Sam's age is 2/5 of his uncle's age now. 5 years ago, his uncle was 25 years old. What will be their total age when Sam's age becomes 7/9 that of his uncle's age?

Let's break this down step by step.

### Step 1: Establish the current ages of Sam and his uncle.
Let:
- \( S \) represent Sam's current age,
- \( U \) represent his uncle's current age.

We are given that Sam's age is \( \frac{2}{5} \) of his uncle's age now, so we can write the equation:
\[
S = \frac{2}{5}U
\]

We are also told that 5 years ago, Sam's uncle was 25 years old. This gives us another equation:
\[
U - 5 = 25
\]
Solving for \( U \):
\[
U = 30
\]
Now, substitute \( U = 30 \) into the first equation to find Sam's current age:
\[
S = \frac{2}{5} \times 30 = 12
\]

### Step 2: Determine their ages when Sam's age becomes \( \frac{7}{9} \) of his uncle's age.
Let \( x \) represent the number of years from now when Sam's age becomes \( \frac{7}{9} \) of his uncle's age. At that time, their ages will be:
- Sam's age: \( S + x = 12 + x \),
- Uncle's age: \( U + x = 30 + x \).

We are told that at that time, Sam's age will be \( \frac{7}{9} \) of his uncle's age, so:
\[
12 + x = \frac{7}{9}(30 + x)
\]
Now, solve this equation for \( x \).

### Step 3: Solve the equation.
First, eliminate the fraction by multiplying both sides by 9:
\[
9(12 + x) = 7(30 + x)
\]
Expanding both sides:
\[
108 + 9x = 210 + 7x
\]
Now, isolate \( x \) by moving terms involving \( x \) to one side and constants to the other:
\[
9x - 7x = 210 - 108
\]
\[
2x = 102
\]
\[
x = 51
\]

### Step 4: Find their total age when Sam's age becomes \( \frac{7}{9} \) of his uncle's age.
In 51 years, Sam's age will be:
\[
12 + 51 = 63
\]
And his uncle's age will be:
\[
30 + 51 = 81
\]
Thus, their total age at that time will be:
\[
63 + 81 = 144
\]

### Final Answer:
The total age of Sam and his uncle when Sam's age becomes \( \frac{7}{9} \) of his uncle's age will be 144 years.

Would you like a more detailed explanation or have any other questions?

Here are 5 related questions:
1. How would the equation change if Sam's current age was 3/5 of his uncle's age?
2. If Sam’s uncle was 35 years old 5 years ago, how would that affect the total age?
3. How can we generalize this problem to other age ratios?
4. What if instead of 5 years ago, the uncle's age was given 10 years ago?
5. How would the total age differ if Sam's age was 5/9 of his uncle's age instead?

**Tip:** Always try to set up relationships between ages using algebraic expressions—it simplifies the problem!

Learn how to solve an age-related algebra problem where Sam's age is 2/5 of his uncle's age, and 5 years ago his uncle was 25 years old. This problem covers proportions, solving linear equations, and calculating total age when Sam's age becomes 7/9 of his uncle's. Suitable for Grades 7-9.

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