Question

COLUMN A: \(37\times\frac{37}{36}\) 
COLUMN B: \(37+\frac{37}{36}\)

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

Hint:

When comparing complex-looking arithmetic expressions, try to find an algebraic relationship. Factoring or finding a common denominator can reveal that the two expressions are actually the same, as in this case where \(a \times \frac{a}{a-1}\) is being compared to \(a + \frac{a}{a-1}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The problem asks us to compare two expressions involving the same numbers: \(37\) and \(\frac{37}{36}\). Specifically, we are asked to compare: \[ \text{Column A: } 37 \times \frac{37}{36},
\text{Column B: } 37 + \frac{37}{36}. \] At first glance, it might seem that a product and a sum would naturally differ, but we need to carefully compute both values to determine which is larger—or if they are equal. Step 2: Key Formula or Approach:
We can approach this problem systematically using the following methods:

Express both Column A and Column B with a common denominator so that we can compare them directly.
Alternatively, use a general algebraic check: for two numbers \(a\) and \(b\), the product \(ab\) is equal to the sum \(a+b\) if \((a-1)(b-1) = 1\). This is derived as follows: \[ ab = a+b \implies ab - a - b = 0 \implies (a-1)(b-1) = 1. \] This formula can help confirm whether the two quantities are equal.
Step 3: Detailed Explanation:
Step 3.1: Evaluate Column A \[ \text{Column A} = 37 \times \frac{37}{36} = \frac{37 \times 37}{36} = \frac{1369}{36}. \] This is a straightforward calculation of the product. Step 3.2: Evaluate Column B \[ \text{Column B} = 37 + \frac{37}{36}. \] To combine these terms, we write 37 as a fraction with denominator 36: \[ 37 = \frac{37 \times 36}{36} = \frac{1332}{36}. \] Now add the second term: \[ \text{Column B} = \frac{1332}{36} + \frac{37}{36} = \frac{1332 + 37}{36} = \frac{1369}{36}. \] Step 3.3: Compare Column A and Column B Both Column A and Column B simplify to the same fraction: \[ \frac{1369}{36}. \] Thus, the two quantities are exactly equal. Step 3.4: Verification Using the General Formula For two numbers \(a\) and \(b\), we can verify equality using: \[ ab = a+b \iff (a-1)(b-1) = 1. \] Here, \(a = 37\) and \(b = \frac{37}{36}\): \[ a-1 = 37-1 = 36,
b-1 = \frac{37}{36} - 1 = \frac{1}{36}. \] Multiply: \[ (a-1)(b-1) = 36 \cdot \frac{1}{36} = 1. \] This confirms that indeed \(ab = a+b\), which agrees with our direct calculation. Step 4: Observations and Conclusion Even though one expression is a product and the other is a sum, in this specific case the two quantities turn out to be equal. This is due to the particular relationship between the numbers: one number is slightly greater than 1 and the other is large enough to satisfy \((a-1)(b-1) = 1\). Step 5: Final Answer: \[ \boxed{\text{Column A = Column B}} \]