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The problem asks for the value of \( t \) for which the vectors \( a = \left( \frac{t}{t+1} \right) \) and \( b = \left( \frac{t}{2} \right) \) are equivalent.

For the vectors to be equivalent, their components must be equal:
\[
\frac{t}{t+1} = \frac{t}{2}
\]

Now, let's solve for \( t \).

1. Start with the equation:
   \[
   \frac{t}{t+1} = \frac{t}{2}
   \]

2. Cross-multiply to eliminate the denominators:
   \[
   2t = t(t+1)
   \]

3. Expand the right-hand side:
   \[
   2t = t^2 + t
   \]

4. Rearrange the equation to form a quadratic equation:
   \[
   t^2 + t - 2t = 0 \quad \Rightarrow \quad t^2 - t = 0
   \]

5. Factor the quadratic equation:
   \[
   t(t - 1) = 0
   \]

6. Solve for \( t \):
   \[
   t = 0 \quad \text{or} \quad t = 1
   \]

Thus, \( t = 0 \) or \( t = 1 \).

Now let's check which option matches these values:
- \( t = 1 \) corresponds to the option **b. \( t = 1 \) og \( t = 3 \)**.

Would you like further clarification on any step?

Here are some follow-up questions you may find helpful:
1. How do we solve equations involving fractions like these more efficiently?
2. What does cross-multiplication achieve in solving equations like this?
3. How can we verify if our solutions for \( t \) are correct?
4. What if we encountered a more complex denominator for \( t \)? How would we solve it?
5. How can factoring help solve quadratic equations quickly?

**Tip:** Always check if solutions make sense by plugging them back into the original equation!

Reiknaðu fyrir hvaða gildi á t, vigurinn a = (t / (t+1)) og b = (t/2) eru jafngildir.

Learn how to solve for t in the equation a = (t / (t+1)) and b = (t/2), where the vectors are equivalent. This problem involves key algebraic concepts like cross-multiplication and factoring quadratic equations. Suitable for students in Grades 10-12.

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