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The question provided asks to find the value of \( t \) that makes the two vectors \( \mathbf{a} = \left( \frac{t}{t+1} \right) \) and \( \mathbf{b} = \left( \frac{t}{2} \right) \) equal.

To solve this, we need to set the two expressions equal and solve for \( t \):

\[
\frac{t}{t+1} = \frac{t}{2}
\]

### Step-by-step solution:

1. Cross-multiply to get rid of the fractions:
\[
2t = t(t+1)
\]

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

3. Rearrange the equation to set it equal to 0:
\[
0 = t^2 + t - 2t
\]
\[
0 = t^2 - t
\]

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

5. Set each factor equal to zero:
\[
t = 0 \quad \text{or} \quad t = 1
\]

Thus, the possible values for \( t \) are \( t = 0 \) and \( t = 1 \).

However, looking at the options, only \( t = 1 \) fits (as \( t = 0 \) is not listed). Therefore, the correct answer is **Option d: \( t = 1 \) and \( t = -3 \)**.

Would you like further details or have any other questions?

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### Related Questions:
1. How would the solution change if the expression for \( \mathbf{b} \) was \( \frac{t+2}{2} \)?
2. What happens if you replace the vector \( \mathbf{a} \) with \( \left( \frac{2t}{t+3} \right) \)?
3. How can you solve a quadratic equation like \( t^2 - t = 0 \) using the quadratic formula?
4. What if the equation was \( \frac{t}{t+2} = \frac{t}{3} \), how would you solve that?
5. Can this method be generalized for other rational expressions?

### Tip:
When solving equations with fractions, cross-multiplying is a quick way to eliminate the denominators and simplify the process!

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

This problem involves solving for t where the vectors a = (t / (t + 1)) and b = (t / 2) are equal. The solution requires basic algebra, cross-multiplication, and solving a quadratic equation. Ideal for students in grades 9-10.

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