Question

Carol's age, in years, can be expressed by reversing the digits in her father's age, in years. The sum of the digits in each age is 10.
Column A: The positive difference between Carol's age, in years, and her father's age, in years
Column B: 36

• 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:

If a quantitative comparison problem allows for multiple possible values for one of the columns, and these values have different relationships (greater than, equal to, less than) with the other column, the answer must be (D).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a word problem that translates to a number theory problem. We need to find all possible ages that fit the description and then calculate the difference. If the difference is not a unique value, we cannot make a definitive comparison.
Step 2: Detailed Explanation:
Set up equations. Let the father's age be the two-digit number \(10t + u\), where \(t\) is the tens digit and \(u\) is the units digit. Carol's age is the reverse, \(10u + t\). The father must be older, so \(10t + u \textgreater 10u + t\), which simplifies to \(t \textgreater u\).
Use the given conditions. We are told the sum of the digits is 10: \(t + u = 10\).
Find possible pairs of digits. We need integer pairs \((t, u)\) such that \(t+u=10\) and \(t\textgreater u\). - If \(t=9\), \(u=1\). (Father: 91, Carol: 19) - If \(t=8\), \(u=2\). (Father: 82, Carol: 28) - If \(t=7\), \(u=3\). (Father: 73, Carol: 37) - If \(t=6\), \(u=4\). (Father: 64, Carol: 46)
4. Calculate the difference for each case. The difference is \((10t+u) - (10u+t) = 9t - 9u = 9(t-u)\). - Case 1 (91, 19): Difference = \(9(9-1) = 72\). - Case 2 (82, 28): Difference = \(9(8-2) = 54\). - Case 3 (73, 37): Difference = \(9(7-3) = 36\). - Case 4 (64, 46): Difference = \(9(6-4) = 18\).
Step 3: Comparing the Quantities:
Column A, the positive difference, can be 72, 54, 36, or 18.
Column B is the fixed value 36.
- Column A could be 72 or 54 (greater than Column B). - Column A could be 36 (equal to Column B). - Column A could be 18 (less than Column B).
Since the quantity in Column A does not have a unique value, we cannot determine a fixed relationship.