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Let $a$ be the number of adults and $c$ be the number of children. Then we have $$25a+12c=1950=25\times 78.$$Rearranging terms gives us $$a=78-\frac{12c}{25}.$$Since the number of adults must be an integer, this tells us that $c$ is a multiple of 25.

The ratio we want close to 1 is $$\frac{a}{c}=\frac{78}{c}-\frac{12}{25}$$If $\frac{a}{c}=1$, then $\frac{78}{c}-\frac{12}{25}=1$, which means that $\frac{78}{c}=\frac{37}{25}$. In other words, $c=\frac{78\cdot 25}{37}$.

The multiple of 25 that comes closest to this is 50, and hence $c$ must be 50. Then, $a=78-\frac{12\cdot 50}{25}=54$. So the ratio of adults to children closest to 1 is $\frac{54}{50}=\boxed{\frac{27}{25}}.$