Ken's monthly take-home pay is \(w\) dollars. After he pays for food and rent, he has \(x\) dollars left.
Column A: \(x\)
Column B: \(w - x\)
Column A: \(x\)
Column B: \(w - x\)
Show Hint
- Quantity A is greater
- Quantity B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given
The Correct Option is D
Solution and Explanation
This problem tests the ability to translate a word problem into an algebraic relationship and understand its implications. We need to analyze the relationship between total pay, expenses, and savings.
Step 2: Detailed Explanation:
Let's define the variables based on the problem statement.
\(w\) = Ken's monthly take-home pay (total income).
Let \(E\) be the total amount spent on food and rent.
\(x\) = The amount of money Ken has left (savings).
The relationship between these quantities is:
\[ \text{Savings} = \text{Total Income} - \text{Expenses} \] \[ x = w - E \] From this equation, we can express the expenses \(E\) as:
\[ E = w - x \] Now let's look at the quantities we need to compare.
Column A: \(x\) (the amount left/saved).
Column B: \(w - x\) (the amount spent on food and rent, which is \(E\)).
So, the problem is asking to compare the amount Ken saves with the amount he spends.
Step 3: Analyzing the Relationship:
The problem gives no information about how much Ken spends relative to how much he saves. We can consider different scenarios.
Scenario 1: Ken spends less than half his pay.
If \(w = \$1000\) and he spends \(E = \$400\) on food and rent, then he has \(x = \$1000 - \$400 = \$600\) left.
In this case, Column A is \(x = 600\) and Column B is \(w - x = 400\). Here, A \textgreater B.
Scenario 2: Ken spends more than half his pay.
If \(w = \$1000\) and he spends \(E = \$700\), then he has \(x = \$1000 - \$700 = \$300\) left.
In this case, Column A is \(x = 300\) and Column B is \(w - x = 700\). Here, B \textgreater A.
Scenario 3: Ken spends exactly half his pay.
If \(w = \$1000\) and he spends \(E = \$500\), then he has \(x = \$1000 - \$500 = \$500\) left.
In this case, Column A is \(x = 500\) and Column B is \(w - x = 500\). Here, A = B.
Step 4: Final Answer:
Since the relationship between Column A and Column B can change depending on Ken's spending habits, we cannot determine a fixed relationship from the information given.
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