The age of a woman is two-digit integer. On reversing this integer, the new integer is the age of her husband who is elder to her. The difference between their ages is one eleventh of their sum. What is the difference between their ages?
9 years
Solving the Age Word Problem with Reversed Digits
Let's break down this interesting age word problem step by step. We are given information about a woman's age and her husband's age, which are related by reversing the digits of a two-digit number.
Representing the Ages Using Digits
A two-digit integer can be represented using two digits, say \(a\) and \(b\), where \(a\) is the tens digit and \(b\) is the units digit.
- Let the woman's age be \(10a + b\). Here, \(a\) must be a digit from 1 to 9 (since it's a two-digit number, the tens digit cannot be 0) and \(b\) must be a digit from 0 to 9.
- When the digits are reversed, the new integer represents the husband's age. So, the husband's age is \(10b + a\).
We are told the husband is elder than the woman. This means \(10b + a > 10a + b\).
Setting Up the Equation Based on Age Difference and Sum
The problem states that the difference between their ages is one eleventh of their sum.
- Difference in ages: \((10b + a) - (10a + b)\)
- Sum of ages: \((10a + b) + (10b + a)\)
The given condition translates to the equation:
\((10b + a) - (10a + b) = \frac{1}{11} \times ((10a + b) + (10b + a))\)
Solving the Algebraic Equation
Let's simplify both sides of the equation.
Simplify the difference side:
\((10b + a) - (10a + b)\)
\(= 10b - b + a - 10a\)
\(= 9b - 9a\)
\(= 9(b - a)\)
Simplify the sum side:
\((10a + b) + (10b + a)\)
\(= 10a + a + b + 10b\)
\(= 11a + 11b\)
\(= 11(a + b)\)
Substitute the simplified expressions back into the main equation:
\(9(b - a) = \frac{1}{11} \times 11(a + b)\)
\(9(b - a) = a + b\)
Now, distribute and rearrange the terms to find the relationship between \(a\) and \(b\).
\(9b - 9a = a + b\)
\(9b - b = a + 9a\)
\(8b = 10a\)
Divide both sides by 2:
\(4b = 5a\)
Finding the Possible Digits \(a\) and \(b\)
We have the equation \(4b = 5a\), where \(a\) is the tens digit of the woman's age (\(a \in \{1, 2, \dots, 9\}\)) and \(b\) is the units digit (\(b \in \{0, 1, \dots, 9\}\)).
For \(4b\) to be equal to \(5a\), \(4b\) must be a multiple of 5, which means \(b\) must be a multiple of 5. Also, \(5a\) must be a multiple of 4, which means \(a\) must be a multiple of 4.
Let's check possible values for \(a\) (multiples of 4 from 1 to 9):
- If \(a=4\), then \(4b = 5 \times 4 = 20\). So, \(b = 20 / 4 = 5\).
- This gives \(a=4, b=5\). Both are valid digits.
- Woman's age: \(10a + b = 10(4) + 5 = 45\).
- Husband's age: \(10b + a = 10(5) + 4 = 54\).
- If \(a=8\), then \(4b = 5 \times 8 = 40\). So, \(b = 40 / 4 = 10\).
- This gives \(a=8, b=10\). \(b=10\) is not a single digit, so this solution is not valid for a two-digit number.
Let's check possible values for \(b\) (multiples of 5 from 0 to 9):
- If \(b=0\), then \(4(0) = 5a\), so \(0 = 5a\), which means \(a=0\).
- This gives \(a=0, b=0\). The woman's age would be 00, which is not a two-digit number. Also, the husband's age would be 00. Not a valid solution.
- If \(b=5\), then \(4(5) = 5a\), so \(20 = 5a\), which means \(a = 20 / 5 = 4\).
- This gives \(a=4, b=5\). Both are valid digits.
- Woman's age: \(10a + b = 10(4) + 5 = 45\).
- Husband's age: \(10b + a = 10(5) + 4 = 54\).
The only combination of digits that works is \(a=4\) and \(b=5\).
Verifying the Ages
Using \(a=4\) and \(b=5\):
- Woman's age: \(45\) years.
- Husband's age: \(54\) years.
Check the conditions:
- Is the woman's age a two-digit integer? Yes, 45.
- Is the husband's age (reversed digits) also an integer? Yes, 54.
- Is the husband elder? \(54 > 45\). Yes.
- Is the difference one eleventh of the sum?
- Difference: \(54 - 45 = 9\).
- Sum: \(54 + 45 = 99\).
- One eleventh of the sum: \(\frac{1}{11} \times 99 = 9\).
The difference (9) is indeed one eleventh of the sum (99). All conditions are met with the ages 45 and 54.
Calculating the Difference in Ages
The question asks for the difference between their ages.
Difference = Husband's age - Woman's age
Difference = \(54 - 45\)
Difference = \(9\)
The difference between their ages is 9 years.
| Concept | Representation | Calculations |
|---|---|---|
| Woman's Age | \(10a + b\) | \(10(4) + 5 = 45\) |
| Husband's Age (Reversed) | \(10b + a\) | \(10(5) + 4 = 54\) |
| Relationship | \(4b = 5a\) | Derived from the difference/sum condition |
| Valid Digits | \(a=4, b=5\) | Found by checking digit constraints |
| Difference | \((10b+a) - (10a+b)\) | \(54 - 45 = 9\) |
| Sum | \((10a+b) + (10b+a)\) | \(45 + 54 = 99\) |
| Check Condition | Difference = \(\frac{1}{11}\) Sum | \(9 = \frac{1}{11} \times 99\) (True) |
Revision Table: Key Concepts in Age Problems
| Concept | Description | How it applies here |
|---|---|---|
| Representing Two-Digit Numbers | A number with tens digit \(a\) and units digit \(b\) is \(10a + b\). | Woman's age is \(10a + b\). |
| Reversing Digits | The number with digits reversed is \(10b + a\). | Husband's age is \(10b + a\). |
| Setting Up Equations | Translate word problem conditions into algebraic equations. | Difference is \(\frac{1}{11}\) of the sum gives the equation \(9(b - a) = \frac{1}{11} \times 11(a + b)\). |
| Solving Diophantine-like Equations | Finding integer solutions for equations involving variables representing digits. | Solving \(4b = 5a\) for single digits \(a, b\) with \(a \ne 0\). |
Additional Information: Digit Reversal Problems
Problems involving reversing digits of numbers are common in number theory and algebra. The key is understanding how place value works.
- A two-digit number \(ab\) is \(10a + b\). The reversed number \(ba\) is \(10b + a\).
- The difference between a two-digit number and its reverse is always a multiple of 9: \((10a + b) - (10b + a) = 9a - 9b = 9(a - b)\).
- The sum of a two-digit number and its reverse is always a multiple of 11: \((10a + b) + (10b + a) = 11a + 11b = 11(a + b)\).
In this specific age problem, we used both of these properties implicitly. The difference being related to the sum allowed us to find a relationship between the digits \(a\) and \(b\), and then identify the unique pair of digits that fit the constraints of being a two-digit number and single digits.
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