Question 696182: mary is 24 years old. mary is twice as old as ana was when mary was as old as ana is now. how old is ana?
Found 3 solutions by Stitch, kentucky, MathTherapy:
Answer by Stitch(470)   (Show Source):
Answer by kentucky(1) (Show Source):
You can put this solution on YOUR website! ^ the answer above is incorrect.
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Let M = Mary
Let N = Ana
Let d = difference between their ages.
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Now this is a very tricky question to wrap your head around, so read the question very carefully.
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From the first sentence "Mary is 24 years old". We have the first equation.
Eq. 1: M = 24
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Now it gets very tricky in the following sentence. "Mary is twice as old as
Ana was when Mary was as old as Ana is now."
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To carefully dissect this statement, let's split in into two parts;
a.) "Mary is twice as old as Ana."
b.) "When Mary was as old as Ana."
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From statement a, we can deduce that M = 2N, and from statement b, we can
deduce that the said statement a is under a premise: "When Mary was as old
as Ana." We can write this into an equation such that;
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M - d = N
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(Mary's age, minus the difference between Mary and Ana's age, is equal to
the current age of Ana.)
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M - (M - N) = N
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(the difference between their ages is equal to (M - N), therefore we can
substitute this value in place of d)
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After figuring out the equation of the premise under under statement a, we
can now deduce the main equation of our problem.
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Eq. of statement a: M = 2N
Eq. of statement b: M - (M - N) = N
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Eq. of statement a, under the premise of statement b:
M - d = 2*(N - d)
M - (M - N) = 2*(N - (M - N))
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If you will notice, we also have to subtract d from N in order to preserve
the passage of time in both sides, since we are comparing their ages from
d years in the past.
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Now we simplify our equation.
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M - d = 2(N - d)
M - (M - N) = 2(N - (M - N)
M - M + N = 2(N - M + N)
N = 2N - 2M + 2N
N = 4N - 2M
2M = 3N
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Now that we have the simplest form of our equation, we can substitute
M = 24 into the equation and finally find the value of N.
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2 (24) = 3N
48 = 3N
48/3 = 3N/3
N = 16 , Ana is 16 years old.
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Now with this answer, the premise "Mary is twice as old as Ann, when Mary
was as old as Ann now", is satisfied.
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When Mary was 16 years old (8years ago; d = 8), Ann will be 16-8 = 8 years
old, which is twice as young as Mary; In other words, Mary was twice as old
as Ann (M = 2N) when M - (M - N) = N.
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(p.s. the answer is not 18 years old, it does not satisfy the premise)
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Edit: In response to MathTherapy's claim, I have decided to expound further why 18 IS WRONG. and 16 is the right answer. 11/26/2021
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First of all with the very obvious rebut. If you will read carefully, it says in the problem that "Mary was twice as old as Ana, when Mary was as old as Ana".
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This means that "d" years ago, M = 2N. If Ana was 18, "d" years ago, then "d" would be equal to 6 since 24 - 18 = 6. (M - N = d)
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24, and 18, 6 years in the past would be 18, and 12. and 18 is not twice of 12. Therefore the problem should rather state "Mary was one and a half times older than Ana was, when Mary was as old as Ana" to satisfy 18 being the correct answer.
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18 is wrong.
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I understand what you guys did but there was something wrong in your assumptions which formed the wrong equation for the problem.
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Your equation was M = 2 (N - d); it is very similar to mine but you forgot about one thing. You did not subtract d from M which was the fatal flaw. We are dealing with time here, and we are comparing their past ages therefore we must subtract d from both variables. M = 2 (N - d) is simply comparing the PRESENT age of Mary to the past age of Ana which is not what is asked in the problem when the problem which states "Mary WAS twice as old as Ana WAS, when Mary was as old as Ana (of the present). If we do realize this and change our equation so that we are comparing their past ages, we get my equation (M - d) = 2 (N - d), and lastly, we will be able to find my answer which satisfies the premise.
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M = 24 (present day age of Mary)
N = unknown (present age of Ana)
M - d = unknown (past age of Mary)
N - d = unknown (past age of Ana)
d = M - N (difference of their ages in the present, and in the past)
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Here is a clearer and more straightforward version of my solution in case of confusion due to my excessive explanations:
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M = Mary
N = Ana
d = difference between their ages (M - N)
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M - d = 2 (N - d)
M - (M - N) = 2 [N - (M - N)]
M - M + N = 2 (N - M + N)
N = 2N - 2M + 2N
N = 4N - 2M
2M = 4N - N
2M = 3N ; now we insert M = 24
48 = 3N
16 = N ; Ana is 16 years old now. she was 8 in the past when Mary was 16, therefore the premise "Mary was TWICE as old as Ana, WHEN Mary was 'AS OLD AS ANA' is satisfied.
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website! mary is 24 years old. mary is twice as old as ana was when mary was as old as ana is now. how old is ana?
The other person's answer, 18-years-old is correct!
Age-related problems like this can be a real HEADACHE, but once you get used to them, they become a "breeze" to figure out.
One KEY piece of information that needs to be derived is who's OLDER. The statement, "when mary was as old as ana is now"
clearly shows that Mary is older.
With Mary being the older person, and with Ana's age being A, the difference in their ages is: M - A, or 24 - A, since Mary is 24.
Mary was Ana's age, "24 - A" years ago, at which time Ana was A - (24 - A), or 2A - 24
Now, since it's stated that, "mary is twice as old as ana was when mary was as old as ana is now," we get
the following equation: 24 = 2(2A - 24)
24 = 4A - 48
24 + 48 = 4A
72 = 4A
Ana, or 
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