#### (A new question of the week)

Some questions we get, while small, raise interesting issues. In a question we got last month, there are several little issues pertaining to how the final answer should be chosen; as is too often the case, it seems that a diligent student who cares about accuracy might be penalized. This has important implications for the design of tests. Let’s take a look.

## The question

Jing Zhu asked:

A question says,

What is the

smallest amount of moneythat a man needs to deposit annually for there to be an amount balance of over $15,000 at 6% interest after 12 years? (Round your answer to the nearest dollar.)According to the formula provided, the answer in decimal is $889.155. When 889 is plugged in the formula, the balance is

lessthan $15,000. When 890 is plugged in the formula, the balance is over $15,000. So, in this case, should the answer be889 according to the instruction statedinside the parentheses or890 according to the context?

To do a quick check before answering, I brought up Excel and used its FV (future value) formula to find the amount. With deposits of $889, the final amount comes to $14,997.38, which is indeed too little; it doesn’t answer the question! With deposits of $890, we end up with $15,014.25, which is enough. Jing Zhu is right (assuming payments are at the end of each year, which we’ll be looking at below).

We haven’t been told what formula was provided; that will be one issue (and is a reason we prefer that you show us the entire problem you are working on, include such context as formulas you use). The main issue is how to round.

## Rounding issues

I replied:

That’s a good question! I think it is a matter of interpretation.

I personally would want to round up to $890, as you suggest, because that is the smallest whole-dollar amount that will have the desired effect.If we take the problem literally, then we might say that

the answer is $889.16(where I rounded up to the nextpenny), but then round that as required to the nearest dollar to get $889. In this case, we are supposing that they don’t want an actual payment amount (which would be to the penny), but just want you toround for the sake of reporting your answer. This is particularly likely in a machine-grading context.I have seen book problems, and perhaps test problems, whose authors apparently just didn’t think that closely about the reality of the problem, but just automatically asked for a rounded answer.

If this is to be

graded by a human, I would state both answers with an explanation; it should get extra credit, for showing how well you are thinking! If it isgraded by a machine, then all you can do is what you think best. In that case, I would probably go with the literal answer, as machines don’t think carefully.And if there is a way to

ask the teacherabout it, I would do so.

One of my pet peeves is how computerized homework and testing can twist the way we are required to think. But even without computers, the same concerns surround multiple-choice questions or other formats where there is no way to show work. When work is shown, I often recommend explaining any assumptions you make (which may result in your solving the wrong problem, but conceivably getting full credit if your interpretation was reasonable and your work correct *for the problem you solved*).

Here the issue is at the other end of the problem: not an initial assumption about its meaning, but how you enter the answer at the end. When you aren’t sure how to round, I recommend prominently showing both the unrounded result and the result rounded as you think is requested or appropriate (or both, if they differ). But when all you can enter is a number, you are stuck in a dilemma.

(Similarly, when I teach, I often tell students clearly that even during a test, they are welcome to raise their hands and ask interpretation questions; this is especially important when I have students for whom English is a second language, or for whom the culture assumed in some “word problems” may be foreign. But it is good for the rest of the class, too! Some may not be familiar with some cultural detail I assume every American knows.)

## A better problem statement

Doctor Rick joined in with a slightly different perspective:

Hi, If I may add my own personal opinion, it is that the answer of

$890 is actually the answer to a different (and arguably better) question:What is the

smallest whole number of dollarsthat a man needs to deposit annually for there to be an amount balance of over $15,000 at 6% interest after 12 years?My view of the question as stated is that

the answer to the questionis $889.18, and then we follow thedirections for, by rounding it to $889.reportingour answerThere is another issue here, having to do with

details that are missing from the problem. Is the interestcompoundedannually? When are the deposits made — at thebeginning or endof each year? You say a formula was provided, so perhaps you didn’t have to know these details in order to choose a formula. Based on your answer, I supposethe formula providedwas some form of the first formula in the “Regular Deposits” section of this Ask Dr. Math FAQ:

This agrees with my thoughts, but says it more plainly: As a teacher, we would rather change the wording of the question so that the correct answer is clearly correct; as a student, we would take the wording literally and round after finding the truly correct answer.

And why does his wording result in a better question? This way, the rounding is an inherent part of the problem, not an afterthought tacked on for the sake of uniform answers. It takes rounding seriously, requiring thought about which way to round in order to fulfill the needs of the problem, which can be important in real life. And in so doing, it prevents the conflict we’ve seen in trying to answer it.

