Investment word problems usually involve simple annual interest — as opposed to compounded interest. Simple interest is earned on the entire investment amount for a given period of time.
This differs from compounded interest, where simple interest is earned for a smaller amount of time (for instance, for one month).
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Then, over the next period of time, the interest is earned on the original investment *plus* the interest that was earned on that first time period. Then, during the third time period, interest is earned on the initial investment amount plus the interest earned during the first two periods. And so forth.
The formula for the simple interest earned on an investment is given by I = Prt.
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In the simple-interest formula I = Prt, the variable I stands for the interest on the original investment, P stands for the amount of the original investment (called the "principal"), r is the interest rate (expressed in decimal form), and t is the time.
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For annual interest, the time t must be in years. If they give you a time of, say, nine months, you must first convert this to years. Otherwise, you'll get the wrong answer.
The time units, in all cases, must match the interest-rate units. For instance, if you got a loan from your friendly neighborhood loan shark, where the interest rate is monthly, rather than yearly, then your time must be measured in terms of months.
Investment word problems are not generally terribly realistic; in real life, interest is pretty much always compounded somehow, and investments are not generally all for whole numbers of years. But you'll get to more practical stuff later in your studies; this topic is just warm-up, to prepare you for later.
In all of these word problems, you will want to substitute all known information into the I = Prt equation, and then solve for whatever is left.
In this case, the investment amount (that is, the principal) is P = $1000, the interest rate (expressed in decimal form) is r = 0.06, and the time is t = 2. Substituting these values into the simple-interest formula, I get:
I = (1000)(0.06)(2) = 120
I will get $120 in interest.
For this exercise, I first need to find the amount of the interest. Since simple interest is added to the principal, and since the principal was P = $500, then the interest is I = $650 − 500 = $150. The time is t = 3. Substituting all of these values into the simple-interest formula, I get:
150 = (500)(r)(3)
150 = 1500r
Of course, I need to remember to convert this decimal to a percentage.
I was getting 10% interest.
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The hard part comes when the exercises involve multiple investments. But there is a trick to these that makes them fairly easy to handle.
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The problem here comes from the fact that I'm splitting that $50,000 in principal into two smaller amounts. Here's how to handle this:
I will make a table. The top row has, as its entries, the variables in the simple-interest formula. The left-hand column labels the two funds and the totals, as needed. (I can't add rates — that's just a thing; rates can't be added — and each of the investments is for one year, so there are no "total" values for the interest rates or the years. Hence, the dashes, to remind me not to try to put anything in there.)
I | P | r | t | |
---|---|---|---|---|
Fund X | ||||
Fund Y | ||||
total | 4,500 | 50,000 | — | — |
I know the interest total that I'm aiming for, and I know the total amount that I'm investing, so I can enter "total" values for the "interest" and "investment" columns.
I know the interest rates and the time (namely, one year) for the two investments, so I can enter these values in the "rate" and "time" columns, in each fund's row.
Putting it all together, I get the following start to my set-up:
I | P | r | t | |
---|---|---|---|---|
Fund X | ? | ? | 0.06 | 1 |
Fund Y | ? | ? | 0.14 | 1 |
total | 4,500 | 50,000 | — | — |
How do I fill in for those question marks? I'll start with the principal P. Let's say that I put x dollars into Fund X, and y dollars into Fund Y. Then x + y = 50,000.
But this doesn't help much, since I only know how to solve equations in one variable. However, I then notice that I can solve x + y = 50,000 to get y = 50,000 − x.
THIS TECHNIQUE IS IMPORTANT! The amount in Fund Y is (the total) less (what I've already accounted for in Fund X), or 50,000 − x.
Any time you have a total that is divided into two parts, you can designate one part as (one part), and the rest will be (the total) minus the (one part), because the second part is whatever is left, from the total, after (one part) is accounted for.
You will need this technique, this "how much is left" construction, in the future, so make sure you understand it now.
So now I have a variable for the Fund X part, and an expression for however much was left to go into the Fund Y part. I can add these to my table:
I | P | r | t | |
---|---|---|---|---|
Fund X | ? | x | 0.06 | 1 |
Fund Y | ? | 50,000 − x | 0.14 | 1 |
total | 4,500 | 50,000 | — | — |
Now I will show you why I set up the table like this. By organizing the columns according to the interest formula, I can now multiply across (in this case, I will multiply the three right-hand columns to get expressions for the interest in the left-hand column) and fill in the "interest" column.
I | P | r | t | |
---|---|---|---|---|
Fund X | 0.06x | x | 0.06 | 1 |
Fund Y | 0.14(50,000 − x) | 50,000 − x | 0.14 | 1 |
total | 4,500 | 50,000 | — | — |
The interest from Fund X and the interest from Fund Y will add up to $4,500. As a result, I can add down the "interest" column, set the sum of the two interest expressions equal to the total interest, and solve the resulting equation for the value of the variable:
0.06x + 0.14(50,000 − x) = 4,500
0.06x + 7,000 − 0.14x = 4,500
7,000 − 0.08x = 4,500
−0.08x = −2,500
x = 31,250
The value of x stands for the amount invested in Fund X. So the amount that is left, from the total invested, is given by 50,000 − 31,250 = 18,750. And this (amount that is left) is the amount that is invested in Fund Y.
I should put $31,250 into Fund X, and $18,750 into Fund Y.
Note that the answer did not involve nice, neat values like $10,000 or $35,000. You should understand that this means that you cannot always expect to be able to use guess-n-check to find your answers. You really do need to know how to do these exercises.
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