Answered

Brice has twice as many stamps in his collection as his friends Anthony and Paul combined. Paul has 32 stamps in his collection. Brice has 96 stamps in his collection. How many stamps does Anthony have in his collection?



The equation $2(x+32)=96$ models this situation. Which choice shows the sequence of operations needed to solve this problem, using both arithmetic and algebra?


A


Arithmetic:		Algebra:


\begin{tabular}[t]{l}
- Divide by 2.

- Subtract 32.
& $\begin{array}{c}

2(x+32)=96 \\

\frac{2(x+32)}{2}=\frac{96}{2} \\

(x+32)=48 \\

x+32-32=48-32 \\

x=16

\end{array}$ \\

\hline

\end{tabular}


B

Arithmetic:		Algebra:

- Subtract 32.		$\begin{array}{l}
2(x+32)=96

\frac{2(x+32)}{2}=\frac{96}{2}
\end{array}$

- Divide by 2.		$\begin{array}{c}
(x+32)=48

x+32-32=48-32

x=16
\end{array}$

Verified Answer :

  • Solve the equation \(2(x + 32) = 96\) by first dividing both sides by \(2\) to get \(x + 32 = 48\).
  • Subtract \(32\) from both sides to find that \(x = 16\).
  • Compare the algebraic steps (Divide by 2, then subtract 32) to the choices provided.
  • Identify that Choice 1 correctly reflects the sequence of operations used in the algebraic solution. The final answer is \(\boxed{\text{Choice 1}}\).

Explanation

  1. Problem Analysis

The problem describes a scenario where Brice's stamp collection consists of twice the sum of Anthony's (\(x\)) and Paul's (\(32\)) collections. We are given that Brice has \(96\) stamps in total. This situation is modeled by the equation \(2(x + 32) = 96\). Our objective is to determine how many stamps Anthony has and identify which choice correctly describes the sequence of operations used to solve the equation.

  1. Solving the Equation

To solve the equation \(2(x + 32) = 96\), we perform algebraic operations in a specific order.
First, we isolate the term in the parentheses by dividing both sides of the equation by \(2\):
\[\frac{2(x + 32)}{2} = \frac{96}{2}\]
\[x + 32 = 48\]
Second, we solve for \(x\) by subtracting \(32\) from both sides of the equation:
\[x + 32 - 32 = 48 - 32\]
\[x = 16\]
Thus, Anthony has \(16\) stamps.

  1. Identifying the Correct Choice

The algebraic steps we followed were:

  1. Divide by \(2\).
  2. Subtract \(32\).

Now, we compare this sequence to the provided choices:

  • Choice 1 lists: "Divide by \(2\)" followed by "Subtract \(32\)". This matches our algebraic steps perfectly.
  • Choice 2 lists: "Subtract \(32\)" followed by "Divide by \(2\)". This does not match our algebraic steps.

Therefore, Choice 1 is the correct answer.

Examples

Mathematical modeling is an essential skill for managing resources in everyday life. For instance, if you and a friend are splitting a total cost of a dinner, including a set service fee, you can use an equation like \(2(x + S) = T\), where \(x\) is your share, \(S\) is the service fee, and \(T\) is the total bill. By following the algebraic steps to solve for \(x\), you ensure both parties pay their fair share accurately. Mastering this process helps in everything from budget planning to distributing tasks efficiently within a group.

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