When we come to use the order of operations, we can encounter various special cases.
Sometimes, these cases will affect the order of operations, and in other cases we can use them to make the solution path easier for ourselves.
Special Cases Order of Operations Practice Problems
Master special cases with 0, 1, reciprocals, and fraction lines in order of operations. Practice problems with step-by-step solutions and examples.
📚Master Special Cases in Order of Operations
- Apply multiplication and division rules with zero and one
- Identify and work with reciprocal numbers in expressions
- Handle fraction lines as implied parentheses in calculations
- Recognize when zero makes entire expressions equal zero
- Use reciprocal shortcuts to simplify complex problems
- Solve expressions with undefined division by zero cases
Understanding Special Cases (0 and 1, Inverse, Fraction Line)
Complete explanation with examples
Special Cases in Order of Operations
The number
Addition and subtraction do not affect the number.
Multiplication by =
Number divided by =
Division by is undefined
number
Multiplication by does not change the number
Division by does not change the number
Reciprocal Numbers
when is not equal to
Division and multiplication of reciprocal numbers
fraction line
Let's treat the arithmetic operation in the numerator as if the numerator is in parentheses.
Example
Solution:
Let's start by solving the numerator:
Let's continue with the parentheses:
Let's continue with multiplication and ignore adding :
Practice Special Cases (0 and 1, Inverse, Fraction Line)
Test your knowledge with 19 quizzes
\( \frac{1}{2}+0+\frac{1}{2}= \) ?
Examples with solutions for Special Cases (0 and 1, Inverse, Fraction Line)
Step-by-step solutions included
Exercise #1
Solve the following exercise:
Step-by-Step Solution
According to the order of operations rules, since the exercise only involves addition and subtraction, we will solve the problem from left to right:
Answer:
9.5
Video Solution
Exercise #2
?
Step-by-Step Solution
According to the order of operations, the exercise is solved from left to right as it contains only an addition operation:
Answer:
0.8
Video Solution
Exercise #3
Solve the following exercise:
Step-by-Step Solution
According to the order of operations, we first multiply and then add:
Answer:
Exercise #4
Step-by-Step Solution
Let's begin by multiplying the numerator:
We obtain the following fraction:
Finally let's reduce the numerator and denominator by 10 and we are left with the following result:
Answer:
Video Solution
Exercise #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:
Video Solution
Frequently Asked Questions
What happens when you multiply any number by zero in order of operations?
+When you multiply any number by zero, the entire expression becomes zero, regardless of how complex the other calculations are. For example, (5855 × 9358) × 0 = 0, so you don't need to calculate what's in the parentheses first.
How do you handle fraction lines in order of operations?
+Treat any arithmetic operations in the numerator as if they're in parentheses - solve them first before dividing. For example, in (32-20)/6 + 3×2, you calculate 32-20 = 12 first, then 12/6 = 2, then 2 + 6 = 8.
What are reciprocal numbers and how do they help with calculations?
+Reciprocal numbers are two numbers that multiply to equal 1, like 2 and 1/2. They help because a ÷ (1/b) = a × b, allowing you to convert division into easier multiplication problems.
Why is division by zero undefined in mathematics?
+Division by zero is undefined because there's no number that, when multiplied by zero, gives you the original number. If you see something like 5 ÷ 0 in a problem, the answer is 'undefined' rather than a numerical value.
Does adding or subtracting zero change the order of operations?
+Adding or subtracting zero doesn't change the value of numbers, so you can often skip these steps to simplify calculations. For example, (4 + 0) × 2 + 1 becomes 4 × 2 + 1 = 9.
How does multiplying or dividing by 1 affect expressions?
+Multiplying or dividing by 1 leaves numbers unchanged, so you can simplify expressions quickly. For instance, (233434 ÷ 1) + 2 + 1 immediately becomes 233434 + 2 + 1 = 233437.
What's the difference between 0 - 7 and 7 - 0?
+0 - 7 = -7 (you get the negative of the number you're subtracting), while 7 - 0 = 7 (subtracting zero doesn't change the original number). Order matters in subtraction with zero.
When should you look for special cases to simplify order of operations problems?
+Look for special cases before doing complex calculations: check for multiplication by zero (makes everything zero), division/multiplication by 1 (no change), reciprocal pairs (equal 1), and fraction lines (treat numerator as parentheses).