Intermath | Workshop Support

Unknown Ages



Problem Statement
Roberto was born on his grandmother's birthday. The two digits of Roberto's age today are the reverse of the digits of his grandmother's age. The grandmother's age is a prime number. Ten years from now, his grandmother will be three times as old as Roberto will be then. How old are Roberto and his grandmother now?

Problem setup

Roberto shares his birthday with his grandmother.  Their current ages are reversed digits of one another, but the grandmother's age is a prime number.  Ten years from now, the grandmother's age is three times Roberto's age.  What are the current ages of Roberto and his grandmother?


Plans to Solve/Investigate the Problem

A brief compilation of prime numbers, from 2 to 100, is helpful in completing this problem.  The prime numbers (ages) then must be evaluated for reasonableness.  The use of Excel is helpful in completing this problem.


Investigation/Exploration of the Problem

First, the set of prime numbers from 2 to 100 were evaluated for reasonableness.  The essential question is if the grandmother's age (the prime number) were reversed, is it reasonable that the resulting number could be Roberto's age?  Since we are speaking of a grandmother, ages should differ reasonably across a two generation span.


For the sake of example, if we selected the prime number 53 as the grandmother's age, is it reasonable that Roberto could be 35 years of age?  53 - 35 = 18.  Is it reasonable that Roberto and his grandmother's ages are 18 years apart?  Not likely.


This process was continued until there were four possible ages remaining of our set of prime numbers:  61, 71, 73 and 83.


I then completed a spreadsheet of these possible ages to discover which combination met the other requirements of the problem.


Grandmother's age Roberto's age (reverse of grandmother's age) Roberto's age in 10 years Grandmother's age in 10 years should be 3 times Roberto's age 
61 16 26 71
71 17 27 81
73 37 47 83
83 38 48 93


If you examine Roberto's age in 10 years and the grandmother's age in 10 years, I am seeking a solution in which the grandmother's age is is three times Roberto's age. 


This relationship is found if the grandmother's current age is 71and Roberto's current age is 17.  In 10 years, grandma is 27 * 3 or 81.


We have a solution!


Extensions of the Problem

Does a solution exist where grandmother's age is twice Roberto's age in ten years?  Four times?

Author & Contact
Angela Gilliam

Link(s) to resources, references, lesson plans, and/or other materials
Click here to see problem listed on InterMath