Problem 77 - Algebra for College Students

Question

Your friend keeps getting an answer of 30 when simplifying \(7+8(2)\). What mistake is he making and how would you help him?

Step-by-Step Solution

Answer

Your friend forgot to multiply before adding; the correct answer is 23 by following the order of operations.

Step 1: Identify the Problem

Your friend is evaluating the expression \(7+8(2)\) and arriving at an answer of 30. This indicates they are not applying the correct order of operations.

Step 2: Introduce Order of Operations

Explain that the correct way to simplify expressions is by using the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Step 3: Apply Parentheses and Multiplication

In the expression \(7+8(2)\), treat the parentheses and the multiplication: first, perform the multiplication inside the parentheses, which is \(8\times2\).

Step 4: Compute the Multiplication

Calculate \(8\times2 = 16\).

Step 5: Perform Addition

Now substitute the multiplication result back into the expression: \(7+16\). Add these together to get \(23\).

Step 6: Verify the Correctness

Finally, verify that \(23\) is indeed the correct result by reapplying the order of operations to confirm the earlier steps were correctly followed.

Key Concepts

PEMDAS

A common issue when simplifying expressions, like in the example of \(7+8(2)\), is not applying the proper order of operations. The mistake often lies in ignoring or misapplying PEMDAS. This acronym stands for:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

By following this sequence, you ensure expressions are simplified the same way every time. Remember, operations must be performed left to right, especially for multiplication/division and addition/subtraction, which share their respective ranks in the hierarchy. Applying PEMDAS correctly avoids errors, producing accurate solutions to arithmetic expressions.

Simplification Errors

Simplification errors can often stem from misinterpretations of the order of operations. In our example, evaluating \(7+8(2)\) and arriving at \(30\) indicates a common error where addition is mistakenly performed before multiplication. To avoid this, apply PEMDAS judiciously:

First, recognize that \(8(2)\) represents a multiplication operation.

Perform the multiplication before attending to the addition, meaning calculate \(8 \times 2 = 16\).

Ignoring these steps could lead to an incorrect and larger result, as multiplication, when delayed, alters final outputs significantly.

Multiplication Techniques

Multiplication is fundamental but can sometimes be overlooked in expressions like \(7+8(2)\). In math, parentheses imply multiplication. This means \(8(2)\) should be computed first.

Various multiplication strategies can aid in simplifying expressions correctly:

• Visualize the operation: Treat the parentheses as a cue to multiply rather than add.
• Double-check each step, especially where multiple operations overlap.

Mastering efficient multiplication techniques, like breaking down larger numbers or using mental math tricks, ensures precision within broader calculations, such as calculating \(8 \times 2 = 16\) correctly before moving on to addition. These checks help cement understanding and avoid superfluous errors.