Explanation:

- Let's suppose the numbers are $x,3x,4x$ based on the given proportion 1 : 3 : 4.

- The sum of these numbers is $x+3x+4x=8x$.

- Two times the sum is $2\times 8x=16x=80$.

- Solving for $x$ gives $x=5$.

- The numbers are $5,15,20$.

- The squares of these numbers are $25,225,$ and $400$.

- Their proportion is $25:225:400$.

- Simplifying, we get $1:9:16$.

Here are the options:

- Option 1: 4: 36: 64

This is incorrect because it doesn't match the simplified proportion.

- Option 2: 25 : 225 : 500

This is incorrect as the third value is not 400 in simplified form.

- Option 3: 1 : 9 : 16

This matches the correct simplified proportion.

- Option 4: 9: 81 : 144

This is incorrect as both first and second numbers are incorrect.