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Chapter 2: Problem 2

Peter and Seija evaluated the expression \(37+8 \cdot \frac{6}{2}\), Peter said the answer was 135 . Seija said it was 61 . Who is correct? What error did the other person make?

Short Answer

Expert verified
Seija is correct; Peter mixed up the order of operations.

Step by step solution

01

Understand the Expression

Before evaluating, understand the order of operations that you must follow: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). This is commonly abbreviated as PEMDAS.
02

Evaluate the Expression Inside the Fraction

The expression inside the fraction is \( \frac{6}{2} \). According to the order of operations, division is performed, resulting in 3.
03

Perform Multiplication

Now with 3 from the previous step, perform the multiplication with 8 as per the expression: \( 8 \cdot 3 = 24 \).
04

Perform the Addition

Take the result from the multiplication step and add it to 37: \( 37 + 24 = 61 \).
05

Compare the Results

Seija is correct because the evaluated answer using the correct order of operations is 61. Peter did not follow the order of operations correctly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

PEMDAS
PEMDAS is an acronym used to remember the order of mathematical operations. It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. This rule is essential to correctly evaluate mathematical expressions and avoid confusion in calculations.

  • Parentheses: Always perform operations inside parentheses first.
  • Exponents: Next, solve exponents (powers and roots).
  • Multiplication and Division: These operations are equal in hierarchy and are solved from left to right as they appear in the expression.
  • Addition and Subtraction: Like multiplication and division, these are performed from left to right.
Understanding PEMDAS is crucial in evaluating expressions correctly, as demonstrated in the problem where Seija correctly applied the order of operations to find the correct result.
Evaluating Expressions
Evaluating mathematical expressions involves following a step-by-step process, typically guided by the PEMDAS rule. Let's use the example from the exercise: the expression "37 + 8 \cdot \frac{6}{2}".

Here's how you evaluate it:
  • First, simplify the division in the fraction: \( \frac{6}{2} = 3 \).
  • Next, perform the multiplication: \( 8 \cdot 3 = 24 \).
  • Finally, add the result to 37: \( 37 + 24 = 61 \).
Following these steps ensures that you arrive at the correct answer, which in this case is 61. Making sure every operation is conducted in the proper order avoids mistakes and provides a correct solution consistently.
Mathematical Errors
Mathematical errors often occur when steps in the order of operations are not followed correctly. Let鈥檚 look at Peter鈥檚 mistake in the given exercise. He provided a result of 135 by possibly adding before finishing the multiplication and division.

Common errors include:
  • Not adhering to PEMDAS: Failing to prioritize operations according to their order can lead to wrong results.
  • Assuming left-to-right is only necessary: While some operations need evaluating from left to right, ignoring the need for prioritization leads to errors.
  • Misplacing numbers or incorrectly calculating: Simple arithmetic mistakes can alter results significantly.
By carefully applying PEMDAS, as Seija did, and being mindful of each step's correctness, errors in calculation are minimized, ensuring the accurate evaluation of expressions.

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