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Chapter 7: Problem 44

A three-foot-long deli sandwich must be split into three pieces so that the middle piece is twice as long as the shortest piece and the shortest piece is 8 in. shorter than the longest piece. How long should the three pieces be?

Short Answer

Expert verified
The pieces should be 7 in., 14 in., and 15 in. long.

Step by step solution

01

- Define Variables

Let the shortest piece be denoted as \(x\) inches.
02

- Express Middle and Longest Pieces

The middle piece is twice the shortest piece, so it is \(2x\) inches long. The longest piece is 8 inches longer than the shortest piece, so it is \(x + 8\) inches long.
03

- Setup the Equation

The total length of the sandwich is 3 feet, which is 36 inches. Therefore, the sum of the lengths of all three pieces is: \[x + 2x + (x + 8) = 36\]
04

- Simplify the Equation

Combine like terms to simplify: \[x + 2x + x + 8 = 36\] \[4x + 8 = 36\]
05

- Solve for x

Subtract 8 from both sides: \[4x = 28\] \[x = 7\]
06

- Calculate the Lengths of Each Piece

The shortest piece is \(x = 7\) inches. The middle piece is \[2x = 2(7) = 14\] inches. The longest piece is \[x + 8 = 7 + 8 = 15\] inches.

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Key Concepts

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

headline of the respective core concept
When approaching algebra word problems, it is crucial to identify and label the unknowns in the problem through defining variables. In this exercise, the shortest piece of the deli sandwich was chosen as the variable, labeled as x. By defining variables, you can translate words into mathematical symbols, making the problem more manageable. Consider what each part of the problem represents using the variable. Here, we defined:
  • Shortest piece: x
  • Middle piece: twice the shortest piece, so 2x
  • Longest piece: 8 inches longer than the shortest piece, so x + 8
Defining variables forms the foundation for setting up equations and solving the problem. Always make sure your variables accurately represent what's asked in the problem.
headline of the respective core concept
Once the variables are defined, the next step is setting up equations to relate those variables logically. In our example, we need an equation that represents the total length of the deli sandwich in terms of the defined variables. We know the entire sandwich is 3 feet long, which converts to 36 inches. By adding the lengths of all three pieces, we set up our equation:
x + 2x + (x + 8) = 36
This equation gathers all parts of the problem into one expression that maintains equality. The left side of the equation adds up all pieces of the sandwich, while the right side shows the total length. Setting up the correct equation is crucial for solving word problems accurately.
headline of the respective core concept
Solving linear equations is the process of finding the value of the variable that makes the equation true. Here, we start simplifying our equation:
x + 2x + x + 8 = 36
Combine like terms:
4x + 8 = 36
Then, isolate x by performing the same operation on both sides of the equation. Subtract 8 from both sides:
4x = 28
Finally, divide both sides by 4 to solve for x:
x = 7
This gives us the length of the shortest piece. Clear and systematic steps help in solving the equations properly. Practice different linear equations to become confident in this vital math skill.
headline of the respective core concept
Expressing relationships mathematically allows you to interpret real-world problems in terms of algebra. Here, we expressed the relationships between different lengths of the deli sandwich using algebraic expressions:
The shortest piece is x inches, the middle piece is 2x inches, and the longest piece is x + 8 inches. Translating the problem into mathematical terms helps in solving it step by step. For clarity, revise that:
  • The shortest piece was 7 inches (x = 7)
  • The middle piece was 14 inches (2x = 2(7))
  • The longest piece was 15 inches (x + 8 = 7 + 8)
Representing real-world scenarios mathematically is essential. It breaks down complex problems into simpler, manageable parts, making the solution possible by applying algebraic skills.

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