The sum of total number of carbonyl groups ( >C=O ) present in the...

Reaction 1:
The reactant is 2,4-dimethyl-3-oxopentanedioic acid, which is a $\beta$-keto acid (both $-\text{CO}_2\text{H}$ groups are at the $\beta$-position with respect to the central ketone group). $\beta$-keto acids readily undergo decarboxylation upon heating. Since there are two such groups, it will undergo double decarboxylation.

$\text{HO}_2\text{C}-\text{CH}(\text{CH}_3)-\text{C}(=\text{O})-\text{CH}(\text{CH}_3)-\text{CO}_2\text{H} \xrightarrow{\Delta} \text{CH}_3-\text{CH}_2-\text{C}(=\text{O})-\text{CH}_2-\text{CH}_3 + 2\text{CO}_2$

The major product $\textbf{X}$ is 3-pentanone.
Number of carbonyl groups ($>\text{C=O}$) in $\textbf{X} = 1$.

Reaction 2:
The reactant is *cis*-3,4-dicarboxycyclopentanone. The two carboxylic acid groups are on adjacent carbons (a 1,2-dicarboxylic acid system). Heating a 1,2-dicarboxylic acid leads to dehydration, forming a stable 5-membered cyclic anhydride.

The major product $\textbf{Y}$ is the cyclic anhydride of the reactant, which is a fused bicyclic system containing the original cyclopentanone ring and a newly formed succinic anhydride ring.
The structure of $\textbf{Y}$ contains one ketone carbonyl group and two anhydride carbonyl groups.
Number of carbonyl groups ($>\text{C=O}$) in $\textbf{Y} = 1 + 2 = 3$.

Sum of total number of carbonyl groups in $\textbf{X}$ and $\textbf{Y} = 1 + 3 = 4$.

Answer: $4$