A man was shopping for a ring for his soon to be wife. The jeweler asked the man how old his wife was to help determine the style of ring to suggest. The man replied:
"Well I'm 27 years old and there was a time when my fiancee was three times as old as me, but the next year, I was only half her age. So suggest a ring based on that."
The man's soon to be wife is 29 years old.
To figure this out, we can do a little bit of guessing, however we can write down what we know in equation form to figure this out in a more mathematical way.
First the man says at one time his wife was three times older than him. Lets say that m is the man's age at this time and w is his fiancee's age.
w = 3(m)
This equation says the woman is 3 times as old as the man.
Now we know that 1 year later, the man is half the age of the woman.
(1/2)(w + 1) = m+1
We make sure to use w+1 and m+1 to show that we are talking about their ages 1 year after the first equation.
we can now substitute 3(m) for w in the second equation since our first equation say w = 3m:
We can first rewrite the equation to get rid of the parentheses:
We can solve the equation for m by first multiplying both sides by 2:
Next we distribute the 2 on the right hand side of the equation:
Now we can subtract 2m from both sides:
Finally we can subtract 1 from both sides:
We find that m = 1. This tells us that when the man was 1 years old, his future wife was 3 since we can plug 1 in for m in our original equation:
w = 3(m)
w = 3(1)
w = 3
So since the woman was 3 when the man was 1, the next year, the woman was 4 and the man was 2 which satisfies the second part of the problem.
We also no that since the man was 1 when the woman was 3, the man's fiancee is 2 years older than him. Since the man is 27, that would make his fiancee 29.