Question

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: \(37 \diamondsuit 71\) 
 

• 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:

In problems with defined operations, especially in quantitative comparisons, look for symmetries or simplify the general formula before plugging in numbers. Often, you can compare the results algebraically without performing the full arithmetic calculation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
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 \(N\).