Understanding the Math Problem and Order of Operations

The problem asks us to find the value of the mathematical expression: \(8 + (\frac{1}{2} + \frac{1}{4}) \times 16\). To solve this correctly, we need to follow the order of operations. A common acronym used for the order of operations is BODMAS or PEMDAS.

• B/P: Brackets or Parentheses first.
• O/E: Orders or Exponents next.
• DM: Division and Multiplication (from left to right).
• AS: Addition and Subtraction (from left to right).

In our expression, we have parentheses, addition, and multiplication. According to the order of operations, we must evaluate the expression inside the parentheses first.

Step-by-Step Calculation of the Expression Value

Let's break down the calculation step by step:

Step 1: Evaluate the expression inside the parentheses

The expression inside the parentheses is the addition of two fractions: \(\frac{1}{2} + \frac{1}{4}\). To add fractions, they must have a common denominator. The least common multiple of 2 and 4 is 4. So, we convert \(\frac{1}{2}\) to an equivalent fraction with a denominator of 4:

\(\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}\)

Now, we can add the fractions:

\(\frac{2}{4} + \frac{1}{4} = \frac{2 + 1}{4} = \frac{3}{4}\)

Step 2: Substitute the result back into the main expression

Now that we have evaluated the parentheses, we substitute the result back into the original expression:

\(8 + (\frac{3}{4}) \times 16\)

Step 3: Perform the multiplication

According to the order of operations, multiplication comes before addition. We need to calculate \(\frac{3}{4} \times 16\). We can think of 16 as \(\frac{16}{1}\):

\(\frac{3}{4} \times 16 = \frac{3}{4} \times \frac{16}{1}\)

Multiply the numerators together and the denominators together:

\(\frac{3 \times 16}{4 \times 1} = \frac{48}{4}\)

Now, simplify the fraction by dividing 48 by 4:

\(\frac{48}{4} = 12\)

Step 4: Perform the addition

Finally, we perform the addition:

\(8 + 12 = 20\)

So, the value of the expression \(8 + (\frac{1}{2} + \frac{1}{4}) \times 16\) is 20.

Summary of Calculation Steps

The final value of the expression is 20.

Revision Table: Key Math Concepts

Additional Information: Understanding BODMAS/PEMDAS

The BODMAS (or PEMDAS) rule is crucial for ensuring everyone gets the same answer when evaluating a mathematical expression. Without it, different people might perform operations in different orders and arrive at different results.

• Parentheses/Brackets: Always solve the operations inside the grouping symbols first. If there are nested parentheses, work from the innermost outwards.
• Exponents/Orders: Evaluate any powers or roots next.
• Multiplication and Division: Perform these operations from left to right as they appear in the expression. They have equal priority.
• Addition and Subtraction: Perform these operations from left to right as they appear in the expression. They also have equal priority.

Applying this rule systematically helps avoid errors in calculations, especially with more complex expressions involving multiple operations and grouping symbols.