Answered

The perimeter of a rectangle is 40 cm. The length is 14 cm. Let $x =$ width of the rectangle.



Ravi says he can find the width using the equation $2(x+14)=40$.

Fran says she can find the width using the equation $2x+28=40$.



Answer the questions to solve the equations and to compare the steps and solutions.



Part A

Which of these is the most helpful first step for solving Ravi's equation, $2(x+14)=40$?


AAdd 14 to both sides.
BSubtract 14 from both sides.
CDivide both sides by 2.
DMultiply both sides by 2.

Verified Answer :

  • The perimeter formula for a rectangle is \(P = 2(L + W)\).
  • Ravi's equation is \(2(x + 14) = 40\).
  • To solve for \(x\), dividing both sides by 2 is the most direct approach, simplifying the equation to \(x + 14 = 20\).
  • Final Answer: \(\boxed{\text{Divide both sides by 2}}\).

Explanation

  1. Problem Analysis

The problem involves a rectangle with a perimeter \(P = 40 \text{ cm}\) and a length \(L = 14 \text{ cm}\). Let \(x\) represent the width. The perimeter of a rectangle is given by the formula \(P = 2(L + W)\). Substituting the known values, we have \(40 = 2(14 + x)\), which is equivalent to Ravi's equation \(2(x + 14) = 40\). Our goal is to identify the most efficient first step to solve this equation for \(x\).

  1. Evaluating the Options

Ravi's equation is \(2(x + 14) = 40\). We want to isolate \(x\).

  1. Adding 14 to both sides gives \(2(x+14)+14 = 54\), which complicates the expression.
  2. Subtracting 14 from both sides results in \(2(x+14)-14 = 26\), which also does not isolate \(x\) directly because of the multiplication by 2.
  3. Dividing both sides by 2 gives \(\frac{2(x + 14)}{2} = \frac{40}{2}\), which simplifies to \(x + 14 = 20\). This step effectively reduces the equation to a much simpler linear form, leaving only one step (subtracting 14) to solve for \(x\).
  4. Multiplying both sides by 2 results in \(4(x+14) = 80\), which does not simplify the equation.

Therefore, dividing both sides by 2 is clearly the most helpful first step.

  1. Final Conclusion

The most helpful first step for solving the equation \(2(x + 14) = 40\) is to divide both sides by 2. This leaves the expression \(x + 14 = 20\), from which we can easily find \(x = 6\). Thus, the correct choice is to divide both sides by 2.

Examples

In real-life scenarios, such as designing a garden bed or partitioning a space for furniture, you often encounter formulas that relate dimensions to a fixed perimeter or area. For instance, if you have a fixed amount of fencing material, say 40 meters, to enclose a rectangular vegetable patch, and you know one side must be 14 meters long, the equation \(2(x + 14) = 40\) represents the constraint. Choosing the most efficient method to solve for the unknown side \(x\)—in this case, dividing by the common factor—allows for quick and accurate planning of your project.

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