David W. answered • 03/30/19
Experienced Prof
There are three points of reference in time (this is important):
Since the problem asks, "How old is the first man now?",
Let x = age of the first man now
y = age of the second man now
The "two men have the same birthday," so we don't have to worry about spanning year boundaries.
Let d = time difference.
The three points of reference are:
[1] d years ago, when the first man was as old as the second man is now
[2] now
[3] d years from now, when the second man will be as old as the first man is now
d = x – y [eq1]
"When the first man was as old as the second man is now" happened d years ago. The first man's age back then was (x-d). The second man's age back then was (y-d).
"the first man was twice as old as the second man was back then"
(x-d) = 2(y-d)
x - d = 2y - 2d
x - 2y + d = 0
x - 2y + (x – y) = 0 [use eq1]
2x – 3y = 0 [eq2]
"When the second man becomes as old as the first man is now,"
Since the first man was y years old d years ago, the second man will become x years old d years from now.
"the sum of their ages will be 119"
So, d years from now:
(age of first man) + (age of second man) = 119
(x+d) + (y+d) = 119
x + y + 2d = 119
x + y + 2(x – y) = 119 [use eq1]
3x – y = 119
9x – 3y = 357 [eq3; 3 * previous equation]
2x – 3y = 0 [use eq2]
--------------------- [elimination; subtract equations]
7x = 357
x = 51 [eq4]
2x = 102 [2* eq4]
2x – 3y = 0 [use eq2]
---------------------------- [elimination; subtract equations]
3y = 102
y =34
d = x-y = 17
The first man is now 51.
Check:
17 years ago, was first man twice as old as second man was”
(54 – 17) = 2*(34 – 17) ?
34 = 2*27 ?
34 = 34 ?yes
17 years from now, will their combined ages be 119”
(51+17) + (34+17) = 119 ?
68 + 51 = 119 ?
119 = 119 ?yes