As for the formula, the FAQ says this:

Suppose you opened an account at a bank which was paying an annual interest rate of i (a fraction, equivalent to 100i%). You make a deposit of Mat the endof each of q equal time periods each year (including the end of the last period). The interest iscompounded once per period. Then the value P of the account at the end of n years is given by P = M([1+(i/q)]^{nq}-1)(q/i)

This formula, expressed more clearly, is $$P=M\left(\left[1+\left(\frac{i}{q}\right)\right]^{nq}-1\right)\frac{q}{i}$$ which can be rearranged slightly to $$P=\frac{Mq}{i}\left(\left[1+\left(\frac{i}{q}\right)\right]^{nq}-1\right)$$

For annual payments, \(q = 1\) (because payments are made once a year), and the formula becomes $$P=\frac{M}{i}\left(\left[1+i\right]^{n}-1\right)$$

This formula is derived from the formula for the sum of a geometric series, which is particularly easy to see in this case of annual payments. If payments are made at the end of each year, for *n* years, then the last payment earns no interest, adding only *M* to the final amount; the next-to-last payment grows for 1 year, adding \(M(1+i)\) to the total; and so on, until the first payment grows over \(n – 1\) years, contributing \(M(1+i)^{n-1}\). The total is therefore $$M+M(1+i)+…+M(1+i)^{n-1} = M\frac{\left(1+i\right)^{n}-1}{(1+i)-1}$$ which leads to our formula.

## The wrong formula?

Jing Zhu answered us, providing the entire problem:

Thank you for your reply. I attached the original question for your reference.

This is an SAT prep question, and because of the word “over” (“over $15,000), I think we should use inequality to solve this problem. As you can see, the final answer turns out to be greater than 889.155. Therefore, I’d argue that the only correct answer should be 890.

The formula is the same as ours, where \(M\) is called \(P_0\), and \(i\) is called \(r\). But …

I replied,

Thanks. Interestingly, though the formula you were given is the same one we give in our FAQ that Dr. Rick referred to,

their description is wrong. This formula (for an “ordinary annuity”) applies when payments are made at theendof each period (year), not at thebeginningas they say (which is called an “annuity due”, and is mentioned after the main discussion in our FAQ). But we have to go by their formula, so we can ignore that discrepancy.

When payments are at the start of each period, each earns an extra period of interest, so the total is multiplied by \(\left(1+\frac{i}{q}\right)\), leading to the FAQ’s formula, $$P=M\left(\left[1+\left(\frac{i}{q}\right)\right]^{nq}-1\right)\left(1+\frac{q}{i}\right)$$ as shown in the FAQ.

The problem is valid if we change their word “beginning” to “end”. So the work will be the same whether we see this error or not. But if a student knew the correct formula, the problem would be still more confusing!

Here is the work to solve problem 38, as I would write it:

$$\frac{P_0}{0.06}\left(\left[1+0.06\right]^{12}-1\right)>15,000$$

$$P_0>\frac{(0.06)15,000}{\left[1.06\right]^{12}-1}$$

$$P_0>\frac{(0.06)15,000}{1.012196}=889.15544$$

So the smallest amount (in pennies) that satisfies this is found by rounding up to $889.16; and the smallest amount (in whole dollars) is found by rounding up to $890.

## Wisdom for test-takers

I continued:

Your use of an inequality, and your work itself, are good. We are left with the same dilemma we discussed: to

take it literallyand round thecorrectanswer of $889.16 to the (technicallyincorrect) answer, $889, just because they told you to; or toanswer Dr. Rick’s modified question, which makes more practical sense, and say $890.If I were you, I would answer $889, which seems to be what they are asking for, knowing that you are wiser than that. They said to find the correct answer, and

thento round that to the nearest dollar, so that is what you do.If this were given on an actual test, and it was a question on which you show work, you could

explain all this in writingand certainly get credit. If you could only give thenumerical answer, then I would enter $890.And it’s quite likely that the actual test would not have such a dilemma built in. (Sometimes there have been bad problems, and they have been discovered and accounted for.)

A similar point was made by teacher CTB in a comment to a post from last year, while I was writing this post:

Having myself also worked as an examiner (for A-level Maths – Stats module), it is unlikely that in an exam you would be penalised for any sensible decision you make to get to an answer. If you write down your assumption somewhere in your solution, so that the examiner can read it, you should get credit for your calculations.

When you can show work, you can be much more confident of being judged fairly, and therefore less anxious about how to answer.

We received this closing comment:

Thank you for your detailed explanation. It was a pleasure to have had these discussions with you. Have a wonderful day! ^^