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Grandpa's Age

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Grandpa:

"My grandson is about as many days as my son is weeks, and my grandson is as many months as I am in years. My grandson, my son and I together are 120 years. Can you tell me my age in years?"
Answer: 72

Solution:

Grandpa:

"This problem is conveniently solved by writing down the necessary equations. Note that there are 12 months in a year, 52 weeks in a year, and 365 days in a year.

Let m be my age in years. If s is my son's age in years, then my son is 52s weeks old. If g is my grandson's age in years, then my grandson is 365g days old. Thus,

365g = 52s.

Since my grandson is 12g months old,

12g = m.

Since my grandson, my son and I together are 120 years,

g + s + m = 120.

The above system of 3 equations in 3 unknowns (g, s and m) can be solved as follows.

m / 12 + 365 m / (52 × 12) + m = 120 or

52 m + 365 m + 624 m = 624 × 120 or

m = 624 × 120 / 1041 = 72.

So, I am 72 years old."

Food for thought:

Why is the word "about" used in the Problem Statement in the sentence "My grandson is about as many days as my son is weeks"? Calculate the son's age and the grandson's age. Then, verify whether the first equation (i.e., 365g = 52s) is exactly satisfied.

An elegant solution is possible on realizing the significance of the word "about" in the Problem Statement.

Elegant Solution:

The first equation (365g = 52s) can be approximated by

7g = s.

As before, the other two equations are

12g = m

g + s + m = 120.

The above system of 3 equations in 3 unknowns (g, s and m) can be simply solved as follows.

g + 7g + 12g = 120 or 20g = 120.

m = 12g = 12 × 120 / 20 = 72.

So, Grandpa is 72 years old.


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