$A$ spring has a spring constant $k$ and original length $l$. If it is cut into pieces in the ratio $\alpha : \beta : \gamma$,find the spring constant of each piece in terms of the original spring constant $k$ (where $\alpha, \beta$,and $\gamma$ are integers).

(N/A) The spring constant $k$ of a spring is inversely proportional to its length $l$,i.e.,$k \propto 1/l$ or $kl = \text{constant}$.
Let the total length be $l = \alpha + \beta + \gamma$.
The lengths of the three pieces are $l_1 = \frac{\alpha}{\alpha+\beta+\gamma} l$,$l_2 = \frac{\beta}{\alpha+\beta+\gamma} l$,and $l_3 = \frac{\gamma}{\alpha+\beta+\gamma} l$.
For the first piece,$k_1 l_1 = kl \implies k_1 = \frac{kl}{l_1} = \frac{kl}{\frac{\alpha}{\alpha+\beta+\gamma} l} = k \frac{(\alpha+\beta+\gamma)}{\alpha}$.
Similarly,for the second piece,$k_2 = k \frac{(\alpha+\beta+\gamma)}{\beta}$.
For the third piece,$k_3 = k \frac{(\alpha+\beta+\gamma)}{\gamma}$.