Let's determine the total capacity of the water tank step-by-step.
### Step 1: Identify the initial and final fullness
When the tank is initially [tex]\(\frac{3}{4}\)[/tex] full, we denote this initial fullness as follows:
[tex]\[ \text{Initial fullness} = \frac{3}{4} \][/tex]
After adding 52 Litres of water, the fullness changes to [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[ \text{Final fullness} = \frac{4}{5} \][/tex]
### Step 2: Let [tex]\( C \)[/tex] be the total capacity of the tank
Let's denote the total capacity of the tank by [tex]\( C \)[/tex] (in Litres).
### Step 3: Formulate the equation
Initially, the amount of water in the tank is:
[tex]\[ \frac{3}{4}C \][/tex]
After adding 52 Litres, the amount of water in the tank becomes:
[tex]\[ \frac{3}{4}C + 52 \][/tex]
Now, this amount of water corresponds to the final fullness of the tank:
[tex]\[ \frac{3}{4}C + 52 = \frac{4}{5}C \][/tex]
### Step 4: Solving for [tex]\( C \)[/tex]
Now, we solve the equation for [tex]\( C \)[/tex]:
[tex]\[
\frac{3}{4}C + 52 = \frac{4}{5}C
\][/tex]
Rearranging the equation to isolate [tex]\( C \)[/tex]:
[tex]\[
52 = \frac{4}{5}C - \frac{3}{4}C
\][/tex]
To combine the terms involving [tex]\( C \)[/tex], we need a common denominator:
[tex]\[
52 = \left( \frac{4 \cdot 4}{5 \cdot 4} \right)C - \left( \frac{3 \cdot 5}{4 \cdot 5} \right)C
\][/tex]
This simplifies to:
[tex]\[
52 = \left( \frac{16}{20} - \frac{15}{20} \right)C
\][/tex]
[tex]\[
52 = \frac{1}{20}C
\][/tex]
Solving for [tex]\( C \)[/tex]:
[tex]\[
C = 52 \times 20
\][/tex]
[tex]\[
C = 1040
\][/tex]
Therefore, the total capacity of the water tank is [tex]\( \boxed{1040 \text{ Litres}} \)[/tex].
