The amount of extension in an elastic string varies directly as the weight hung on it.

Let x gm be the weight that would produce an extension of 17.4 cm.

Since the amount of extension in an elastic string and the weight hung on it are in direct variation, we have:

$\frac{150}{x}=\frac{2.9}{17.4}$

$\Rightarrow 17.4 \times 150=2.9 \times x$

$\Rightarrow x=\frac{17.4 \times 150}{2.9}$

$=\frac{2610}{2.9}$

$=900$

Thus, the required weight will be $900 \mathrm{gm}$.