1. Understand the Concepts

* Newton's Law of Universal Gravitation: The force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The equation is:

F = G * (m1 * m2) / r²

Where:

* F is the gravitational force

* G is the gravitational constant (6.674 × 10⁻¹¹ N⋅m²/kg²)

* m1 and m2 are the masses of the objects

* r is the distance between their centers

* Total Mass: The sum of the individual masses of the two objects.

2. Set Up the Equations

We have two unknowns (m1 and m2) and two equations we can use:

* Equation 1 (Force of Gravity): 2.5 × 10⁻¹⁰ N = G * (m1 * m2) / (25 m)²

* Equation 2 (Total Mass): m1 + m2 = 4.0 kg

3. Solve for the Masses

* Solve for one mass in terms of the other:

* From Equation 2, we get: m1 = 4.0 kg - m2

* Substitute this into Equation 1:

* 2.5 × 10⁻¹⁰ N = G * ((4.0 kg - m2) * m2) / (25 m)²

* Simplify and solve for m2:

* 2.5 × 10⁻¹⁰ N * (25 m)² = G * (4.0 kg * m2 - m2²)

* 1.5625 × 10⁻⁶ = (6.674 × 10⁻¹¹ N⋅m²/kg²) * (4.0 kg * m2 - m2²)

* 23400 = 4.0 * m2 - m2²

* m2² - 4.0 * m2 + 23400 = 0

* Use the quadratic formula to solve for m2:

* m2 = (-b ± √(b² - 4ac)) / 2a

* Where a = 1, b = -4, and c = 23400

* This will give you two possible values for m2. One will be a realistic mass, and the other will be very large and unrealistic.

* Find m1:

* Once you have m2, plug it back into either Equation 1 or Equation 2 to solve for m1.

Let me know if you need help solving the quadratic equation to get the final values for m1 and m2.