Age Word Problem

[chieftang]

Today, Bill is 1/4 the age of his father. 12 years ago, Bill was 1/12 the age of his father. How old will Bill's father be one decade from today?

A. 58
B. 66
C. 72
D. 76
E. 82

Happy New Years, BTG!!

[GMATGuruNY]

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Today, Bill is 1/4 the age of his father. 12 years ago, Bill was 1/12 the age of his father. How old will Bill's father be one decade from today?

A. 58
B. 66
C. 72
D. 76
E. 82

Happy New Years, BTG!!

We can plug in the answers, which represent the age of Bill's father 10 years from today.

Since 12 years ago Bill was 1/12 the current age of his father, when 10 is subtracted from the correct answer, the result must be a multiple of 12.
The only viable answer choices are A (58-10 = 48) and E (82-10 = 72).
Eliminate B, C and D.

Answer choice A:
Bill's age = (1/4)48 = 12.
Too small: 12 years ago Bill was not yet born, so he could not have been 1/12 the age of his father.

The correct answer is E.

Notice that we were able to determine the correct answer by plugging in only ONE answer choice -- a very efficient way to solve the problem.

Answer choice E:
Bill's age = (1/4)72 = 18.
12 years ago, Bill's age = 18-12 = 6, which 1/12 of 72.
Success!

[chieftang]

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Answer choice E:
Bill's age = (1/4)72 = 18.
12 years ago, Bill's age = 18-12 = 6, which 1/12 of 72.
Success!

Mitch, there's a bit of a gotcha in this one that is subject to interpretation: 1/12 of 72 is 1/12 of the father's age today, but 12 years ago both Bill and his father were 12 years younger! So, Bill's father would have been 60 years old 12 years ago if the answer were E. That's not how you've interpreted it and your value plug-ins clearly bear out the correct interpretation.

But, if we were to set this one up algebraically, then how would we know from the outset to interpret "12 years ago, Bill was 1/12 the age of his father" as "12 years ago, Bill's age was 1/12 of his father's current age" rather than "12 years ago, Bill was 1/12 the age of his father (12 years ago)" ? It feels like you have to solve it both ways and then see which one works out. Incidentally, answer choices for each interpretation are given (both interpretations yield integers for the father's age)! And it's only after solving for the father's age and then plugging it in to determine Bill's age that you can see one interpretation gives a fraction for Bill's age and the other interpretation gives an integer for both Bill and his father!!

Thanks.

[user123321]

Today, Bill is 1/4 the age of his father. 12 years ago, Bill was 1/12 the age of his father. How old will Bill's father be one decade from today?

A. 58
B. 66
C. 72
D. 76
E. 82

Happy New Years, BTG!!

There is ambiguity in this question for sure. As we need to interpret 12 years ago bill was 1/12 the age of his father as, B-12 = (F-12)/12
But unfortunately the current answer seems not present among the choices.
But if we take B-12 = F/12 then we are getting answer as 82.
is it from a standard source?

user123321

[GMATGuruNY]

=

Answer choice E:
Bill's age = (1/4)72 = 18.
12 years ago, Bill's age = 18-12 = 6, which 1/12 of 72.
Success!

Mitch, there's a bit of a gotcha in this one that is subject to interpretation: 1/12 of 72 is 1/12 of the father's age today, but 12 years ago both Bill and his father were 12 years younger! So, Bill's father would have been 60 years old 12 years ago if the answer were E. That's not how you've interpreted it and your value plug-ins clearly bear out the correct interpretation.

But, if we were to set this one up algebraically, then how would we know from the outset to interpret "12 years ago, Bill was 1/12 the age of his father" as "12 years ago, Bill's age was 1/12 of his father's current age" rather than "12 years ago, Bill was 1/12 the age of his father (12 years ago)" ? It feels like you have to solve it both ways and then see which one works out. Incidentally, answer choices for each interpretation are given (both interpretations yield integers for the father's age)! And it's only after solving for the father's age and then plugging it in to determine Bill's age that you can see one interpretation gives a fraction for Bill's age and the other interpretation gives an integer for both Bill and his father!!

Thanks.

Note the phrases in red:
Today, Bill is 1/4 the age of his father. 12 years ago, Bill was 1/12 the age of his father.
Generally, the same phrasing should refer to the same value -- in this case, the father's current age -- unless we are explicitly told otherwise.

If the second sentence wished to refer to the age of the father 12 years ago, I would expect wording along the following lines:
Twelve years ago, Bill was 1/12 as old as his father WAS THEN.

That being said, I agree that the intent of the problem is subject to interpretation.

[chieftang]

Today, Bill is 1/4 the age of his father. 12 years ago, Bill was 1/12 the age of his father. How old will Bill's father be one decade from today?

A. 58
B. 66
C. 72
D. 76
E. 82

Happy New Years, BTG!!

There is ambiguity in this question for sure. As we need to interpret 12 years ago bill was 1/12 the age of his father as, B-12 = (F-12)/12
But unfortunately the current answer seems not present among the choices.
But if we take B-12 = F/12 then we are getting answer as 82.
is it from a standard source?

user123321

I have a buddy who took the GMAT many years ago and recorded questions that he missed during his studies. He gave me that list. So this might be OG9?? I'm not sure. But he only studied from official sources. He didn't put any trust in non-GMAC questions.

Anyhow, under one interpretation (1/12 of the father's age 12 years ago):
Let F = the father's age in 10 years, or F-10 = the father's age today
Let B = Bill's age today
Then:
B = (F-10)/4
B-12 = (F-10-12)/12
Substitute and solve for F:
F = 76 (Answer D)

Under the other apparently correct interpretation (1/12 of the father's current age):
Let F = the father's age in 10 years, or F-10 = the father's age today
Let B = Bill's age today
Then:
B = (F-10)/4
B-12 = (F-10)/12
Substitute and solve for F:
F = 82 (Answer E)

Now plugging F back in to the original equations, we see that 82-10 works, but 76-10 does not. So the answer must be E.

Very tedious, and very easy to miss!

[chieftang]

=

Today, Bill is 1/4 the age of his father. 12 years ago, Bill was 1/12 the age of his father. How old will Bill's father be one decade from today?

A. 58
B. 66
C. 72
D. 76
E. 82

Happy New Years, BTG!!

We can plug in the answers, which represent the age of Bill's father 10 years from today.

Since Bill today is 1/4 the age of his father, when 10 is subtracted from the correct answer, the result should be a multiple of 4.
The only viable answer choices are A (58-10 = 48) and E (82-10 = 72).
Eliminate B, C and D.

Thanks again for the feedback. Hopefully this type of ambiguity doesn't pop up often.

By the way, answer choice B was initially viable too and shouldn't have been eliminated immediately... B (66-10 = 56). 56 is a multiple of 4, and would make Bill 14 years old today. But it doesn't work out since it isn't divisible by 12.

[GMATGuruNY]

Thanks again for the feedback. Hopefully this type of ambiguity doesn't pop up often.

By the way, answer choice B was initially viable too and shouldn't have been eliminated immediately... B (66-10 = 56). 56 is a multiple of 4, and would make Bill 14 years old today. But it doesn't work out since it isn't divisible by 12.

Right you are. The reason B is not viable is that it yields a current age for the father (56) that is not a multiple of 12. I've amended my post above in order to make the reasoning clear.

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B=F/4
(B-12)=1/12(F-12)
This gives: F=66
10 YRS HENCE, F=76.