Problem 5 - Fundamentals of Mathematics

Question

Determine the value of each of the following. $$(33 \div 3)-11$$

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Perform Division

The expression is \((33 \div 3) - 11\). We first perform the division operation: \(33 \div 3 = 11\).

Step 2: Perform Subtraction

With the result from Step 1, the expression now becomes \(11 - 11\). We subtract: \(11 - 11 = 0\).

Key Concepts

Understanding Division

Division is one of the four basic arithmetic operations. It involves splitting a number into equal parts. In simpler terms, it asks, "How many times can one number fit into another?" In the expression

• \(33 \div 3\),

we are trying to determine how many times the number 3 fits into 33.
Imagine having 33 apples and wanting to put them into baskets of 3 apples each. You'll find out that you can make 11 complete baskets, meaning the result of \(33 \div 3\) is 11.

Remember, the first number, 33, is called the "dividend," and the second number, 3, is the "divisor." The result, 11, is the "quotient." Division helps to simplify expressions by breaking them into smaller numbers that are easier to work with.

Understanding Subtraction

Subtraction is another fundamental arithmetic operation. It tells us how much one quantity differs from another. In terms of practical use, it's like taking away a part from the whole.
In the expression

• \(11 - 11\),

we subtract 11 from 11. This operation will show us what remains when you remove 11 from 11. It’s like having 11 candies and giving all 11 away—there are none left.

The result is 0, a special kind of number called 'zero', representing no quantity or value.
Subtraction is intuitive; however, remember it is directional. Subtracting gives a different outcome based on the order of numbers. Therefore, \(11 - 11\) is different from \(11 + 11\).

So, by performing subtraction, we finalize the given expression's calculations and find the result.

Understanding Order of Operations

When solving mathematical problems, the order in which you perform operations is crucial. This concept is known as the "order of operations." It ensures mathematical expressions are solved correctly and consistently.

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This sequence can be easily remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

In the given exercise, \((33 \div 3) - 11\), the division is performed before the subtraction based on PEMDAS.
Firstly, 33 divided by 3 is calculated, giving a quotient of 11. Then, 11 is subtracted from the result.

Following this order helps us determine that the expression simplifies to 0. Adhering to the order of operations provides a clear and accurate path to solve complex mathematical problems.