SOLUTION: 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.

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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(52894)   (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.
~~~~~~~~~~~~~~~~~~~~~ 


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.


=======================


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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