# Order or Hierarchy of Operations with Fractions - Examples, Exercises and Solutions

## 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:

1. Parentheses
2. Multiplications and divisions in the order they appear in the exercise
3. Additions and subtractions in the order they appear in the exercise

## Practice Order or Hierarchy of Operations with Fractions

### Exercise #1

$100+5-100+5$

### Step-by-Step Solution

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

10

### Exercise #2

Solve:

$3-4+2+1$

### Step-by-Step Solution

We will use the substitution property to arrange the exercise a bit more comfortably, we will add parentheses to the addition operation:
$(3+2+1)-4=$
We first solve the addition, from left to right:
$3+2=5$

$5+1=6$
And finally, we subtract:

$6-4=2$

2

### Exercise #3

Solve:

$9-3+4-2$

### Step-by-Step Solution

According to the rules of the order of operations, we will solve the exercise from left to right since it only has addition and subtraction operations:

$9-3=6$

$6+4=10$

$10-2=8$

8

### Exercise #4

Solve:

$-5+4+1-3$

### Step-by-Step Solution

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

### Answer

$-3$

### Exercise #5

$3+4-1+40=$

### Step-by-Step Solution

According to the rules of the order of arithmetic operations, we solve the exercise from left to right since it only has addition and subtraction operations:

$3+4=7$

$7-1=6$

$6+40=46$

### Answer

$46$

### Exercise #1

$9+3-1=$

### Step-by-Step Solution

According to the rules of the order of arithmetic operations, we will work to solve the exercise from left to right:

9+3=11

11-1=10

And this is the solution!

### Answer

$11$

### Exercise #2

$-7+5+2+1=$

### Step-by-Step Solution

According to the rules of the order of arithmetic operations, we solve the exercise from left to right since it only has addition and subtraction operations:

$-7+5=-2$

$-2+2=0$

$0+1=1$

### Answer

$1$

### Exercise #3

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

### 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 #4

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

### 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 #5

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

### 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,

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.

No solution

### Exercise #1

$5-2\times\frac{1}{2}+1=$

### Step-by-Step Solution

בשלב הראשון של התרגיל יש לחשב את הכפל.

$2\times\frac{1}{2}=\frac{2}{1}\times\frac{1}{2}=\frac{2}{2}=1$

מכאן ניתן להמשיך לשאר פעולות החיבור והחיסור, מימין לשמאל.

$5-1+1=5$

5

### Exercise #2

$8:2(2+2)=$

### Step-by-Step Solution

Let's start with the part inside the parentheses.

$2+2=4$
Then we will solve the exercise from left to right

$8:2=4$
$4 × (4)=16$

The answer: $16$

16

### Exercise #3

$19\times(20-4\times5)=$

### Step-by-Step Solution

First, we solve the exercise in the parentheses

$(20-4\times5)=$

According to the order of operations, we first multiply and then subtract:

$20-20=0$

Now we obtain the exercise:

$19\times0=0$

0

### Exercise #4

$20-(1+9:9)=$

### Step-by-Step Solution

First, we solve the exercise in the parentheses

$(1+9:9)=$

According to the order of operations, we first divide and then add:

$1+1=2$

Now we obtain the exercise:

$20-2=18$

### Answer

$18$

### Exercise #5

What is the missing number?

$23-12\times(-6)+?\colon7=102$

### Step-by-Step Solution

First, we solve the multiplication exercise:

$12\times(-6)=-72$

Now we get:

$23-(-72)+x\colon7=102$

Let's pay attention to the minus signs, remember that a negative times a negative equals a positive.

We multiply them one by one to be able to open the parentheses:

$23+72+x\colon7=102$

We reduce:

$95+x:7=102$

We move the sections:

$x:7=102-95$

$x:7=7$

$\frac{x}{7}=7$

Multiply by 7:

$x=7\times7=49$

49

### Topics learned in later sections

1. Multiplicative Inverse