How many `beta`-ketoacids on heating undergo decarboxylation to give 2-methylcyclohexanone?

Explanation

To determine how many beta-keto acids can undergo decarboxylation to yield 2-methylcyclohexanone, we need to follow these steps:

### Step 1: Understand the Structure of 2-Methylcyclohexanone
2-Methylcyclohexanone has a cyclohexane ring with a ketone group and a methyl group at the second carbon. Its molecular formula is C7H12O.

### Step 2: Identify the Structure of Beta-Keto Acids
Beta-keto acids have the general structure RCOCH2COOH, where the keto group (C=O) is at the beta position relative to the carboxylic acid group (COOH).

### Step 3: Analyze Decarboxylation
When beta-keto acids undergo decarboxylation (removal of CO2), they typically form a ketone. The decarboxylation reaction can be represented as:
\[ RCOCH_2COOH \rightarrow RCOCH_3 + CO_2 \]

### Step 4: Determine the Required Beta-Keto Acid
To produce 2-methylcyclohexanone, we need to find beta-keto acids that, upon decarboxylation, will yield a structure that matches 2-methylcyclohexanone.

### Step 5: Construct Possible Beta-Keto Acids
1. The simplest beta-keto acid that can yield 2-methylcyclohexanone would have a structure that, after losing CO2, results in the desired ketone.
2. We can consider beta-keto acids with different alkyl groups that can lead to the formation of the cyclohexanone structure.

### Step 6: Count the Valid Beta-Keto Acids
After analyzing the structures, we can find the following beta-keto acids that can decarboxylate to yield 2-methylcyclohexanone:
- 4-Methyl-3-oxobutanoic acid
- 3-Methyl-3-oxobutanoic acid

### Conclusion
In total, there are **two beta-keto acids** that can undergo decarboxylation to give 2-methylcyclohexanone.

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