Question

\(6 \textless x \textless 7\)
\(y = 8\)
Column A: \(\frac{x}{y}\)
Column B: 0.85

• A. Quantity A is greater
• B. Quantity B is greater
• C. The two quantities are equal
• D. The relationship cannot be determined from the information given

Hint:

When dealing with inequalities in quantitative comparisons, determine the full possible range of the quantity in question. If the value in the other column falls within that range, the answer is almost always (D). If it falls outside the range, you can determine a relationship.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves inequalities. We are given a range for the variable \(x\) and a fixed value for \(y\). We need to determine the possible range for the expression \(\frac{x}{y}\) and compare it to a fixed value.
Step 2: Detailed Explanation:
We are given the inequality \(6 \textless x \textless 7\).
We are also given \(y = 8\).
To find the range for \(\frac{x}{y}\), we can divide the entire inequality for \(x\) by the value of \(y\). Since \(y=8\) is a positive number, the direction of the inequality signs will not change.
\[ \frac{6}{y} \textless \frac{x}{y} \textless \frac{7}{y} \] Substituting \(y = 8\):
\[ \frac{6}{8} \textless \frac{x}{y} \textless \frac{7}{8} \] Now, let's convert these fractions to decimals to make the comparison easier.
\[ \frac{6}{8} = \frac{3}{4} = 0.75 \] \[ \frac{7}{8} = 0.875 \] So, the range for the quantity in Column A is:
\[ 0.75 \textless \frac{x}{y} \textless 0.875 \] Step 3: Comparing the Quantities:
Column A: A value strictly between 0.75 and 0.875.
Column B: 0.85
The value 0.85 lies within the possible range for Column A.
This means that the quantity in Column A could be less than, equal to, or greater than 0.85.
Scenario 1: If \(x = 6.8\), then \(\frac{x}{y} = \frac{6.8}{8} = 0.85\). In this case, A = B.
Scenario 2: If \(x = 6.4\), then \(\frac{x}{y} = \frac{6.4}{8} = 0.8\). In this case, B \textgreater A.
Scenario 3: If \(x = 6.9\), then \(\frac{x}{y} = \frac{6.9}{8} = 0.8625\). In this case, A \textgreater B.
Step 4: Final Answer:
Since the value in Column A could be less than, equal to, or greater than the value in Column B, the relationship cannot be determined from the given information.