The ages of Old and Young total 48. Old is twice as old as Young was
when Old was half as old as Young will be when Young is three times as
old as Old was when Old was three times as old as Young.
How old is Old?
Let x = Old's present age
Let y = Young's present age
I will insert some parentheses to help clarify:
1. The (present) ages of Old (x years old) and Young (y years old) total 48.
2. Old (now) is twice as old as Young was (z years ago)
3. (z years ago was) when Old was half as old as Young will be (w years from
now)
4. (w years from now will be) when Young is three times as old as Old was (v
years ago)
5. (v years ago was) when Old was three times as old as Young (was v years ago).
Now for each of those five sentences we'll set up five equations in five
unknowns, x, y, z, w, and v:
1. >>...The (present) ages of Old (x years old) and Young (y years old) total
48...<<
x + y = 48
2. >>...Old is (now) twice as old as Young was (z years ago)...<<
Old is now x, and z years ago, Young was y - z, so
x = 2(y - z)
3. >>...(z years ago was) when Old was half as old as Young will be (w years
from now)...<<
z years ago, Old was x - z
w years from now, Young will be y + w, so
x - z = 1/2(y + w)
4. >>...(w years from now will be) when Young is three times as old as Old was
(v years ago)...<<
w years from now, Young will be y + w
v years ago, Old was x - v, so
y + w = 3(x - v)
5. >>...(v years ago was) when Old was three times as old as Young (was v years
ago)...<<
v years ago, Old was x - v
v years ago, Young was y - v, so
x - v = 3(y - v)
So we have this system of 5 equations and 5 unknowns:
x + y = 48
x = 2(y - z)
x - z = 1/2(y + w)
y + w = 3(x - v)
x - v = 3(y - v)
We can simplify these to give this system
x + y + 0z + 0w + 0v = 48
x - 2y + 2z + 0w + 0v = 0
2x - y - 2z - w + 0v = 0
-3x + y + 0z + w + 3v = 0
x - 3y + 0z + 0w + 2v = 0
The best way to solve this system is with a TI-83, finding the
"rref" (row-reduced echelon form) of this augmented 5×6 matrix:
[ 1 1 0 0 0 | 48]
[ 1 -2 2 0 0 | 0]
[ 2 -1 -2 -1 0 | 0]
[-3 1 0 1 3 | 0]
[ 1 -3 0 0 2 | 0]
which is
[ 1 0 0 0 0 | 30]
[ 0 1 0 0 0 | 18]
[ 0 0 1 0 0 | 3]
[ 0 0 0 1 0 | 36]
[ 0 0 0 0 1 | 12]
So x is 30 and y is 18. z = 3, w = 36, v = 12
You were asked "How old is Old?", so the answer is 30.
---------------------------------------------------------
Now let's check:
1. The (present) ages of Old (30 years old) and Young (18 years old) total 48.
2. Old (now 30) is twice as old as Young was (3 years ago, when Young was 15)
3. (3 years ago was) when Old was half as old as Young will be (36 years from
now)
[3 years ago Old was 27 and that is half as old as Young will be 36 years from
now when he will be 54]
4. (36 years from now will be) when Young is three times as old as Old was (12
years ago)
[36 years from now Young will be 54. 12 years ago Old was 18, and 3 times 18 is
54]
5. (12 years ago was) when Old was three times as old as Young (was 12 years ago)
[12 years ago, Old was 18 and Young was 6, and 18 is 3 times 6]
Edwin
AnlytcPhil@aol.com
