Problem of the Week
Problem C and Solution
Mmmm, I Love Tortoises!
Problem
Jonathan the tortoise is 183 years old according to several sources (BBC.com, Wikipedia, live
science.com, to name a few). That makes Jonathan the oldest living terrestrial animal. Two tortoises,
Terrance and his grandfather Tyrone share the same birthday. Tyrone is older than 100 but not as old
as Jonathan. This year, Tyrone’s age is exactly fifteen times Terrance’s age. How old will Terrance
and Tyrone be when Tyrone’s age is exactly eleven times Terrance’s age?
Solution
Solution 1
Since Tyrone’s age is fifteen times Terrance’s age, this means Tyrone’s age is a multiple of 15 that is
greater than 100 and less than 183. Tyrone’s possible ages are 105, 120, 135, 150, 165 and 180. The
corresponding possible ages for Terrance are 7, 8, 9, 10, 11 and 12.
At some time in the future, Tyrone’s age will be eleven times Terrance’s age. This means that
Tyrone’s future age is a multiple of 11. The possibilities are 110, 121, 132, 143, 154, 165, 176, 187 and
(possibly) 198.
We will look at the possible cases.
1. Could Terrance’s age now be 7 and Tyrone’s age now be 105?
The closest multiple of 11 greater than 105 is 110. This is 5 years from now. Terrance would
then be 12 and 110 ÷ 12 < 11. At this point, Tyrone’s age is less than eleven times Terrance’s
age. It is not possible for Terrance to be 7 and Tyrone to be 105.
2. Could Terrance’s age now be 8 and Tyrone’s age now be 120?
The closest multiple of 11 greater than 120 is 121. This is 1 year from now. Terrance would then
be 9 and 121 ÷ 9 > 11. At this point, Tyrone’s age is still more than eleven times Terrance’s age.
Let’s check the next multiple of 11 past 120. That is, consider 132. This is 12 years from now.
Terrance would then be 20 and 132 ÷ 20 < 11. At this point, Tyrone’s age is less than eleven
times Terrance’s age. It is not possible for Terrance to be 8 and Tyrone to be 120.
3. Could Terrance’s age now be 9 and Tyrone’s age now be 135?
The closest multiple of 11 greater than 135 is 143. This is 8 years from now. Terrance would
then be 17 and 143 ÷ 17 < 11. At this point, Tyrone’s age is less than eleven times Terrance’s
age. It is not possible for Terrance to be 9 and Tyrone to be 135.
4. Could Terrance’s age now be 10 and Tyrone’s age now be 150?
The closest multiple of 11 greater than 150 is 154. This is 4 years from now. Terrance would
then be 14 and 154 ÷ 14 = 11. At this point, Tyrone’s age is eleven times Terrance’s age. (We
need to check the two remaining possibilities for completeness.)
5. Could Terrance’s age now be 11 and Tyrone’s age now be 165?
The closest multiple of 11 greater than 165 is 176. This is 11 years from now. Terrance would
then be 22 and 176 ÷ 22 < 11. At this point, Tyrone’s age is less than eleven times Terrance’s
age. It is not possible for Terrance to be 11 and Tyrone to be 165.
6. Could Terrance’s age now be 12 and Tyrone’s age now be 180?
The closest multiple of 11 greater than 180 is 187. This is 7 years from now. Terrance would
then be 19 and 187 ÷ 19 < 11. At this point, Tyrone’s age is less than eleven times Terrance’s
age. It is not possible for Terrance to be 12 and Tyrone to be 180.
We have considered all of the possible cases. Terrance’s age will be one-eleventh of Tyrone’s age in 4
years when Terrance is 14 and Tyrone is 154. Although it was not asked, today Terrance is 10 and
Tyrone is 150.
Solution 2
This solution is much more algebraic and may be beyond your present experience.
Let c represent Terrance’s age today and g represent Tyrone’s age today. Since Tyrone’s age is fifteen
times Terrance’s age, it follows that g = 15c.
Let n be the number of years until Tyrone’s age is eleven times Terrance’s age. In n years, Terrance’s
age will be c + n and Tyrone’s age will be g + n. Then
g + n = 11(c + n)
But g = 15c so
15c + n = 11(c + n)
15c + n = 11c + 11n
4c = 10n
2c = 5n
2c
= n
5
Since both c and n are whole years, it follows that c is divisible by 5. But in the first solution, the
possible ages for Terrance were 7, 8, 9, 10, 11 and 12. The only multiple of 5 in this list is 10.
Therefore c = 10 and then n = 4. But g = 15c so g = 15 × 10 = 150. Terrance is 10 today and Tyrone
is 150 today. In 4 years, Terrance will be 14, Tyrone will be 154 and Tyrone’s age will be eleven times
Terrance’s age.