Originally posted by yoyoma
What is the exact equation to solve Kalah's first riddle? I've read on a couple of posts in the other forums how they boasted it was all easy (and it probably is for someone smarter than by diminuitive IQ) but no one would post how they solved when asked. Trivial thing but it's annoying the heck out of me, heh.
@decadurabolin
I don't get what you are saying.
@Yoyoma
I assume you are talking about the mathematical genie at the bridge?
If so, the question is as follows:
“A princess is as old as the prince will be when the princess is twice as old as the prince was when the princess’ age was half the sum of their present ages…so what are the present ages of the prince and princess?”
Let:
x1 = her present age
y1 = his present age
Working out further:
x2 = her past age = (x1 + y1)/2
y2 = his past age
(i.e. when he’s y2, she’s (x1 + y1)/2)
x3 = her future age = 2y2
y3 = his future age
(i.e. when he’s y3, she’s twice his early age (y2))
z = their age difference, which doesn’t change = (x1 – y1) = (x2 – y2) = (x3 – y3)
Solution:
We need to end up with an equation that has only variables x1 and y1 in it, from which we can get a proportion of her present age to his present age, and then select the Genie’s answer that fits this proportion.
The Genie tells you that her present age is equal to his future age, or:
…x1 = y3
since z = x1 – y1
…z = y3 – y1 (call this equation #1)
since z = x3 – y3
…z = 2y2 – y3 (equation #2)
adding equations #1 and #2:
…2z = 2y2 – y1 (equation #3) (this gets rid of the y3 term)
since z = x2 – y2 = ((x1 + y1)/2) – y2
…2z = x1 + y1 – 2y2 (equation #4) (multiplied both sides by 2)
adding equations #3 and #4:
…4z = x1
since z = x1 – y1
…4(x1 – y1) = x1
…4x1 – 4y1 = x1
…3x1 = 4y1
…x1/y1 = 4/3 = the proportion of (her present age) divided by (his present age)
The only answer that the Genie offers that fits this proportion is the Princess presently being 40 and the Prince presently being 30.
Now you can fill in the other unknowns:
age difference = z = x1 – y1 = 40 – 30 = 10 years
her past age = x2 = (x1 + y1)/2 = (40 + 30)/2 = 35 years old
his past age = y2 = x2 – z = 35 – 10 = 25 years old
her future age = x3 = 2y2 = 2(25) = 50 years old
his future age = y3 = x3 – z = 50 – 10 = 40 years old
Hence, working out their present ages, the prince is 30 and the princess 40.
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