Karim is the father of Abdul. Karim's age 10 years ago was three times the age of his son. 10 years hence, Karim's age will be two times the age of Abdul. Find the ratio of their present ages
5:4
9: 6
8:6
7: 30

Let's represent Karim's and Abdul's present ages by:

- \( K \) = Karim's present age
- \( A \) = Abdul's present age

### Step 1: Formulate the equations
#### Equation 1: 10 years ago
Karim's age 10 years ago was three times Abdul's age 10 years ago.
\[
K - 10 = 3(A - 10)
\]
Simplifying this equation:
\[
K - 10 = 3A - 30
\]
\[
K = 3A - 20 \quad \text{(Equation 1)}
\]

#### Equation 2: 10 years from now
In 10 years, Karim's age will be twice Abdul's age.
\[
K + 10 = 2(A + 10)
\]
Simplifying this equation:
\[
K + 10 = 2A + 20
\]
\[
K = 2A + 10 \quad \text{(Equation 2)}
\]

### Step 2: Solve the system of equations
We now have two equations:
1. \( K = 3A - 20 \)
2. \( K = 2A + 10 \)

Set them equal to each other:
\[
3A - 20 = 2A + 10
\]
Solve for \( A \):
\[
3A - 2A = 10 + 20
\]
\[
A = 30
\]

Now, substitute \( A = 30 \) into one of the equations (use Equation 2):
\[
K = 2(30) + 10 = 60 + 10 = 70
\]

### Step 3: Find the ratio
Karim's present age \( K = 70 \) and Abdul's present age \( A = 30 \). The ratio of their present ages is:
\[
\frac{K}{A} = \frac{70}{30} = \frac{7}{3}
\]

Thus, the ratio of their present ages is **7:3**, which isn't one of the given options. However, it seems the closest plausible choice might have been **7:30**, possibly due to a typo in the question.

Do you want more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How can we adjust the problem if the ratio was 5:4?
2. What if the time periods (10 years ago and 10 years hence) were changed?
3. How would the system of equations change if the conditions involved percentages instead of multiples of ages?
4. What if Karim's future age was three times Abdul's instead of two?
5. How do such ratio problems relate to simultaneous equations?

**Tip:** When solving age-related problems, clearly defining the present age in terms of variables helps simplify the process of setting up equations.

Karim is the father of Abdul. Karim's age 10 years ago was three times the age of his son. 10 years hence, Karim's age will be two times the age of Abdul. Find the ratio of their present ages.

This problem involves solving a system of linear equations to find the ratio of the present ages of Karim and Abdul. Learn how to set up equations from word problems involving age and solve for their ages. Suitable for Grades 8-10 students interested in algebra and real-life application problems.

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