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I am asked whether dimethyl malonate or methyl acetoacetate decarboxylate when hit with acid, water, and extreme temperatures.

I believe the answer is both because both are esters, and esters, upon being hit by such conditions, return to their respective carboxylic acids.

From there, both carboxylic acids can undergo decarboxylation because both resulting carboxylic acids are beta-keto acids. Technically, dimethyl malonate becomes malonic acid, which can still undergo decarboxylation (but malonic acid isn't of the beta-keto class; rather the dicarboxylic acid class).

Acid and heat treatment of methyl acetoacetate initially gives 3-oxobutanoic acid but this can decarboxylate under the heat as well.

Is my analysis correct?

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Yes, your analysis is correct. This is a common sequence in the malonate synthesis of carboxylic acids and acetoacetate synthesis of methyl ketones. After taking advantage of the acidic, doubly alpha-proton and using the resulting enolate as a nucleophile, the ester(s) are hydrolyzed with acid. A carboxylic acid with a carbonyl in the beta-position can be thermally decarboxylated to give an enol, which typically tautomerizes to the keto form.

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