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I have 3 minutes to solve questions like these:

Someone invested $\$2,000$ in a fund with an interest rate of $1\%$ a month for $24$ months. Consider it to be compounded interest. What will be the accumulated value of the investment after $24$ months?

Remember, I am not allowed to use a calculator or any electronic device. Only pencil and pen.

Anyone have an idea how to solve this very fast within 3 minutes?

Edit [by SN]:

  1. Logarithm tables are not allowed either.

  2. This is a multiple choice question; the options are:

    • $2,437.53$
    • $2,465.86$
    • $2,539.47$
    • $2,546.68$
    • $2,697.40$
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1  
How much precision do you require? And do you have a logarithm table? –  Henning Makholm Oct 21 '11 at 14:34
2  
What is the time period after which the principal gets compounded? –  Eisen Oct 21 '11 at 14:35
9  
... and then studies are made to figure out why people both hate math and do not learn anything :/ –  Mariano Suárez-Alvarez Oct 21 '11 at 14:42
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@HenningMakholm: Yes, the options are:(A) 2.437,53 (B) 2.465,86 (C) 2.539,47 (D) 2.546,68 (E) 2.697,40 –  jvb Oct 21 '11 at 14:46
7  
-1 for leaving out the information that it is a multiple choice question. –  Phira Oct 21 '11 at 14:49
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6 Answers

up vote 9 down vote accepted

You must compute $C \; (1 + I)^N = 2000 \; (1+0.01)^{24}$ The power is difficult to compute without a calculator. But if the interest is low (more to the point, if $N \; I \ll 1$) , you can approximate it by a first or second order approximation of the binomial formula:

$(1+0.01)^{24} \approx 1 + 24 \times 0.01 = 1.24$ $\hspace{20px} C_f = 2480$ (first order)

$(1+0.01)^{24} \approx 1 + 24 \times 0.01 + \frac{24 \times 23}{2}\times 0.01^2 = 1.2676$ $\hspace{20px} C_f = 2535.20$ (second order)

In this case, the second order appproximation is just decent, if not very precise (real value: 1.2697346..., $C_f=2539.47$)

(Update: and for the multiple-choice decision, it's enough).

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1  
+1. When I saw the question I thought of this approach and did a third-order approximation on paper. It took about 45 seconds (including the doubling at the end) and was precise enough to determine the answer within 20 cents or so. –  Charles Oct 21 '11 at 15:13
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Perhaps this is where they (used to--before calculators) use the "Rule of 78s"

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Develop $2000(1+0.01)^{24}$ using the binomial theorem and start to calculate and add the first terms until you see that the summands are small enough. Four terms should be sufficient.

I don't know if you can do this calculation in 3 minutes. It is certainly doable.

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I might add that the advantage of this calculation over the repeated squaring is that if you do not finish it, you are still quite close to the final result and can make an educated guess on the multiple choise quiz. –  Phira Oct 21 '11 at 14:54
    
"I don't know if you can do this calculation in 3 minutes. It is certainly doable." I have seen students pull out a calculator to compute $0.5\times 2$ and so your statement is dead on target. –  Dilip Sarwate Oct 21 '11 at 16:42
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It is fastest to evaluate $(1+0.01)^8$ and cube it: $$ ( 1+ 0.01)^8 \approx 1 + \frac{8}{100} + \frac{8 \times 7}{2} \frac{1}{100^2} = 1.0828 $$ Raising this to the third power $(1+y)^3 \approx 1 + 3y(1+y) = 1 + \frac{3 \times 828}{10,000} \times 1.08$, thus $$ \begin{eqnarray} 2000 \times \left( 1 + 0.01 \right)^{24} &\approx& 2000 + 2000 \times \frac{3 \times 828}{10,000} \times 1.08 = 2000 + \frac{6 \times 828}{10} \times 1.08 \\ &=& 2000 + 496.8 + 496.8 \times 0.08 \approx 2000 + 496.8 + 39.8 = 2536.6 \end{eqnarray} $$ And from the way we did the approximation we should round up, which leaves you with the third choice.

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If the interest is compounded monthly, which the wording of your question suggests, then you have to calculate $2000 \times 1.01^{24}$, which is not possible in three minutes without log tables. Did the question perhaps specify that the interest was compounded annually?

Edited to add: Now the OP reveals that this was a multiple-choice question. Suddenly it's doable! (As many people have pointed out.)

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3  
Computing $1.01^{24}$ by repeated squaring using pencil and paper with 3 decimal places in each intermediate took me 130 seconds, giving 1.267, reasonably close to the true value of 1.26973. –  Henning Makholm Oct 21 '11 at 14:49
    
But it is not close enough when you want to mulitply it with 2000 and have two decimals (and yes, I know, this is a multiple choice question, so much less precision is actually required ...). –  Phira Oct 21 '11 at 14:52
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By the way, I agree that taking the first few terms of the binomial expansion is probably smarter. –  Henning Makholm Oct 21 '11 at 15:02
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In decimal, 1% of X is just X right shifted two places. Thus, X + 1% of X is just X added with itself shifted two places, which you should easily be able to do in place, starting at the rightmost digit and adding the digit two to its left. The first few months give this:

2000.0000
2020.0000
2040.2000
2060.6020
2081.2080
...

Each if these takes a second or two, so 24 months in 3 minutes should be easy.

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