The operation \(\diamondsuit\) is defined for all positive numbers \(r\) and \(t\) by r ♢ t = \(\frac{(r-t)^2 + rt}{t}\).
Column A: \(71 \diamondsuit 37\)
Column B: \(37 \diamondsuit 71\)
The operation \(\diamondsuit\) is defined for all positive numbers \(r\) and \(t\) by r ♢ t = \(\frac{(r-t)^2 + rt}{t}\).
Column A: \(71 \diamondsuit 37\)
Column B: \(37 \diamondsuit 71\)
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 A
Solution and Explanation
This is a "defined operation" problem. We need to apply the given rule to the numbers in each column and then compare the results. A good strategy is to first simplify the general formula for the operation.
Step 2: Key Formula or Approach:
The defined operation is \(r \diamondsuit t = \frac{(r-t)^2 + rt}{t}\).
Let's simplify the numerator of the expression:
\[ (r-t)^2 + rt = (r^2 - 2rt + t^2) + rt = r^2 - rt + t^2 \] So, the simplified rule for the operation is:
\[ r \diamondsuit t = \frac{r^2 - rt + t^2}{t} \] Step 3: Detailed Explanation:
Calculating Column A:
Here, \(r = 71\) and \(t = 37\).
Using the simplified formula:
\[ 71 \diamondsuit 37 = \frac{71^2 - (71)(37) + 37^2}{37} \] Calculating Column B:
Here, \(r = 37\) and \(t = 71\).
Using the simplified formula:
\[ 37 \diamondsuit 71 = \frac{37^2 - (37)(71) + 71^2}{71} \] Step 4: Comparing the Quantities:
Let's look at the numerators of both expressions.
Numerator of A: \(71^2 - (71)(37) + 37^2\)
Numerator of B: \(37^2 - (37)(71) + 71^2\)
The numerators are identical. Let's call this common value \(N\). Since \(r\) and \(t\) are positive, \(r^2-rt+t^2\) will be positive. (It can be written as \((r-t/2)^2 + 3t^2/4 \textgreater 0\)).
So we have:
Column A: \(\frac{N}{37}\)
Column B: \(\frac{N}{71}\)
Since \(N\) is a positive number, and we are dividing it by two different positive numbers, the fraction with the smaller denominator will be larger.
Because \(37 \textless 71\), it follows that:
\[ \frac{N}{37} \textgreater \frac{N}{71} \] Therefore, the quantity in Column A is greater than the quantity in Column B. We do not need to calculate the exact value of \(N\).
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