Calculate Yard Work Earnings: A Math Problem

by Alex Johnson 45 views

Let's break down this math problem step-by-step to understand how Cooper and his friends calculated their earnings from yard work. We'll explore the expression [(16 x 4) + 8] ÷ 3 and see which statement accurately reflects the amount each boy earned.

Understanding the Expression

The expression [(16 x 4) + 8] ÷ 3 is the key to solving this problem. It tells us exactly what calculations were performed to determine the earnings. Let's dissect it:

  • (16 x 4): This part signifies that a certain amount, $16, was multiplied by 4. In the context of yard work, this could represent earning $16 per yard for 4 yards.
  • + 8: This indicates that an additional $8 was added to the result of the multiplication. This could represent earning $8 for a specific task, such as edging.
  • ÷ 3: This means that the total amount calculated in the previous steps was divided by 3. Since Cooper and his two friends are involved, there are a total of three people, so this division represents the equal distribution of earnings among them.

Therefore, the expression [(16 x 4) + 8] ÷ 3 can be interpreted as: earning $16 per yard for 4 yards, plus an additional $8 for edging, and then dividing the total earnings equally among three people.

Analyzing the Statements

Now, let's analyze the given statement and determine if it accurately reflects the expression [(16 x 4) + 8] ÷ 3.

The statement is:

  • $16 per yard for 4 yards and $8 for edging, and then separated equally

This statement perfectly aligns with our interpretation of the expression. It describes the following steps:

  1. Earning $16 per yard for 4 yards (16 x 4).
  2. Earning an additional $8 for edging (+ 8).
  3. Dividing the total earnings equally among three people (÷ 3).

Therefore, this statement accurately represents the amount each boy earned.

Alternative Scenarios

To further illustrate the importance of accurately interpreting the expression, let's consider some alternative scenarios and see how they would translate into mathematical expressions.

  • Scenario 1: Cooper and his friends earned a flat rate of $16 for each of the 4 yards they worked on, and an additional $8 for supplies. They then split the earnings equally.

    • Expression: (16 + 8) x 4 ÷ 3. In this scenario, it means that the $16 and $8 are added and then multiplied by 4. And after spliting the earnings.

  • Scenario 2: Cooper earned $16, one friend earned $4, and the other earned $8. They want to calculate each person earned in total.

    • Expression: (16 + 4 + 8) ÷ 3. They are simply adding the individual earnings and dividing by 3.

  • Scenario 3: Cooper and his friends earned $8 for 3 hours of yard work. In addition, Cooper earned $16 for his extra work.

    • Expression: (8 + 16) ÷ 3. Adding what Cooper earned to the total amount, and calculating each person earned in total.

These scenarios highlight how different wording can lead to different mathematical expressions and, consequently, different results. The order of operations and the specific tasks performed are crucial in determining the correct expression.

Why Understanding Order of Operations is Important

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. Misunderstanding or ignoring the order of operations can lead to incorrect results.

In the expression [(16 x 4) + 8] ÷ 3, the parentheses indicate that the multiplication (16 x 4) should be performed first. Then, the addition (+ 8) should be performed. Finally, the division (÷ 3) should be performed.

If we were to ignore the order of operations and perform the addition before the multiplication, we would get a different result:

(16 x 4) + 8 ÷ 3 = 64 + 8 ÷ 3 = 64 + 2.67 = 66.67 (approximately)

This is different from the correct result, which is:

[(16 x 4) + 8] ÷ 3 = (64 + 8) ÷ 3 = 72 ÷ 3 = 24

As you can see, following the correct order of operations is essential for accurate calculations. In real-world scenarios, such as calculating earnings or expenses, accuracy is paramount.

Real-World Applications

Understanding mathematical expressions and order of operations is not just an academic exercise; it has numerous real-world applications. Here are a few examples:

  • Budgeting: When creating a budget, you need to calculate your income, expenses, and savings. This often involves using mathematical expressions to represent different financial scenarios.
  • Cooking: Recipes often involve measurements and ratios that need to be scaled up or down. Understanding mathematical expressions can help you adjust recipes to suit your needs.
  • Construction: Construction projects require precise measurements and calculations to ensure that structures are built safely and accurately. Architects and engineers rely on mathematical expressions to design and plan construction projects.
  • Programming: Computer programming involves writing code that performs calculations and manipulates data. Understanding mathematical expressions is essential for writing effective and efficient code.

These are just a few examples of how mathematical expressions are used in real-world situations. By developing a strong understanding of mathematical concepts, you can improve your problem-solving skills and make informed decisions in various aspects of your life.

Conclusion

In conclusion, the statement "$16 per yard for 4 yards and $8 for edging, and then separated equally" accurately represents the amount each boy earned, as it aligns perfectly with the expression [(16 x 4) + 8] ÷ 3. By understanding the order of operations and the meaning of each component in the expression, we can confidently determine the correct interpretation and solve the problem.

For further learning about mathematical expressions and order of operations, you can visit Khan Academy for comprehensive lessons and practice exercises.