Problem 60 - Elementary Algebra

Question

\(12 \div 6 \cdot 2\)

Step-by-Step Solution

Answer

The solution to the expression is 4.

Step 1: Understand the Expression

Identify the arithmetic operations involved in the expression. The given expression involves division and multiplication: \[ 12 \div 6 \cdot 2 \]

Step 2: Apply the Order of Operations

According to the order of operations (PEMDAS/BODMAS), division and multiplication should be performed from left to right. The first operation to perform is division: \[ 12 \div 6 = 2 \]

Step 3: Complete the Expression

Use the result of the division in the next operation, which is multiplication: \[ 2 \cdot 2 = 4 \]

Step 4: Verify the Result

Recheck each step to confirm the final result is correct. The expression simplifies to 4.

Key Concepts

Division

Division is one of the four basic operations in arithmetic. It involves splitting a number into equal parts. In the expression \(12 \div 6 \cdot 2\), division is the first operation we perform. This is because, according to the order of operations, division and multiplication are performed from left to right.
For instance, when dividing 12 by 6, we are essentially asking how many times 6 fits into 12, which equals 2.

Some key points about division:

• It is the inverse operation of multiplication.
• If the divisor is larger than the dividend, the result is a fraction.
• Division by zero is undefined.

Understanding the concept of division is crucial for solving many mathematical problems correctly.

Multiplication

Multiplication is another fundamental operation in arithmetic. It involves adding a number to itself a certain number of times. In the expression \(12 \div 6 \cdot 2\), multiplication comes after division when following the left-to-right rule.
After dividing 12 by 6 to get 2, we then multiply this result by 2, which gives us 4.

Some important aspects of multiplication:

• It is the inverse operation of division.
• Any number multiplied by 1 remains unchanged.
• Any number multiplied by zero results in zero.

Multiplication is a key operation used in various areas of mathematics and everyday calculations.

PEMDAS/BODMAS

PEMDAS/BODMAS is a mnemonic that helps remember the order of operations in mathematics. It stands for:

• Parentheses or Brackets
• Exponents or Orders (such as squares and square roots)
• Multiplication
• Division
• Addition
• Subtraction

This order ensures that mathematical expressions are solved correctly.
For example, in the expression \(12 \div 6 \cdot 2\), PEMDAS/BODMAS tells us to perform multiplication and division from left to right. Therefore, we first divide 12 by 6 to get 2, and then multiply 2 by 2 to get 4.

Following this rule avoids confusion and ensures consistency in solving mathematical problems. It's crucial for students to master this concept.