A solid iron ball is melted and 64 smaller solid balls of equal size are made using the entire volume of iron. What is the ratio of the surface area of the larger ball to the sum of the surface areas of all the smaller balls ? 

Formula used:

Volume of sphere = (4/3) π (radius)³

Surface  Area of sphere = 4π(radius)² 

Calculation:

Let's assume the radius of the larger solid iron ball is R, and the radius of

each of the smaller solid balls is r.

By using the above formula

V_larger = (4/3)πR³

V_smaller = (4/3)πr³

Since the entire volume of iron is used to make the smaller balls, we have:

V_larger = 64V_smaller

(4/3)πR³ = 64 × (4/3)πr³

R³ = 64r³

R = 4r

Now, let's calculate the surface area of the larger ball and the sum of the surface areas of the smaller balls.

Ratio = (4πR2) / (64 (4πr2)

Ratio = (4π(4r)2) / (256πr2)

Ratio = (64πr²) / (256πr²)

Ratio = 64/256 = 1/4

∴ The required ratio is 1/4.