A survey of 40 households recorded the number of children in each — 0 children: 4 households, 1 child: a households, 2 children: 12 households, 3 children: b households, 4 children: 6 households. The mean number of children per household is 2.2. Find a and b.

Two facts give two equations. The households total 40, so $4+a+12+b+6=40$ and $a+b=18$. The children total $40\times 2.2=88$, so $0(4)+a+2(12)+3b+4(6)=88$, that is $a+3b=40$. Subtracting the first from the second gives $2b=22$, so $b=11$ and $a=7$. Check: the frequencies 4, 7, 12, 11, 6 add to 40 and Σfx = 0 + 7 + 24 + 33 + 24 = 88. Swapping the answers gives a mean of 80/40 = 2.0, splitting the 18 evenly gives 84/40 = 2.1, and $a=6$ with $b=12$ gives 90/40 = 2.25 — only one pair meets BOTH conditions, which is why the total frequency alone cannot settle the question.