Five years ago, a person's age was 2/3 ^ r0 her mother's age, The sum of their present ages 120. Find the difference (in years) of their present ages.



Let the present age of the person be \( x \) years and the present age of the mother be \( y \) years.

### Step 1: Set up the equations based on the information given.
1. **Five years ago**:
   - The person's age was \( x - 5 \) years.
   - The mother's age was \( y - 5 \) years.
   - According to the problem, five years ago, the person's age was \( \frac{2}{3} \) of her mother's age.
   \[
   x - 5 = \frac{2}{3} \times (y - 5)
   \]
   
2. **Sum of their present ages**:
   - The sum of their present ages is 120.
   \[
   x + y = 120
   \]

### Step 2: Solve the equations.
1. Start with the equation from five years ago:
   \[
   3(x - 5) = 2(y - 5)
   \]
   Expanding and simplifying:
   \[
   3x - 15 = 2y - 10
   \]
   \[
   3x - 2y = 5
   \]

2. Now, use the sum of their present ages:
   \[
   x + y = 120
   \]
   From this, solve for \( y \):
   \[
   y = 120 - x
   \]

### Step 3: Substitute \( y \) in the first equation.
Substitute \( y = 120 - x \) into \( 3x - 2y = 5 \):
\[
3x - 2(120 - x) = 5
\]
Simplify and solve for \( x \):
\[
3x - 240 + 2x = 5
\]
\[
5x = 245
\]
\[
x = 49
\]

### Step 4: Find \( y \) and the difference in their ages.
Now, substitute \( x = 49 \) back into the equation \( y = 120 - x \):
\[
y = 120 - 49 = 71
\]

So, the present ages are 49 years for the person and 71 years for the mother.

### Step 5: Calculate the difference in their present ages.
The difference in their present ages is:
\[
y - x = 71 - 49 = 22 \text{ years}
\]

### Final Answer:
The difference in their present ages is **22 years**.

Would you like more details on any step or have any other questions?

### Related Questions:
1. What would be the equation if the person's age was half of her mother's age five years ago?
2. How would the solution change if their age difference was given instead of the sum of their ages?
3. If the sum of their ages was 130 instead of 120, what would be the difference in their ages?
4. What if the ratio of their ages five years ago was 3:4 instead of 2:3?
5. How does changing the ratio of their ages five years ago affect the difference in their present ages?
6. What is the sum of their ages 10 years from now?
7. What if the mother's age was given as three times the person's age five years ago?
8. How does the solution approach change if we know their current age difference instead of their age sum?

### Tip:
When solving age-related word problems, translating the problem into equations is key. Always double-check the conditions given, especially the time frame, like "five years ago" or "in ten years," to ensure correct setup.

Learn how to solve age-related problems using linear equations. This problem involves finding the age difference between a person and their mother based on their current ages and a past relationship. Suitable for Grades 7-9.

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