A spring of force constant 15 ~N / m is cut into two pieces. If the...

Spring force constant is inversely proportional to length: $k \propto \frac{1}{L}$
So $kL =$ constant for a given spring material and structure.
Original spring: $k = 15$ N/m, length $= L$.
Smaller piece has length $\frac{L}{4}$ (since length ratio is 1:3, smaller is $\frac{1}{4}$ of total).
For smaller piece: $k_1 \times \frac{L}{4} = 15 \times L$
$k_1 = 15 \times 4 = 60$ N/m