A hydraulic press can lift 100 kg when a mass 'm' is placed on the smaller piston. It can lift _____kg when the diameter of the larger piston is increased by 4 times and that of the smaller piston is decreased by 4 times keeping the same mass 'm' on the smaller piston.

Answer

To solve the problem, we will use Pascal's law, which states that pressure applied to a confined fluid is transmitted undiminished in every direction throughout the fluid.

### Step-by-Step Solution:

1. **Understanding the Initial Setup**:
- Let the area of the smaller piston be \( A_1 \) and the area of the larger piston be \( A_2 \).
- The hydraulic press can lift a mass of 100 kg when a mass \( m \) is placed on the smaller piston. The force exerted by the mass \( m \) is given by:
\[ F_1 = mg = 100g \quad (\text{where } g \text{ is the acceleration due to gravity}) \]

2. **Using Pascal's Law**:
- According to Pascal's law:
\[ \frac{F_1}{A_1} = \frac{F_2}{A_2} \]
- Here, \( F_2 \) is the force exerted by the larger piston, which can lift a mass \( M \):
\[ F_2 = Mg \]

3. **Setting Up the Equation**:
- Substituting the forces into Pascal's law gives:
\[ \frac{100g}{A_1} = \frac{Mg}{A_2} \]
- Simplifying this, we get:
\[ \frac{100}{A_1} = \frac{M}{A_2} \]
- Rearranging gives us:
\[ M = \frac{100 A_2}{A_1} \]

4. **Changing the Areas**:
- The diameter of the larger piston is increased by 4 times, and the diameter of the smaller piston is decreased by 4 times.
- The area of a circle is proportional to the square of its diameter:
\[ A_2' = \pi \left(2d\right)^2 = 4\pi d^2 = 16A_2 \quad (\text{since } A_2 = \pi d^2) \]
\[ A_1' = \pi \left(\frac{d}{4}\right)^2 = \frac{\pi d^2}{16} = \frac{A_1}{16} \quad (\text{since } A_1 = \pi d^2) \]

5. **Substituting the New Areas**:
- Now substituting \( A_1' \) and \( A_2' \) into the equation for \( M \):
\[ M = \frac{100 A_2'}{A_1'} = \frac{100 \times 16 A_2}{\frac{A_1}{16}} = 100 \times 16 \times 16 \frac{A_2}{A_1} \]

6. **Finding the New Mass**:
- Since we know that \( \frac{A_2}{A_1} = \frac{100}{100} = 1 \) (from the initial condition), we can simplify:
\[ M = 100 \times 16 \times 16 = 25600 \text{ kg} \]

### Final Answer:
The hydraulic press can lift **25600 kg** when the diameter of the larger piston is increased by 4 times and that of the smaller piston is decreased by 4 times while keeping the same mass \( m \) on the smaller piston.