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!