Step 1: Write the structure of cumene.
Cumene is the common name for isopropylbenzene. Its molecular formula is $C_9H_{12}$ and its structure is a benzene ring ($C_6H_5-$) attached to an isopropyl group $(-CH(CH_3)_2)$. The full structure is $C_6H_5-CH(CH_3)_2$.
Step 2: Identify the carbon atoms in the benzene ring.
The benzene ring contains 6 carbon atoms. Each carbon in the benzene ring forms: one bond with a hydrogen (or with the isopropyl group for the attachment carbon), one bond with each adjacent ring carbon, and participates in the delocalised $\pi$ system. This means each ring carbon has 3 sigma bonds and participates in pi bonding, so all 6 ring carbons are $sp^2$ hybridised.
Step 3: Count the sp2 hybridised carbons.
All 6 carbons of the benzene ring are $sp^2$ hybridised. Total $sp^2$ carbons = 6.
Step 4: Identify the carbon atoms in the isopropyl group.
The isopropyl group $-CH(CH_3)_2$ contains: one central carbon $(-CH-)$ bonded to the ring, one H, and two methyl groups; and two methyl carbons ($-CH_3$) each bonded to 3 hydrogens and the central carbon. All three carbons form only single (sigma) bonds with no pi bonds.
Step 5: Count the sp3 hybridised carbons.
The central $-CH-$ carbon: 1 carbon (sp3). The two $-CH_3$ carbons: 2 carbons (sp3). Total $sp^3$ carbons = $1 + 2 = 3$.
Step 6: State the final answer.
In cumene, there are 6 $sp^2$ hybridised carbons and 3 $sp^3$ hybridised carbons. \[ \boxed{sp^2 : sp^3 = 6 : 3} \]