Question

If the length of Wanda's telephone call was rounded up to the nearest whole minute by her telephone company, then Wanda was charged for how many minutes for her telephone call?
(1) The total charge for Wanda's telephone call was \$6.50.
(2) Wanda was charged \$0.50 more for the first minute of the telephone call than for each minute after the first.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

In word problems involving variables, count the number of independent equations you can form and the number of unknown variables. If the number of variables is greater than the number of equations, you generally cannot find a unique solution.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
We need to find the number of minutes Wanda was charged for. Let's denote this by \(M\). The charging structure involves a rate for the first minute and a different rate for subsequent minutes.
Step 2: Key Formula or Approach:
Let \(C_1\) be the charge for the first minute and \(C_s\) be the charge for each subsequent minute. The total charge \(T\) for \(M\) minutes is given by: \[ T = C_1 + (M-1)C_s \] Step 3: Detailed Explanation:
Analyze Statement (1): The total charge for Wanda's telephone call was \$6.50.
This gives us the equation: \(6.50 = C_1 + (M-1)C_s\).
We have one equation with three unknowns (\(M\), \(C_1\), \(C_s\)). We cannot solve for \(M\). For example, if the call was 1 minute (\(M=1\)), the cost for the first minute would be \$6.50. If the call was 2 minutes (\(M=2\)), we have \(6.50 = C_1 + C_s\), which is possible for many different rate combinations. Statement (1) is not sufficient.
Analyze Statement (2): Wanda was charged \$0.50 more for the first minute of the telephone call than for each minute after the first.
This gives us the relationship: \(C_1 = C_s + 0.50\).
This statement provides a relationship between the rates but gives no information about the total cost or the number of minutes. Statement (2) is not sufficient.
Analyze Both Statements Together:
We have two equations: 1) \(6.50 = C_1 + (M-1)C_s\) 2) \(C_1 = C_s + 0.50\) Substitute the second equation into the first: \[ 6.50 = (C_s + 0.50) + (M-1)C_s \] \[ 6.50 = C_s + 0.50 + MC_s - C_s \] \[ 6.50 = MC_s + 0.50 \] \[ 6.00 = MC_s \] We are left with one equation, \(MC_s = 6\), with two unknowns (\(M\) and \(C_s\)). We cannot find a unique value for \(M\). For example:

If \(C_s = \$1.00\), then \(M=6\) minutes.
If \(C_s = \$0.50\), then \(M=12\) minutes.
If \(C_s = \$0.75\), then \(M=8\) minutes.
Since we cannot find a unique value for \(M\), the statements together are not sufficient.
Step 4: Final Answer:
Even with both pieces of information, the number of minutes charged cannot be uniquely determined.