division of the term in the parentheses (the denominators), therefore, we will start simply with this term, in this term a subtraction operation is performed between terms that strengthens the division between terms, therefore the calculation of its numerical value is carried out first followed by the multiplication of the terms and continue to perform the subtraction operation:

$\frac{1}{9}\cdot((4^2-3\cdot2):2+4) =\\ \frac{1}{9}\cdot((16-3\cdot2):2+4) =\\ \frac{1}{9}\cdot((16-6):2+4) =\\ \frac{1}{9}\cdot(10:2+4)\\$Note that there is no prohibition to calculate their numerical values of the term that strengthens as in the term in the parentheses in contrast to the multiplication that in the term in the parentheses, this from a concept that breaks in separate terms, also for the sake of good order we performed this step after step,

We continue simply with the terms in the parentheses that were left, we remember that division precedes subtraction and therefore we will start from calculating the outcome of the multiplication in the term, in the next step the division is performed and finally the multiplication in the break that multiplies the term in the parentheses:

$\frac{1}{9}\cdot(10:2+4)\\ \frac{1}{9}\cdot(5+4)=\\ \frac{1}{9}\cdot9=\\ \frac{1\cdot9}{9}=\\ \frac{\not{9}}{\not{9}}=\\ 1$In the last steps we performed the multiplication of the number 9 in the break, this we did while we remember that the multiplication in the break means the multiplication in the amount of the break,

We finished simply with the terms that appear on the left in the given problem, we will summarize the steps of the simplification:

We received that:

$\frac{1}{9}\cdot((4^2-3\cdot2):2+4) =\\ \frac{1}{9}\cdot(10:2+4)\\ \frac{1}{9}\cdot(5+4)=\\ \frac{9}{9}=\\ 1$

B. We will continue and simplify the terms that appear on the right in the given problem:

$4^2-(3\cdot2:2+4)\cdot\frac{1}{7}$In this part to be done in the first part we simplify the terms within the framework of the order of operations,

In this term a multiplication operation is performed on the term in the parentheses, therefore, we will simplify first this term, we remember that multiplication and division precede subtraction, therefore, we will calculate first the numerical values of the first term from the left in this term, noting that the concept that between multiplication and division there is no predetermined precedence in the order of operations, the operations in this term are performed one after the other according to the order from left to right, which is the natural order of operations, in contrast we will calculate the numerical values of the term that strengthens:

$4^2-(3\cdot2:2+4)\cdot\frac{1}{7} =\\ 16-(6:2+4)\cdot\frac{1}{7} =\\ 16-(3+4)\cdot\frac{1}{7} =\\ 16-7\cdot\frac{1}{7}\$We will continue and perform the multiplication in the break, this within that we remember that the multiplication in the break means the multiplication in the amount of the break, in the next step the division operation of the break (by the compression of the break) is performed and in the last step the remaining subtraction operation, this in accordance with the order of operations:

$16-7\cdot\frac{1}{7}=\\ 16-\frac{7\cdot1}{7}=\\ 16-\frac{\not{7}}{\not{7}}=\\ 16-1=\\ 15$We finished simply with the terms that appear on the right in the given problem, we will summarize the steps of the simplification:

We received that:

$4^2-(3\cdot2:2+4)\cdot\frac{1}{7} =\\ 16-(3+4)\cdot\frac{1}{7} =\\ 16-1=\\ 15$We will return to the original problem, and we will present the outcomes of the simplifications that were reported in A and B:

$\frac{1}{9}\cdot((4^2-3\cdot2):2+4)\textcolor{red}{☐}4^2-(3\cdot2:2+4)\cdot\frac{1}{7} \\ \downarrow\\ 1 \textcolor{red}{☐}15$As a result that is established that:

$1 \text{ }\textcolor{red}{\neq}15$Therefore, the correct answer here is answer B.