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
Column A: The positive difference between Carol's age, in years, and her father's age, in years
Column B: 36
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 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.
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