# When would this employee retire?

I cant seem to solve this question

A certain company retirement plan has "a rule of 70" provision that allows an employee to retire when the employees age plus years of employment with the company total at least 70. In what year could a female employee hired in 1986 on her 32and birthday first be eligible to retire under this provision ? (Answer: $2005$)

I cant seem to figure how much experience she has with this company. Any suggestions on how to solve this problem

Edit: Could anyone please tell me what I am doing wrong here:

At 1986 she was 32 and joined the company.

Now she needs to work for the company for $\geqslant 70-32 = 38$ more Years

So $1986 + 38 = 2024$ which is wrong. Why is this method wrong?

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Hint: In 1986, she is 32 with 0 years of employment. In 1987, she is 33 with 1 year of employment getting her closer to the goal of 70 by 2: 33+1 = 34. In 1988, she is 34 with 2 years of employment getting her to 34+2 = 36, and so on. Can you figure out when she will get to a total of 70 if she adds 2 every year to the total, one year of age and one year of employment? – Dilip Sarwate Aug 30 '12 at 11:36

If she is hired at age 32 and works for the company $n$ years, then her age will be $32+n$. And she has to retire when the sum of these is 70. So$$n+32+n=2n+32=70.$$ we can solve this for $n$ and get $n=19$. So she can only work for the company 19 years. If she was hired in 1986, she must retire in year $1986+19=2005$.

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I understand how you got $32+n$ but where did the other n come from ? – MistyD Aug 30 '12 at 11:48
She has worked for $n$ years and her age is $32+n$. From my understanding, it was the sum of these that had to be 70. – Daryl Aug 30 '12 at 11:59
could you look at my edit in the question. – MistyD Aug 30 '12 at 12:00
So your 38 years is the sum of both how much older she gets and how long she works for the company. Since both of these are equal from the information in the question, then the number of extra years is half of 38 or 19. – Daryl Aug 30 '12 at 12:04

The employee was hired on her 32nd birthday, so her age at any given moment is 32 years more than the number of years she's spent at the company. Letting the latter be denoted by $x$, we thus need to solve the following equation $$x + (x + 32) = 70$$ to get the number of years she must be employed at the company to be eligible for retirement. Then add that to the year of employment, 1986, to get the first year in which she's eligible to retire.

Re: your edit, 2024 is the year when the employee turns 70. (She turned 32 in 1986, so she was born in 1954.)

At that point she'd be eligible to retire immediately, even if she'd just joined the company! Since she joined already back in 1985, she must've accumulated several employment years, which would've allowed her to retire earlier.

As Daryl already explained, once the employee is hired, every year she accumulates both one year of age and one year of employment time, so the sum of these grows by two years for every year she works at the company.

Thus, after being hired at the age of 32 and working for 19 years, she'll be 32 + 19 = 51 years old and will have 19 years of employment at the company, for a total of 51 + 19 = 70 years, which makes her eligible for retirement. Since she was hired in 1986, that happens in the year 1986 + 19 = 2005.

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Just beat me to it. – Daryl Aug 30 '12 at 11:37
Thanks for the explanation. I was missing the part "every year she accumulates both one year of age and one year of employment time" so the best way is $x+(x+32)=70$ – MistyD Aug 30 '12 at 14:11