To solve the problem, we need to establish the relationships between the constants \( a \), \( b \), and \( c \) based on the given equations involving mean, median, and mode.
### Step-by-Step Solution:
1. **Understanding the Standard Relation**:
The standard relation between mean, median, and mode is given by:
\[
3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean}
\]
This will be our reference point for deriving the values of \( a \), \( b \), and \( c \).
2. **Finding \( a \)**:
The first equation given is:
\[
\text{Mean} = a \times (3 \times \text{Median} - \text{Mode})
\]
From the standard relation, we can rearrange it to express Mean:
\[
2 \times \text{Mean} = 3 \times \text{Median} - \text{Mode}
\]
Dividing both sides by 2 gives:
\[
\text{Mean} = \frac{1}{2} \times (3 \times \text{Median} - \text{Mode})
\]
Thus, we can conclude:
\[
a = \frac{1}{2}
\]
3. **Finding \( b \)**:
The second equation given is:
\[
\text{Mean} - \text{Mode} = b \times (\text{Mean} - \text{Median})
\]
Rearranging the standard relation, we have:
\[
3 \times \text{Median} - \text{Mean} = \text{Mode} - \text{Mean}
\]
Multiplying both sides by -1 gives:
\[
\text{Mean} - \text{Mode} = 3 \times (\text{Mean} - \text{Median})
\]
Therefore, we can conclude:
\[
b = 3
\]
4. **Finding \( c \)**:
The third equation given is:
\[
\text{Median} = \text{Mode} + c \times (\text{Mean} - \text{Mode})
\]
From the standard relation, we can express Median as:
\[
\text{Median} = \frac{1}{3} \times \text{Mode} + \frac{2}{3} \times \text{Mean}
\]
Rearranging gives:
\[
\text{Median} = \text{Mode} - \frac{2}{3} \times \text{Mode} + \frac{2}{3} \times \text{Mean}
\]
This implies:
\[
c = \frac{2}{3}
\]
5. **Establishing the Inequality**:
Now we have:
\[
a = \frac{1}{2}, \quad b = 3, \quad c = \frac{2}{3}
\]
We can compare these values:
\[
\frac{1}{2} < \frac{2}{3} < 3
\]
Hence, we conclude:
\[
a < c < b
\]
### Final Result:
The relationships between \( a \), \( b \), and \( c \) are:
\[
a < c < b
\]
