before 3 years thomas age was 4 times the age his son .After 10 years thomas age will be 5 more than 2 times the sons's age , find their present age?

Let's solve the problem step by step.

### Step 1: Define the variables
Let:
- \( T \) be the present age of Thomas.
- \( S \) be the present age of his son.

### Step 2: Set up the equations based on the problem statement

**First condition (3 years ago):**
- 3 years ago, Thomas's age was \( T - 3 \).
- 3 years ago, his son's age was \( S - 3 \).
- According to the problem, 3 years ago, Thomas's age was 4 times his son's age:
  \[
  T - 3 = 4(S - 3)
  \]

**Second condition (10 years in the future):**
- 10 years from now, Thomas's age will be \( T + 10 \).
- 10 years from now, his son's age will be \( S + 10 \).
- According to the problem, 10 years from now, Thomas's age will be 5 more than 2 times his son's age:
  \[
  T + 10 = 2(S + 10) + 5
  \]

### Step 3: Solve the system of equations

**Equation 1:**
\[
T - 3 = 4(S - 3)
\]
Expanding the equation:
\[
T - 3 = 4S - 12
\]
Adding 3 to both sides:
\[
T = 4S - 9
\]

**Equation 2:**
\[
T + 10 = 2(S + 10) + 5
\]
Expanding the equation:
\[
T + 10 = 2S + 20 + 5
\]
Simplifying:
\[
T + 10 = 2S + 25
\]
Subtracting 10 from both sides:
\[
T = 2S + 15
\]

Now, we have the two equations:
1. \( T = 4S - 9 \)
2. \( T = 2S + 15 \)

### Step 4: Solve for \( S \) (the son's age)

Since both expressions are equal to \( T \), we can set them equal to each other:
\[
4S - 9 = 2S + 15
\]
Subtract \( 2S \) from both sides:
\[
2S - 9 = 15
\]
Add 9 to both sides:
\[
2S = 24
\]
Divide by 2:
\[
S = 12
\]

### Step 5: Find \( T \) (Thomas's age)

Substitute \( S = 12 \) into one of the original equations. Using \( T = 4S - 9 \):
\[
T = 4(12) - 9 = 48 - 9 = 39
\]

### Final Answer:
- Thomas's present age \( T = 39 \) years.
- His son's present age \( S = 12 \) years.

Would you like any additional details or explanations? Here are some related questions:

1. How do you check the solution for consistency?
2. What if the age difference was different in the future condition?
3. How would the solution change if the son's age was given instead?
4. How can such age-related problems be generalized?
5. What is the significance of creating equations from word problems?

**Tip:** Always verify your solutions by substituting the values back into the original conditions to ensure consistency.

Explore how to solve age-related word problems with a step-by-step solution. This problem involves finding the present ages of Thomas and his son based on given conditions. Suitable for students in Grades 7-9.

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