Order or Hierarchy of Operations with Fractions

Fractions do not influence the order of operations, therefore, you should treat them like any other number in the exercise.

The correct order of mathematical operations is as follows:

Detailed explanation

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Test yourself on special cases (0 and 1, inverse, fraction line)!

What is the result of the following equation?

$36-4\div2$

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Order or Hierarchy of Operations with Fractions

The order of mathematical operations with fractions is no different from the order of operations without fractions.
This means that if you know how to correctly solve a certain exercise based on the order of mathematical operations, you will also know how to solve an exercise with fractions in the same way.
Let's remember the order of operations:

Parentheses - We always start by solving what is inside the parentheses, regardless of the type of operation it is.
Multiplications and divisions – The exercise is read from left to right. Multiplications and divisions have the same hierarchy, therefore, we will resolve them according to their order of appearance in the exercise, from left to right.
Additions and subtractions - After having solved the operations that were in parentheses and those of multiplying and dividing, we will continue with addition and subtraction.
They also share the same hierarchy, therefore, we will resolve them according to their order of appearance in the exercise, from left to right.

Note - We have not given any importance to fractions, nor have we mentioned them.
We will treat fractions like any other number, whether it is a common fraction or a decimal number, it's all the same.

Examples

Exercise 1

$3+6 \times \frac{1}{3}=$

Solution:

Multiplication comes before addition, therefore, we will first solve all the multiplications.
We will obtain:

$3+\frac{6}{3}$
Now we will add and get:
$3+2=5$

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Question 1

$8\times(5\times1)=$

Question 2

$\frac{6}{3}\times1=$

Question 3

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

Exercise 2

$\frac{2}{5} \times (1+3)+4=$

Solution:
Parentheses come first, so we will start by solving what's inside them.

We will obtain:
$\frac{2}{5} \times 4+4=$
Multiplication comes before addition, so we will continue with the multiplication.

We will obtain:
$\frac{8}{5}+4$
Now we will add and get:

$4\frac{8}{5}=5\frac{3}{5}$

Exercise 3

$0.3+(0.4+0.1) \times 4=$

Solution:

We will start with the expression inside the parentheses.

We will solve and obtain:
$0.3+0.5 \times 4=$

Multiplication is resolved before addition, so we will continue with it.

We will obtain:

$0.3+2=$

We will add and get:

$0.3+2=2.3$

Do you know what the answer is?

Question 1

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

Question 2

$(5+4-3)^2:(5\times2-10\times1)=$

Question 3

$$ 100+5-100+5 $$

Exercise 4

$8-9:18 \times 6+5=$

Solution:

We know that if there are no parentheses we start with multiplication and division.
But in what order?
According to the order of appearance in the exercise, from left to right.
We start reading the exercise and we come across a division, therefore, we will start with it.

We will obtain:

$8-\frac{9}{18} \times 6+5=$

We will continue with the multiplication. We will realize that $9 \over 18$ is, in fact, $1 \over 2$
We will obtain:
$8-\frac{1}{2} \times 6+5=$

$8-3+5=$

Now we will continue with the addition and subtraction operations according to the order of appearance.
When we start reading the exercise from the beginning we come across a subtraction, therefore, we will resolve it first. We will obtain:

$5+5=10$

Exercise 5

$5 \times 3-\frac{4}{8} \times 2-3=$

Solution:

There are no parentheses, so we will start with the multiplication and division operations according to their order of appearance in the exercise.
We will start with the first multiplication on the left.

We will obtain:

$15-\frac{4}{8} \times 2-3=$

We will continue with the next multiplication and obtain:

$15-\frac{8}{8}-3=$

We will realize that $8 \over 8$ is $1$ .

We will subtract from left to right according to the order of appearance and obtain:

$15-1-3=$

$14-3=11$

Check your understanding

Question 1

Solve:

$$ 3-4+2+1 $$

Question 2

Solve:

$$ 9-3+4-2 $$

Question 3

Solve:

$$ -5+4+1-3 $$

examples with solutions for order or hierarchy of operations with fractions

Exercise #1

$8\times(5\times1)=$

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the expression in parentheses:

$5\times1=5$

Now we multiply:

$8\times5=40$

Answer

Exercise #2

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

Video Solution

Step-by-Step Solution

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.

Answer

$1$

Exercise #3

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

Video Solution

Step-by-Step Solution

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.

Answer

$0$

Exercise #4

$(5+4-3)^2:(5\times2-10\times1)=$

Video Solution

Step-by-Step Solution

In the given expression, the establishment of division between two sets of parentheses, note that the parentheses on the left indicate strength, therefore, in accordance to the order of operations mentioned above, we start simplifying the expression within those parentheses, and as we proceed, we obtain the result derived from simplifying the expression within those parentheses with given strength, and in the final step, we divide the result obtained from the simplification of the expression within the parentheses on the right,

We proceed similarly with the simplification of the expression within the parentheses on the left, where we perform the operations of multiplication and division, in strength, in contrast, we simplify the expression within the parentheses on the right, which, according to the order of operations mentioned above, means multiplication precedes division, hence we first perform the operations of multiplication within those parentheses and then proceed with the operation of division:

$(5+4-3)^2:(5\cdot2-10\cdot1)= \\ (-2)^2:(10-10)= \\ 4:0\\$ We conclude that the sequence of operations within the expression that is within the parentheses on the left yields a smooth result, this result we leave within the parentheses, these we raised in the next step in strength, this means we remember that every number (positive or negative) in dual strength gives a positive result,

As we proceed, note that in the last expression we received from establishing division by the number 0, this operation is known as an undefined mathematical operation (and this is the simple reason why a number should never be divided by 0 parts) therefore, the given expression yields a value that is not defined, commonly denoted as "undefined group" and use the symbol :

$\{\empty\}$ In summary:

$4:0=\\ \{\empty\}$ Therefore, the correct answer is answer A.

Answer

No solution

Exercise #5

$100+5-100+5$

Video Solution

Step-by-Step Solution

$100+5-100+5=105-100+5=5+5=10$

Answer

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