The general explanation of the burning process is a reaction with an oxygen, which is presented in the air. To put out the fire, you have to prevent the access of the oxygen or air in general. And on the other hand, to increase the intensity of the fire you have to provide even more oxygen or heat.
Let's look into the chemical behaviour of these two compounds. Sodium carbonate in general is a heat resistant chemical, but at elevated temperatures it's possible to decompose it. The reaction is
Na2CO3(s)=Na2O(s)+CO2(g)
So even if the decomposition occurred, formed CO2 (which is inert gas) immediately stops the burning. And if the reaction has not occurred, Sodium carbonate just covered the fire, to prevent oxygen, and also stops fire.
In the case of Sodium chlorate, it can easily decompose by following reaction :
2NaClO3(s)=2NaCl(s)+3O2(g)
In this case, oxygen has been produced, which can even increase the power of the fire, to cause burns.
Because Sally dumped Sodium carbonate, she succeed to put out the fire, and Dave got burnt, because he decomposed Sodium chlorate and liberated oxygen
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