Question 1205573: Hi
Alex says to Sarah when I was as old as you today the sum of our ages was 27 years. Sarah replied when I will be as old as you are today the sum of our ages will be 55 years.
How old are each of them.
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52884)   (Show Source):
You can put this solution on YOUR website! .
Alex says to Sarah when I was as old as you today the sum of our ages was 27 years.
Sarah replied when I will be as old as you are today the sum of our ages will be 55 years.
How old  is each of them.
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Let A be the Alex's age; S be the Sarah's age.
First sentence "Alex says to Sarah when I was as old as you today the sum of our ages was 27 years"
is translated to this equation
[A - (A-S)] + [S - (A-S)] = 27 (it happened A-S years ago).
This equation is equivalent to
S + 2S - A = 27, or 3S - A = 27. (1)
Second sentence "Sarah replied when I will be as old as you are today the sum of our ages will be 55 years"
is translated to this equation
[S + (A-S)] + [A + (A-S)] = 55 (it will happen in A-S years).
This equation is equivalent to
A + A + A - S = 55, or 3A -S = 55. (2).
So, we have the system of two equations (1) and (2).
To solve it, from equation (1) express A = 3S - 27 and substitute it into equation (2).
You will get
3*(3S-27) - S = 55,
9S - 81 - S = 55,
9S - S = 55 + 81
8S = 136
S = 136/8 = 17.
Then A = 3S - 27 = 3*17 - 27 = 24.
ANSWER. Alex is 24 years old; Sarah is 17 years old.
Solved.
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The solution by Theo is incorrect, since his setup equations are incorrect.
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! x = the age of alex today
y = the age of sarah today.
a = the differencd in their ages.
since alex is older than sarah, you get x - a = y and you get y + a = x
alex is assumed to be older because of the statements:
alex says when i was as old as you today.
sarah says when i will be as old as you today.
both of these statements indicate that alex is older than sarah.
since the difference of their ages is a constant, then the letter a is used to represent that difference.
any letter besides x and y would do.
i just chose "a".
you have two equations that need to be solved simultaneously.
they are:
x - a + y = 27
y + a + x = 55
subtract the first equation from the second to get:
2a = 28
solve for a to get:
a = 14
the difference in their ages is 14.
since alex is older, you get x - 14 = y and y + 14 = x
your two equations that need to be solved simultaneously now become:
x - 14 + y = 27
y + 14 + x = 55
if you add these two equations together, you get 2x + 2y = 82
divide both sides of that equation by 2 to get x + y = 41.
since x - 14 = y, then replace y with x - 14 in that equation to get x + x - 14 = 41.
combine like terms to get 2x - 14 = 41
add 14 to both sides to get 2x = 55
solve for x to get x = 27.5
since y is 14 less than x, you get y = 27.5 - 14 = 13.5
you have x = 27.5 and y = 13.5
this gets you x + y = 41.
your two equations that needed to be solved simultaneously are true when x = 27.5 and y = 13.5
x - 14 + y = 27 becomes 27.5 - 14 + 13.5 = 27 which becomes 27 = 27.
y + 14 + x = 55 becomes 13.5 + 14 + 27.5 = 55 which becomes 55 = 55.
this confirms the values of x and y are good.
your solution is that alex is 27.5 years old today and sarah is 13.5 years old today.
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