A wood that is 120 inches in length is to be cut into two parts such that the ratio of the parts constitutes the Golden Ratio. What must be the lengths of the wood?

Question

A wood that is 120 inches in length is to be cut into two parts such that the ratio of the parts constitutes the Golden Ratio. What must be the lengths of the wood?

is 74.16 and 45.84 correct?

Answer 1

Since the problem requires a practical response (location of cut), φ = 1.618

1.618x = 120

x = 74.166

Answer 2

The Golden Ratio is (1+sqrt(5))/2. For an exact answer, we can create a system of linear equations. Let x be the longer piece and y be the shorter piece. From the setup, we have

x+y=120

Then, from the golden ratio, we have x/y=(1+sqrt(5))/2. Note that the longer piece, x, corresponds to the larger term on the other side of the equation.

We now have a system of equations that you can solve using your favorite method (row reduction, substitution, Wolfram, etc.).

The exact solution is x=(120(1+sqrt(5))/(3+sqrt(5)), y=60(1+sqrt(5))/(2+sqrt(5)), which is just your answer without the decimal truncation.