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