Problem 120 - Elementary Algebra

Question

\(4+20-(7+12 \div(2+1))\)

Step-by-Step Solution

Answer

The solution is 13.

Step 1: Solve the Parentheses

First, handle the parentheses inside the main expression. Calculate the value of the innermost parentheses: \( 2 + 1 = 3 \)

Step 2: Perform Division

Next, perform the division within the parentheses: \( 12 \, \text{div} \, 3 = 4 \)

Step 3: Simplify the Parentheses

Now by replacing \( 12 \, \text{div} \, 3 \) with 4, the expression becomes: \( 4 + 20 - (7 + 4) \)

Step 4: Simplify Further Parentheses

Simplify the remaining parentheses: \( 7 + 4 = 11 \)

Step 5: Perform Final Operations

Finally, replace \( 7 + 4 \) with 11 and perform the remaining operations: \( 4 + 20 - 11 \) This simplifies to: \( 24 - 11 = 13 \)

Key Concepts

Parentheses

In any mathematical expression, the first step is to solve the parentheses. Parentheses help to group parts of the expression that should be treated as a single unit. By focusing on the innermost parentheses and working outward, we ensure that we are calculating the expression in the correct order.

Let's solve the parentheses step-by-step:

• Identify and solve the innermost parentheses first: \( 2 + 1 = 3 \)
• Replace the inner calculation within the original expression: \( 4+20-(7+12 \div 3) \)

This step is crucial as it sets up the expression for the correct sequence of operations in the following steps.

Division

After handling parentheses, we must perform division next. Division is prioritized in the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).

Here's how to handle the division in our example:

• The expression within the parentheses is now \( 12 \div 3 \)
• Perform the division: \( 12 \div 3 = 4 \)

Once the division is calculated, we can proceed to simplify the expression by replacing \( 12 \div 3 \) with 4 in the original expression.

Simplification

Simplification is the final step where we solve the reduced expression step by step until we reach a single value. This involves combining like terms and performing the required arithmetic operations.

Following the division step, our expression now looks like this: \( 4 + 20 - (7 + 4) \)

Here's how to simplify:

• Simplify the remaining parentheses: \( 7 + 4 = 11 \)
• Replace \( 7 + 4 \) with 11 in the original expression: \( 4 + 20 - 11 \)
• Add and subtract sequentially from left to right: \( 4 + 20 = 24 \), then \( 24 - 11 = 13 \)

Through simplification, we reach the final answer: 13. Simplification ensures that all parts of the expression are fully resolved in the correct order.