A solution contains `15%` of salt, from which 30 kg of water is removed. The quantity of salt remains `20%` in that solution. Find the initial quantity of solution.

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript while providing clear explanations for each step.

### Step-by-Step Solution:

1. **Assume the Initial Quantity of Solution**:
Let the initial quantity of the solution be \( 100x \) kg.

**Hint**: When dealing with problems involving percentages, it's often useful to express the total quantity in terms of a variable.

2. **Calculate the Quantity of Salt**:
Since the solution contains \( 15\% \) of salt, the quantity of salt in the solution is:
\[ \text{Quantity of salt} = 15\% \text{ of } 100x = \frac{15}{100} \times 100x = 15x \text{ kg} \]

**Hint**: To find the quantity of a substance in a solution, multiply the percentage (as a decimal) by the total quantity.

3. **Remove Water from the Solution**:
When \( 30 \) kg of water is removed, the new quantity of the solution becomes:
\[ \text{New quantity of solution} = 100x - 30 \text{ kg} \]

**Hint**: Always keep track of changes in the total quantity when substances are added or removed.

4. **Set Up the Equation for New Salt Concentration**:
After removing the water, the quantity of salt remains the same (still \( 15x \) kg), but the concentration changes to \( 20\% \). Therefore, we can set up the equation:
\[ \text{New concentration of salt} = 20\% \text{ of the new solution} \]
This gives us:
\[ 15x = 20\% \text{ of } (100x - 30) = \frac{20}{100} \times (100x - 30) = \frac{1}{5} \times (100x - 30) \]

**Hint**: When the concentration changes, relate the amount of the substance to the new total quantity.

5. **Simplify the Equation**:
Now we can simplify the equation:
\[ 15x = \frac{1}{5} (100x - 30) \]
Multiplying both sides by \( 5 \) to eliminate the fraction:
\[ 75x = 100x - 30 \]

**Hint**: When solving equations, try to eliminate fractions to make calculations easier.

6. **Rearranging the Equation**:
Rearranging gives:
\[ 100x - 75x = 30 \]
\[ 25x = 30 \]

**Hint**: Always isolate the variable to solve for it.

7. **Solve for \( x \)**:
Dividing both sides by \( 25 \):
\[ x = \frac{30}{25} = \frac{6}{5} \]

**Hint**: Simplifying fractions can help in finding the value of variables more easily.

8. **Find the Initial Quantity of Solution**:
Since we assumed the initial quantity of the solution to be \( 100x \):
\[ \text{Initial quantity} = 100 \times \frac{6}{5} = 120 \text{ kg} \]

**Hint**: Always substitute back to find the original quantity after solving for the variable.

### Final Answer:
The initial quantity of the solution is \( 120 \) kg.