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The problem states:

"Kate's father is three times her age. How old are Kate and her father now if, in 4 years' time, the sum of their ages is 68? Use \( K \) to represent Kate's age."

### Solution:

1. Let \( K \) represent Kate's current age.
2. Kate's father's current age is \( 3K \), since he is three times her age.
3. In 4 years, Kate's age will be \( K + 4 \), and her father's age will be \( 3K + 4 \).

In 4 years, the sum of their ages will be 68. Thus, the equation becomes:
\[
(K + 4) + (3K + 4) = 68
\]
Simplifying this:
\[
K + 4 + 3K + 4 = 68
\]
\[
4K + 8 = 68
\]
Subtract 8 from both sides:
\[
4K = 60
\]
Now, divide both sides by 4:
\[
K = 15
\]

So, Kate is 15 years old.

4. Kate's father is \( 3 \times 15 = 45 \) years old.

### Final Answer:
- Kate is 15 years old.
- Her father is 45 years old.

Would you like a detailed explanation of any step, or have any questions?

Here are 5 related questions you might explore:
1. If Kate’s father was four times her age, how would that affect their current ages?
2. How old will Kate and her father be in 10 years?
3. What would happen if the sum of their ages in 4 years was 80 instead of 68?
4. Can you create a similar problem with different age relationships?
5. If the age difference between Kate and her father was constant, what would that be?

**Tip:** When solving word problems, always define variables clearly and write an equation that represents the relationship between the given values.

Kate's father is three times her age. How old are Kate and her father now if, in 4 years time, the sum of their ages is 68? Use K to represent Kate's age.

Learn how to solve the age problem where Kate’s father is three times her age, and in 4 years, their ages sum to 68. This detailed step-by-step solution covers algebraic concepts suitable for students in Grades 6-8.

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