`t_(1//2)` is the half of a substance then `t_(3//4)` is the time in which substance

Options

A. substance decays /12
B. substance decays (3/4) th
C. substance remains 1/2
D. substance remains (3/4) th

Explanation

To solve the problem, we need to understand the concepts of half-life and decay of a substance.

### Step-by-Step Solution:

1. **Understanding Half-Life (t_(1/2))**:
- The half-life of a substance, denoted as \( t_{1/2} \), is the time required for half of the substance to decay. After one half-life, 50% of the original substance remains.

2. **Decay Over Time**:
- If we denote the initial amount of the substance as \( N_0 \), after one half-life (\( t_{1/2} \)), the remaining amount \( N \) can be calculated as:
\[ N = \frac{N_0}{2} \]

3. **Calculating for \( t_{3/4} \)**:
- The time \( t_{3/4} \) refers to the time taken for 75% of the substance to decay. This means only 25% of the original substance remains.
- If we denote the remaining amount after \( t_{3/4} \) as \( N \), we have:
\[ N = \frac{N_0}{4} \]

4. **Finding the Relationship**:
- To find the time \( t_{3/4} \), we can use the concept of half-lives. In order to go from \( N_0 \) to \( N_0/4 \), we need to go through two half-lives:
- After the first half-life (\( t_{1/2} \)): \( N = \frac{N_0}{2} \)
- After the second half-life (\( t_{1/2} \)): \( N = \frac{N_0}{4} \)

5. **Calculating Total Time**:
- Therefore, the total time \( t_{3/4} \) is:
\[ t_{3/4} = 2 \times t_{1/2} \]

### Final Answer:
Thus, the time \( t_{3/4} \) in which 75% of the substance has decayed is \( 2 \times t_{1/2} \).

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