To solve this problem, we need to determine the ages of Mr. Manning's three daughters based on the given information.
First, we are told that the product of their ages is 72. To find the possible combinations of three numbers that multiply to equal 72, we can make a list:
1, 1, 72
1, 2, 36
1, 3, 24
1, 4, 18
1, 6, 12
2, 2, 18
2, 3, 12
2, 6, 6
3, 3, 8
4, 6, 3
Next, we know that the sum of the ages is Mr. Manning's house number. However, the census-taker initially states that this information is not enough to determine the ages. This suggests that there must be multiple possible combinations of ages with the same sum.
The key hint we are given is that the oldest daughter likes chocolate milk. Based on this, we can conclude that there must be more than one possible combination with the same sum. If there were only one possible combination, the census-taker would have been able to immediately determine the ages based on the sum alone.
Since the census-taker is still able to figure out the ages after the additional clue, the sum of the ages must be unique to a specific combination.
Now, let's look at the possible combinations with unique sums:
1, 6, 12 (sum = 19)
2, 2, 18 (sum = 22)
3, 3, 8 (sum = 14)
We know that the census-taker was able to determine the ages based on the house number after the additional clue was given. Hence, the house number must correspond to one of these unique sums.
Therefore, the Manning girls' ages are 1, 6, and 12, and the sum of their ages is 19. This is because the census-taker was able to deduce the ages after Mr. Manning hinted that his oldest daughter likes chocolate milk, indicating that there is a unique sum among the possible combinations.
So, the census-taker figured out the ages by considering the product and sum of the daughters' ages, as well as the additional clue provided by Mr. Manning about his oldest daughter's preference for chocolate milk.