Monday, January 22, 2007

One Easy, One Hard

Two new math problems...and BlogPoints available to the first correct answers of each. I'm not saying which is easy and which is hard. But the easy one is worth 2 blog points and the hard one is worth 4.

1) An intelligent, but eccentric census taker (eccentric means he never questions unnecessarily) came to a house where the three residents were not at home. The housekeeper answered the door. Here is their conversation about the three occupants:
CT: What is the product of their ages?
HK: 1296
CT: What is the sum of their ages?
HK: The same as the house number.
CT: I still can't figure out their ages. Are any of them older than you? (He must know the house keeper's age)
HK: No.

How old are the three occupants?

2) If it takes James 2 hours to mow a lawn and it takes Maya 5 hours to mow the same lawn. How long would it take to mow the lawn if James started feeling sorry for Maya after she had been working for 90 minutes and he started helping? (You can assume they each have their own lawnmower)


mikem said...

re 1), wouldn't we need to know the house number? ie, it could consist of a 9 year old and two 12 year olds, two 9 year olds and a 16 year old, a 2 year old, an 8 year old and an 81 year old, ...

(although an 81 year old census taker should certainly consider retiring)

mikem said...

re 2) lawn cutting would cease after 60mins.

when James comes to help Maya, she is only 30% done (90mins/300mins). After helping for an hour, James is 50% done (60mins/120mins) and Maya is also 50% done (150mins/300mins)

Kimota94 aka Matt said...

1) The product answer provided multiple answers, so the census taker proceeded to ask more questions. Since the product is 3*3*3*3*2*2*2*2 (as the product of primes), there are lots and lots of combinations such as:

81, 16, 1
81, 8, 2
81, 4, 4
72, 18, 1
72, 9, 2
72, 6, 3
54, 24, 1
54, 12, 2
54, 8, 3
and so on

Moving to the 2nd question asked by the census taker: The sum equalled the house #, which the census taker knew, and yet that still didn't answer his question definitively. So this suggests that there are at least two product possibilities that add up to the same sum.

I don't know how to solve that other than through inspection, which lead me to two possible answers (up to this point):

81, 8, 2
72, 18, 1

In each case, the product is 1296 and the sum is 91.

Since the third question was "are any of them older than you?" and the answer was "No", then I'd have to say the housekeeper was at least 72 but less than 81 years of age, allowing the census taker to conclude the right ages are:

72, 18, and 1.

Kimota94 aka Matt said...

2) And I agree with MikeM, that an additional hr after Maya's 90 minute head start would be required in order to finish the whole lawn. Maya would do 30% by herself, then 20% with James while James is also doing 50% (since he mows at 2.5 times the speed of Maya).

I know I don't get the Blog Point for it, but I still like to work it through!

vishrut said...

1) without looking at other peoples solutions:

1296 = 9x9x4x4

Lots of different combinations possible, and few which add up to the same number. (hence insufficient for the CT to find each age).
if the HK was 17 then the only possible combination is 16 , 9 , 9.

Jimmy said...

Excellent work guys.

Kimota gets 4 points for the hard question solved perfectly.

Mikem gets 2 for the lawn mowing question (even though he forgot to add the initial 90 minutes to his answer it was explained below).