Problem 43 - Prealgebra

Question

Copy each sentence. Then insert parentheses to make each sentence true. $$12 \times 3 \div 1+2=12$$

Step-by-Step Solution

Answer

Insert parentheses: \((12 \times 3) \div (1 + 2) = 12\).

Step 1: Evaluate the Original Expression

First, evaluate the given expression without any parentheses: \(12 \times 3 \div 1 + 2\). Applying the order of operations, first we perform the multiplication: \(12 \times 3 = 36\). Next, we divide: \(36 \div 1 = 36\). Finally, we add: \(36 + 2 = 38\). The original expression equals 38.

Step 2: Identify the Intent

The aim is to use parentheses to modify the expression to make it equal to 12. This involves changing the order of operations. We'll evaluate different positions to place parentheses.

Step 3: Insert Parentheses for Solution

Try inserting parentheses at different places. Let's insert them around \(3 \div 1 + 2\): \(12 \times (3 \div 1 + 2)\). Evaluate the expression: Inside the parentheses, solve \(3 \div 1 + 2 = 3 + 2 = 5\). Thus, the expression becomes \(12 \times 5 = 60\), which is not equal to 12.

Step 4: Test Another Parentheses Position

Place parentheses around \(12 \times 3\): \((12 \times 3) \div 1 + 2\). This time, evaluate inside the parentheses first: \(12 \times 3 = 36\). Then divide: \(36 \div 1 = 36\). Add: \(36 + 2 = 38\). Still, it's not equal to 12.

Step 5: Correct Parentheses Placement

Insert parentheses strategically to achieve the correct total. Try \((12 \times 3) \div (1 + 2)\). Evaluate each part separately: inside the second parentheses, calculate \(1 + 2 = 3\). Now perform the first operation: \(12 \times 3 = 36\). Next, divide: \(36 \div 3 = 12\). This matches the intended result.

Key Concepts

Order of Operations

Understanding the order of operations is crucial in mathematics, especially when handling multiple calculations in an expression. When you see a mathematical expression like \(12 \times 3 \div 1 + 2\), how do you know where to start? That's where PEMDAS comes in. PEMDAS stands for Parentheses, Exponents, Multiplication, Division (from left to right), Addition, and Subtraction (from left to right). This is the standard sequence we follow to solve expressions.

For example, in the expression \(12 \times 3 \div 1 + 2\), you first perform the multiplication \(12 \times 3\), then division \( \div 1\), and finally the addition \(+ 2\). Without using the proper order, the result of the expression can be incorrect. Each step should respect this sequence unless parentheses suggest a different path. Using PEMDAS helps ensure that the calculations are performed in a consistent and accurate manner.

Parentheses in Mathematics

In mathematics, parentheses are powerful tools that let you change how an expression is solved. They indicate that certain calculations should be done first. For example, if you change \(12 \times 3 \div 1 + 2\) to \((12 \times 3) \div (1 + 2)\), the expression is solved differently.

Without parentheses, the calculations respect the standard order of operations. Yet, when parentheses are involved, the operations inside them take priority. This means you solve those calculations first, altering the final outcome. Parentheses can lead to completely different results, making them essential for guiding math operations where the sequence can change the solution.

Always be careful about how and where you place parentheses to achieve the desired outcome.

Evaluating Expressions

Evaluating expressions involves performing all the mathematical operations needed to find the value of an expression. It's like solving a puzzle, where every piece must be placed in the correct order and manner.

When you encounter an expression such as \((12 \times 3) \div (1 + 2)\), you break it down:

• First, solve inside the parentheses \(1 + 2\), which equals 3.
• Next, calculate \(12 \times 3\), resulting in 36.
• Finally, divide 36 by 3.

This structured approach ensures accuracy in each step until the expression is fully evaluated.

Evaluating expressions correctly requires attention to every detail and respect for mathematical rules. It's not just about solving quickly but also about understanding each part thoroughly before moving to the next.