Order of Operations with Parentheses

Complete the following exercise:

$\frac{(5-3)\cdot15+3}{5+6}-\frac{2\cdot8}{3+1}=$

This simple example demonstrates the order of operations, which states that exponentiation precedes multiplication and division, which precede addition and subtraction, and that operations enclosed in parentheses precede all others,

Let's note that when a fraction (every fraction) is involved in a division operation, it means we can relate the numerator and the denominator to the fraction as whole numbers involved in multiplication, in other words, we can rewrite the given fraction and write it in the following form:

$\frac{(5-3)\cdot15+3}{5+6}-\frac{2\cdot8}{3+1}= \\ \downarrow\\ \big((5-3)\cdot15+3\big):(5+6)-(2\cdot8):(3+1)$We emphasize this by stating that fractions involved in the division and in their separate form , are actually found in multiplication,

Returning to the original fraction in the problem, in other words - in the given form, and simplifying, we separate the different fractions involved in the division operations and simplify them according to the order of operations mentioned, and in the given form:

$\frac{(5-3)\cdot15+3}{5+6}-\frac{2\cdot8}{3+1}= \\ \frac{2\cdot15+3}{11}-\frac{2\cdot8}{4}= \\ \frac{30+3}{11}-\frac{16}{4}=\\ \frac{33}{11}-\frac{16}{4}\\$$$In the first step, we simplified the fraction involved in the division from the left, in other words- we performed the multiplication operation in the division, in contrast, we performed the division operation involved in the fractions, in the next step we simplified the fraction involved in the division from the left and assumed that multiplication precedes division we started with the multiplication involved in this fraction and only then calculated the result of the division operation, in contrast, we performed the multiplication involved in the second division from the left,

We continue and simplify the fraction we received in the last step, this is done again according to the order of operations mentioned, in other words- we start with the division operation of the fractions (this is done by inverting the fractions) and in the next step calculate the result of the subtraction operation:

$\frac{33}{11}-\frac{16}{4}=\\ \frac{\not{33}}{\not{11}}-\frac{\not{16}}{\not{4}}=\\ 3-4=\\ -1$ We conclude the steps of simplifying the fraction, we found that:

$\frac{(5-3)\cdot15+3}{5+6}-\frac{2\cdot8}{3+1}= \\ \frac{2\cdot15+3}{11}-\frac{2\cdot8}{4}= \\ \frac{33}{11}-\frac{16}{4}=\\ 3-4=\\ -1$Therefore, the correct answer is answer d.

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$(3\times5-15\times1)+3-2=$

This simple rule is the order of operations which states that exponentiation precedes multiplication and division, which precede addition and subtraction, and that operations enclosed in parentheses precede all others,

Following the simple rule, multiplication comes before division and subtraction, therefore we calculate the values of the multiplications and then proceed with the operations of division and subtraction

$3\cdot5-15\cdot1+3-2= \\ 15-15+3-2= \\ 1$ Therefore, the correct answer is answer B.

$1$

$(5\times4-10\times2)\times(3-5)=$

This simple rule is the order of operations which states that multiplication precedes addition and subtraction, and division precedes all of them,

In the given example, a multiplication occurs between two sets of parentheses, thus we simplify the expressions within each pair of parentheses separately,

We start with simplifying the expression within the parentheses on the left, this is done in accordance with the order of operations mentioned above, meaning that multiplication comes before subtraction, we perform the multiplications in this expression first and then proceed with the subtraction operations within it, in reverse we simplify the expression within the parentheses on the right and perform the subtraction operation within them:

What remains for us is to perform the last multiplication that was deferred, it is the multiplication that occurred between the expressions within the parentheses in the original expression, we perform it while remembering that multiplying any number by 0 will result in 0:

Therefore, the correct answer is answer d.

$0$

$(3+2-1):(1+3)-1+5=$

This simple rule is the order of operations which states that multiplication and division come before addition and subtraction, and operations enclosed in parentheses come first,

In the given example of division between two given numbers in parentheses, therefore according to the order of operations mentioned above, we start by calculating the values of each of the numbers within the parentheses, there is no prohibition against calculating the result of the addition operation in the given number, for the sake of proper order, this operation is performed later:

$(3+2-1):(1+3)-1+5= \\ 4:4-1+5$In continuation of the principle that division comes before addition and subtraction the division operation is performed first and then the operations of subtraction and addition which were received in the given number and in the last stage:

$4:4-1+5= \\ 1-1+5=\\ 5$Therefore the correct answer here is answer B.

$5$

Complete the following exercise:

$\frac{(7-8)+3}{2}:1+(5-4)=$

This simple equation emphasizes the order of operations, indicating that exponentiation precedes multiplication and division, which come before addition and subtraction, and that operations within parentheses take precedence over all others,

Let's start by discussing the given equation, the first step from the left is division by the number 1, remember that dividing any number by 1 always yields the same number, so we can simply disregard the division by 1 operation, which essentially leaves the equation (with the division by 1 operation, or without it) unchanged, namely:

$\frac{(7-8)+3}{2}:1+(5-4)= \\ \downarrow\\ \frac{(7-8)+3}{2}+(5-4)=$

Continuing with this equation,

Let's note that both the numerator and the denominator in a fraction (every fraction) are equations (in their entirety) between which a division operation is performed, namely- they can be treated as the numerator and the denominator in a fraction as equations that are closed, thus we can rewrite the given equation and write it in the following form:

$\frac{(7-8)+3}{2}+(5-4)= \\ \downarrow\\ \big((7-8)+3\big):2+(5-4)$We highlight this to emphasize that fractions which are the numerator and similarly in its denominator should be treated separately, indeed as if they are closed,

Returning to the original equation, namely - in the given form, and simplifying, we simplify the equation that is in the numerator of the fraction and, this is done in accordance with the order of operations mentioned above and in a systematic manner:

$\frac{(7-8)+3}{2}+(5-4)= \\ \frac{-1+3}{2}+1= \\ \frac{2}{2}+1$In the first stage, we simplified the equation that is in the numerator of the fraction, this in accordance with the order of operations mentioned above hence we started with the equation that is closed, and only then did we perform the multiplication operation that is in the numerator of the fraction, in contrast, we simplified the equation that is in closed parentheses,

Continuing we simplify the equation in accordance with the order of operations mentioned above,thus the division operation of the fraction (this is done mechanically), and continuing we perform the multiplication operation:

$\frac{\not{2}}{\not{2}}+1 =\\ 1+1 =\\ 2$In this case, the simplification process is very short, hence we won't elaborate,

Therefore, the correct answer is option B.

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