As an undergraduate, I concentrated in Finance. When I tell people this, they make two unwarranted assumptions: the first is that I work in Finance (I don’t), and the second is that I am a glutton for mathematical punishment (debatable).
The reason people are intimidated by the kinds of compound interest equations we encounter in finance classes is that they look complicated. GMAT test-takers get anxious whenever I introduce this topic in class. But, as with most seemingly abstruse topics, these concepts are far less difficult than they appear at first glance.
Here’s all we really need to know about interest equations: if we’re talking about simple interest, the interest will be the same in every time period, and the equation you assemble will end up being straightforward linear algebra (if you choose to do algebra, that is). If we’re talking about compound interest, we’re really talking about an exponent question. The rest involves a bit of logic and algebraic manipulation.
Look at this official question that many of my students have initially struggled with:
An investment of $1000 was made in a certain account and earned interest that was compounded annually. The annual interest rate was fixed for the duration of the investment, and after 12 years the $1000 increased to 4000 by earning interest. In how many years after the initial investment was made would the 1000 have increased to 8000 by earning interest at that rate?
Looking at this question, the first instinct of most test-takers is to start frantically rummaging through their memory banks for that compound interest formula – there’s no need. Take a deep breath and remind yourself that these questions are just exponent questions involving a bit of algebra. With this in mind, let’s call the factor that the principal is multiplied by in each time period “x”. (If you’re accustomed to working with the formula, “x” is basically standing in for your standard (1 + r/100.) If you’re not accustomed to this formula, feel free to retroactively erase this parenthetical from your memory banks.)
If the principal is getting multiplied by “x” each year, then after one year, the investment will be 1000x. After two years the investment will be 1000x^2. After three years, it will be 1000x^3… and so on. In our problem, we’re talking about an investment after 12 years, which would be 1000x^12. If this value is 4000, we get the following equation: 1000x^12 = 4000 (and file away for now that the exponent represents the number of years elapsed).
Ultimately, we want to know what the exponent should be when the investment is at $8000. If you’re looking at the answer choices now and think that 24 seems just a little too easy, your instincts are sound.
We need to work with 1000x^12 = 4000. Let’s simplify:
Divide both sides by 1000 to get x^12 = 4. Solving for x seems unnecessarily complicated, so let’s consider our options. x^12 = 4 is the same as x^12 = 2^2, so if we take the square root of both sides, we will get x^6 = 2.
Essentially, this means that every 6 years (the exponent) the investment is doubling, or multiplied by 2. But we want to know how long it will take for that initial $1000 to become $8000, or to be multiplied by a factor of 8.
What can we do to x^6 = 2 so that we have an 8 on the right side? We can cube both sides!
(x^6)^3 = 2^3
x^18 = 8
This means that it will take 18 years to increase the investment by a factor of 8. Therefore, our answer is B.
Alternatively, once we see that the investment doubles every 6 years, we can ask ourselves how many times we need to double an investment to go from $1000 to $8000. Doubling once gets us to $2000. Doubling twice gets us to $4000. Doubling a third time gets us to $8000. So if we double the investment every 6 years, and we need the investment to double 3 times, it will take a total of 6*3 = 18 years.
Takeaway: There are plenty of formulas that could come in handy on the GMAT – just know that a little logic and conceptual understanding will allow you to solve many of the questions that seem to require a particular formula. Memorization has limits that logic and mental agility don’t.
*GMATPrep question courtesy of the Graduate Management Admissions Council.