{"id":4313,"date":"2014-01-20T09:00:34","date_gmt":"2014-01-20T17:00:34","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=4313"},"modified":"2020-01-15T10:48:46","modified_gmt":"2020-01-15T18:48:46","slug":"compound-interest-on-the-gmat","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/compound-interest-on-the-gmat\/","title":{"rendered":"Compound Interest on the GMAT"},"content":{"rendered":"

First, a few practice problems.\u00a0 Remember: no calculator<\/a>!<\/p>\n

1) If $5,000,000 is the initial amount placed in an account that collects 7% annual interest, which of the following compounding rates would produce the largest total amount after two years?<\/p>\n

(A) compounding annually<\/p>\n

(B) compounding quarterly<\/p>\n

(C) compounding monthly<\/p>\n

(D) compounding daily<\/p>\n

(E) All four of these would produce the same total<\/p>\n

2) If A is the initial amount put into an account, R is the annual percentage of interest written as a decimal, and the interest compounds annually, then which of the following would be an expression, in terms of A and R, for the interest accrued in three years?<\/p>\n

\"ciotgb_img1\"<\/a><\/p>\n

3) At the beginning of January 2003, Elizabeth invested money in an account that collected interest, compounding more frequently than a year. Assume the annual percentage rate of interest remained constant.\u00a0 What is the total amount she has invested after seven years?<\/p>\n

Statement #1<\/span>: her initial investment was $20,000<\/p>\n

Statement #2<\/span>: the account accrued 7% annual interest<\/p>\n

4) Sarah invested $38,700 in an account that paid 6.2% annual interest, compounding monthly.\u00a0 She left the money in this account,\u00a0 collecting interest for a full three-year period.\u00a0 Approximately how much interest did she earn in the last month of this period?<\/p>\n

(A) $239.47<\/p>\n

(B) $714.73<\/p>\n

(C) $2793.80<\/p>\n

(D) $7,888.83<\/p>\n

(E) $15,529.61<\/p>\n

Solutions to these will be given at the end of the article.<\/p>\n

 <\/p>\n

Simple Interest<\/h2>\n

In grade school, you learn about simple interest, largely because we want to teach little kids something about the idea of interest, and that’s the only kind of interest that children can understand.\u00a0 No one anywhere in the real world actually uses simple interest: it’s a pure mathematical fiction.<\/p>\n

Here’s how it works.\u00a0 There’s an initial amount A, and an annual percentage P.\u00a0 At the end of each year, you get interest in the amount of P percent of A — the same amount every year.\u00a0 Suppose the initial amount is $1000, and the annual percentage is 5%.\u00a0 Well, 5% of $1000 is $50, so each year, you would get the fixed value of $50 in interest.\u00a0 If you plotted the value of the account (principle + interest) vs. time, you would get a straight line.<\/p>\n

\"ciotgb_img2\"<\/a><\/p>\n

Again, this is a fiction we teach children, the mathematical equivalent of Santa Claus.\u00a0 This never takes place in the real world.<\/p>\n

 <\/p>\n

Compound Interest, compounding annually<\/h2>\n

With compound interest, in each successive year or period, you collect more interest not merely on the principle but on all the interest you have accrued up to that point in time.\u00a0\u00a0 Interest on interest: that’s the big idea of compound interest.<\/p>\n

Here’s how it plays out.\u00a0 Again, there’s an initial amount A, and an annual percentage P, and we also have to know how frequently we are compounding.\u00a0 For starters, let’s just say that we are compounding annually, once a year at the end of the year.\u00a0\u00a0 In fact, let’s say we have $1000, and the annual interest rate is 5%.\u00a0 Well, in the first year, we would earn five percent on $1000, and gain $50 in interest.\u00a0\u00a0 The first year is exactly the same as the simple interest scenario.\u00a0 After that first year, we now have $1050 in the account, so at the end of the second year, we gain 5% of $1050, or $52.50, for a new total of $1102.50.\u00a0 Now, that’s our new total, so at the end of the third year, we gain 5% of $1102.50, or $55.12 (rounded down to the nearest penny), for a new total of $1157.62.\u00a0 At the end of three years, the simple interest scenario would give us $1150, so the compound interest gains us an extra $7.62 —- not much, but then again, $1000 is not a lot to have invested.\u00a0 You can see that, with millions or billions of dollars, this would be a significant difference.<\/p>\n

Here, I was demonstrating everything step-by-step for clarity, but if we wanted to calculate the total amount after a large number of years, we would just use a formula.\u00a0 We know that each year, the amount increased by 5%, and we know that 1.05 is the multiplier<\/a> for a 5% increase.\u00a0 After 20 years, the amount in the account would have experienced twenty\u00a05% increases, so the total amount would be<\/p>\n

\"ciotgb_img3\"<\/a><\/p>\n

We don’t have to do it step-by-step: we can just jump to the answer we need, using multipliers.\u00a0 Of course, for this exact value, we would need a calculator, and you don’t get a calculator on the GMAT Quantitative section.\u00a0 Sometimes, though, the GMAT lists some answers in “formula form”, and you would just have to recognize this particular expression, \"1000*(1.05)^20\", as the right formula for this amount.<\/p>\n

If we graph compound interest against time, we get an upward curving graph (purple), which curves away from the simple interest straight line (green):<\/p>\n

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The curve of the graph, that is to say, the multiplying effect of the interest, gets more pronounces as time goes on.<\/p>\n

BIG IDEA #1<\/span>: as long as there is more than one compounding period, then compound interest always earns more than simple interest. \u00a0<\/b><\/p>\n

 <\/p>\n

Other compounding periods<\/h2>\n

A year is a long time to wait to get any interest.\u00a0\u00a0 Historically, some banks have compounded over shortened compounding period.\u00a0 Here is a table of common compounding periods:<\/p>\n

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Technically, the fraction for “compounding daily” would be 1\/365 in a non-leap year and 1\/366 in a leap year; alternatively, one could use 1\/365.25 for every year.<\/p>\n

Now, how does this work?\u00a0 Let’s say the bank gives 5% annual, compounding quarterly.\u00a0 It would be splendid if the bank wanted to give you another 5% each and every quarter, but that’s not how it works.\u00a0 The bank takes the percentage rate of interest and multiplies it by the corresponding fraction.\u00a0 For 5% annual, compounding quarterly, we would multiply (5%)*(1\/4) = 1.25%.\u00a0 That’s the percentage increase we get each quarter.\u00a0\u00a0 The multiplier for a 1.25% increase is 1.0125.\u00a0 Suppose we invest $1000 initially and keep the money in this account for seven years: that would be 7*4 = 28 compounding periods, so there are twenty-eight times in that period in which the account experiences a 1.25% increase.\u00a0 Thus, the formula would be<\/p>\n

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For compounding quarterly, we divide the annual rate by four and compound four times each year.\u00a0 For compounding monthly, we divide the annual rate by twelve and compound twelve times a year.\u00a0\u00a0 Similarly, for daily or any other conceivable compounding period.<\/p>\n

How do the amounts of interest accrued compare for different compounding periods?\u00a0 To compare this, let’s pick a larger initial value, $1,000,000, and collect over a longer period, 20 years.\u00a0 Below are the total amounts, after twenty years, on an initial deposit of one million dollars compound at 5% annual:<\/p>\n