# The “Ages of Three Children” Puzzle

A census taker approaches a woman leaning on her gate and asks about her children. She says “I have three children and the product of their ages is 36. The sum of their ages is the number on this gate.” The census taker does some calculation and (correctly) claims not to have enough information. The woman enters her house, but before slamming the door tells the census taker, “I have to see to my eldest child, who is in bed with measles”. The census taker then departs, satisfied, knowing the answer.

This question tests your ability to factor a number.  Whenever a problem discusses “products” or “factors”, or “divisibility” or a “remainder of zero”, that problem likely involves breaking down a number into its factors.

The factors of 36 are 1 and 36; 2 and 18; 3 and 12; 4 and 9; and 6.  (1, 2, 3, 4, 6, 9, 12, 18, 36). So, each child’s age must be one of those factors, and they must multiply to 36 and add up to the mysterious “number on the gate”, which is not given.

The possible ages of the children, then, are:

1, 1, and 36 (and the number on the gate would be 38)

1, 2, and 18 (and the number on the gate would be 21)

1, 3, and 12 (and the number on the gate would be 16)

1, 4, and 9 (and the number on the gate would be 14)

1, 6, and 6 (and the number on the gate would be 13)

2, 2, and 9 (and the number on the gate would be 13)

3, 3, and 4 (and the number on the gate would be 10)

Based on the above, the census taker must not have been able to figure out the ages using the number on the gate — so the number on the gate must be 13 (the only sum that could come from more than one sets of factors). So, the census taker knows that the children are either 6-year-old twins and a 1-year old, or else a 9-year-old and 2-year-old twins

Once the woman mentions an “eldest child”, the census taker knows that the correct answer is the 9-year-old and 2-year-old twins.

The GMAT Test includes many questions that test your ability to break down and solve problems involving probability.  If you like problems like this, you might enjoy taking the GMAT and earning an MBA.  Contact me for information about classes and tutoring, either locally in Austin or online via The Princeton Review’s LiveOnline classrooms!

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