I am currently studying for my SAT2 Subject Test in Mathematics Level 2 and was check my answers to a practice test when I can across this (below) question.
Problem:
George invests $\$1000$ into an account that he hopes will earn $12\%$ interest annually. How many years (rounded to the nearest year) will it take his investment to double in value? (A) $4$ (B) $6$ (C) $7$ (D) $8$ (E) $12$
So, I will first show you my solution, then the book's solution. The goal for this MSE question is to verify that my answer is right and the book's is wrong to this problem.
My answer:
$$2000=1000(1.12)^n$$ $$2=(1.12)^n$$ $$\log_{1.12}2=n\approx6.12$$ $$n \ \text{(rounded up in terms of years)} \ = 7$$ $$\boxed{\text{(C)} \ 7}$$
Book's Answer:
(B) George's investment doubles to $\$2000$. Therefore, $$2000=1000(1+0.12)^t,$$ or $$2=1.12^t.$$ Take the log is both sides to get $$t=\log_{1.12}2\approx6.12\approx6.$$
What doesn't make sense to me is why they rounded down. Since our takes more than $6$ years to double, don't you round up?
If you plug in $t=6$ into the original equation, you get $$A=1000(1.12)^6\approx\$1973.82.$$
As you can observe, $$\frac{1973.82}{1000}\ngeq 2.$$ So the does not satisfy the question, "how many years (rounded to the nearest year) will it take his investment to double in value?"
Thus proving that the answer cannot be (B) $6$. Since when $t=7$ makes $A>2$, that means that (C) $7$ is the closest estimate for the given problem.
Quite simply, the reason why the official solution is $6$ is because the instructions specifically state to round the result to the nearest (whole) year. There is no implied directive to do the rounding in the context of the minimum time needed for the investment to actually achieve a doubling of its value. You are simply asked to calculate the time, and subsequently round the calculated value.
Had the question asked instead "what is the minimum number of whole years required for the investment to double in value," then your answer of $7$ years would be correct. Note that such a wording would not substantively change the difficulty or problem-solving requirements of the problem, yet it would be unambiguous.
It is an unfortunate defect of the ambiguities of language that occasionally crops up when posing mathematical word problems. This tends to happen more frequently in practice exams that were not authored by the official test-administering body; usually ETS (the administrators of the SAT exams) is more careful in its choice of wording so as to minimize ambiguities in interpretation.