Problem 57 - Prealgebra

Question

Find the value of each expression. $$4(20-13)+4 \times 5$$

Step-by-Step Solution

Answer

The value of the expression is 48.

Step 1: Solve the Expression Inside the Parenthesis

Begin by solving the expression inside the parenthesis: \(20 - 13\). Subtract 13 from 20 to get 7. Now, substitute this value back into the expression, giving you \(4 \times 7 + 4 \times 5\).

Step 2: Multiply 4 by the Result Inside the Parenthesis

Now, multiply 4 by the value obtained in Step 1. \(4 \times 7 = 28\). This reduces your expression to \(28 + 4 \times 5\).

Step 3: Multiply 4 by 5

Next, calculate the product of 4 and 5. \(4 \times 5 = 20\). Substitute this result back, and the expression becomes \(28 + 20\).

Step 4: Add the Results

Finally, add the products from Steps 2 and 3. \(28 + 20 = 48\). So, the value of the expression is 48.

Key Concepts

Understanding Parentheses

Parentheses are used to group parts of an expression that need to be solved first. They indicate the priority when dealing with mathematical operations, guiding us to start solving whatever is within them before moving on to other operations. In our exercise, the parentheses contain the expression \(20 - 13\).

• **Step in Order of Operations**: Parentheses are the first step in the common order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• **Importance**: Ignoring parentheses can lead to incorrect results, as they dictate the order of calculations.

To solve \(20 - 13\), first subtract 13 from 20 to get 7. It’s crucial to solve inside the parentheses fully before averaging out to the remaining expression. Once calculated, substitute the rest of the expression with this result, turning \(4(20-13)+4 \times 5\) into \(4 \times 7 + 4 \times 5\).

The Role of Multiplication

Multiplication is one of the basic operations in arithmetic used to calculate the product of numbers. Once parentheses are resolved, multiplication is the next step. This follows the rule of operations sequence as it comes before addition.

• **Working with Values**: In the current expression, we first encounter multiplication with \(4 \times 7\) because of the previous step results.
• **Calculating Products**: Calculating \(4 \times 7\) gives us 28. After this, move to the second multiplication, \(4 \times 5\), yielding 20.

Both products 28 and 20 will be essential for the next operation, addition. Remember that each segment where multiplication takes place should be handled with care, as multiplication must be completed before any addition or subtraction.

Importance of Addition

Addition is a fundamental operation that is carried out after both parentheses and multiplication in order of operations problems. Our expression after solving for parentheses and multiplying becomes \(28 + 20\).

• **Final Step**: Addition is often the last operation performed in an expression unless the expression includes subsequent subtraction.
• **Combining Results**: At this stage, you simply add the results of previous calculations together. In our expression, this is \(28 + 20\), which yields a final result of 48.

By performing addition last, you ensure that all prior operations have been addressed, providing accurate results. Knowing the role and order for each operation helps avoid missteps and ensures precision in mathematical expressions.