In a colony, 82% of the homes have a ceiling fan and 65% have a cooler. At least what per cent of the homes must have both?

The two shares add to 82 + 65 = 147%, but a home can be counted at most twice and the colony is only 100%. The overlap is therefore at least 147 - 100 = 47%. Picture it as pushing the two groups as far apart as they will go: the 18% with no fan can absorb at most 18 of the 65 cooler-owners, leaving 47 that must fall inside the fan group. 65% is the LARGEST the overlap could be, not the smallest. 17% is the 82 - 65 gap, which measures nothing here, and 53% is simply 100 - 47.