A census-taker rings Mr. Manning's bell and asks how many children he has.
"Three daughters," he replies.
"And how old are they, in whole numbers?" asks the census-taker.
"Well, I'll tell you this:the product of their ages is 72, and the sum of their ages is my house number."
"But that isn't enough information!" complains the census-taker.
"Okay, my oldest daughter(in years)likes chocolate milk," replies Mr. Manning.
With that, the census-taker nods and writes down the three ages.
How old are the Manning girls, and how did the census-taker figure it out?Include a full explanation! *My teacher said "But that isn't enough information!" complains the census-taker. was a big hint but i don't get it please help
The census-taker was able to figure out the ages of the Manning girls by using the information given. Since the product of their ages is 72 and the sum of their ages is Mr. Manning's house number, the census-taker can use algebra to solve the problem.
Let x, y, and z represent the ages of the Manning girls. Then, the equation can be written as:
x * y * z = 72
x + y + z = Mr. Manning's house number
The census-taker can then use substitution to solve the equation. Since Mr. Manning said that his oldest daughter likes chocolate milk, the census-taker can assume that x is the oldest daughter's age. Then, the equation can be rewritten as:
x * (x + y + z - x) * (x + y + z - x - y) = 72
x * y * z = 72
By substituting x for the oldest daughter's age, the census-taker can solve for the other two ages. After solving the equation, the census-taker can determine that the Manning girls are 8, 6, and 4 years old.
To solve this problem, we need to find three whole numbers whose product is 72 and whose sum is equal to Mr. Manning's house number.
Let's start by finding all possible combinations of three numbers whose product is 72:
1 * 1 * 72 = 72
1 * 2 * 36 = 72
1 * 3 * 24 = 72
1 * 4 * 18 = 72
1 * 6 * 12 = 72
1 * 8 * 9 = 72
2 * 2 * 18 = 72
2 * 3 * 12 = 72
2 * 4 * 9 = 72
2 * 6 * 6 = 72
3 * 3 * 8 = 72
3 * 4 * 6 = 72
Now, we need to find the sum of each combination:
1 + 1 + 72 = 74
1 + 2 + 36 = 39
1 + 3 + 24 = 28
1 + 4 + 18 = 23
1 + 6 + 12 = 19
1 + 8 + 9 = 18
2 + 2 + 18 = 22
2 + 3 + 12 = 17
2 + 4 + 9 = 15
2 + 6 + 6 = 14
3 + 3 + 8 = 14
3 + 4 + 6 = 13
Now, let's consider Mr. Manning's statement that his "oldest daughter (in years) likes chocolate milk." This means that there is only one combination of ages that satisfies this condition, as there must be one oldest daughter. We know that the census-taker initially thought the information provided was insufficient, which implies that there must be two possible combinations that result in the same sum.
From the calculations above, we notice that there are two combinations that result in a sum of 14: (2, 6, 6) and (3, 3, 8). However, since Mr. Manning mentioned that his oldest daughter likes chocolate milk, we can conclude that the ages of the Manning girls are 3, 3, and 8.
Therefore, the census-taker figured out the ages of the Manning girls by considering the product and sum of their ages, along with the additional clue about the oldest daughter's preference for chocolate milk.
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.