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12. A water tank is [tex]\frac{3}{4}[/tex] full and after adding 52 liters, it is [tex]\frac{4}{5}[/tex] full. Determine its total capacity in cubic meters.

(4 Marks)

Sagot :

GinnyAnswer
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].
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