Problem 20 - Prealgebra

Question

Find the value of each expression. $$12(11)-56$$

Step-by-Step Solution

Answer

The value of the expression is 76.

Step 1: Understand the Expression

The expression given is \[ 12 \cdot 11 - 56 \] . We are multiplying 12 by 11 and then subtracting 56 from the result.

Step 2: Perform the Multiplication

First, we calculate the product of 12 and 11.\[ 12 \times 11 = 132 \]

Step 3: Perform the Subtraction

Next, we subtract 56 from the result of the multiplication.\[ 132 - 56 = 76 \]

Key Concepts

Multiplication

Multiplication is a fundamental arithmetic operation, widely represented by the symbol \(\cdot\). It involves adding a number (called the multiplicand) repeatedly, as many times as the other number (called the multiplier) indicates. For instance, when we multiply 12 by 11, it is equivalent to adding 12 a total of 11 times. This can be expressed as:

• 12 + 12 + 12 + ... (11 times) = 132

In the context of our exercise, we perform the multiplication step by multiplying 12 and 11 to get 132. This is the first critical step before performing any other operations in the expression. Multiplication needs to precede subtraction because of the order of operations.
When dealing with multiple operations in an expression, it's crucial to follow a specific sequence known as the "PEMDAS" rule. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This order ensures that operations are carried out logically and correctly.

Subtraction

Subtraction, represented by the minus sign \(-\), is the process of taking one quantity away from another. It is one of the core arithmetic operations and involves finding the difference between numbers. In our exercise, after determining the product of 12 and 11, which is 132, we proceed to subtract 56 from 132.

• Start with 132.
• Subtract 56 from it.
• The result is 76.

Subtraction follows multiplication in our expression because, according to the order of operations (PEMDAS), multiplication must be completed before subtraction. Understanding this sequence helps to avoid mistakes and ensures accuracy in calculations.
When performing subtraction, it’s often helpful to write the numbers vertically, lining them up by their rightmost digits. This method can make it easier to see whether you need to "borrow" from a higher digit, which occurs when the top digit is smaller than the bottom digit.

Evaluating Expressions

Evaluating expressions is about simplifying an arithmetic expression down to a single numerical value. It requires following the order of operations rigorously. In the expression \((12 \cdot 11) - 56\), we started by simplifying the multiplication within the parentheses, achieving a result of 132. Next, we performed the subtraction to complete the evaluation and arrived at 76.

In general, to evaluate any expression:

• Identify and resolve parentheses first, dealing with any inner computations.
• Solve any exponents following the parentheses.
• Conduct all multiplications and divisions from left to right.
• Complete any additions and subtractions from left to right.

By following these steps systematically, students can simplify any expression. Evaluating expressions accurately demands careful attention to each operation and sequence, ensuring a thorough understanding of fundamental arithmetic practices.