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Chapter 7: Problem 58

Tom is from the U.S. Census Bureau and greets Mary at her door. They have the following conversation: Tom: I need to know how old your three kids are. Mary: The product of their ages is 36 . Tom: I still don't know their ages. Mary: The sum of their ages is the same as my house number. Tom: I still don't know their ages. Mary: The younger two are twins. Tom: Now I know their ages! Thanks! How old are Mary's kids and what is Mary's house number?

Short Answer

Expert verified
Answer: Mary's children's ages are 1, 6, and 6, and her house number is 13.

Step by step solution

01

Find the factors of 36 and possible age combinations

To find the possible age combinations, we will first find all the factors of 36 and list them in sets of three. 1 × 2 × 18 1 × 3 × 12 1 × 4 × 9 1 × 6 × 6 2 × 2 × 9 2 × 3 × 6 3 × 3 × 4 There are 7 possible sets of ages that meet the condition of having a product of 36.
02

Exclude age combinations that reveal the children's ages

Since Tom still does not know their ages after Mary told him the sum is the same as her house number, there must be more than one combination with the same sum. So, we will exclude age combinations with unique sums: (1, 2, 18) has a sum of 21. (1, 3, 12) has a sum of 16. (1, 4, 9) has a sum of 14. (1, 6, 6) has a sum of 13. (2, 2, 9) has a sum of 13. (2, 3, 6) has a sum of 11. (3, 3, 4) has a sum of 10. We can eliminate (1, 2, 18), (1, 3, 12), (1, 4, 9), (2, 3, 6), and (3, 3, 4). This leaves us with two possible age combinations: (1, 6, 6) and (2, 2, 9), both of which have a sum of 13.
03

Use the twin information to find the correct ages

Mary told Tom that the younger two kids are twins, which means they are of the same age. In our remaining sets, we can see that (1, 6, 6) has younger twins. Thus, the ages of Mary's children are 1, 6, and 6.
04

Find Mary's house number

As the sum of the children's ages is the same as Mary's house number, the house number is 1+6+6 = 13. So, Mary's children's ages are 1, 6, and 6, and Mary's house number is 13.

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