First, some definitions:
Principal – money that is invested in an account
Interest – money that is paid out for investing principal
Simple interest – interest that is calculated using the formula Interest=(Principal)× (Rate)× (Time). This formula often is abbreviated I=PRT. If the time is equal to one year, the formula becomes I=PR.
In most interest problems, we will be trying to find the
amount of money (principal) invested in each of two bank accounts. To help
us find these things, we will be told the total amount of money invested
and the total interest paid on the two accounts.
Let’s look at an example of a typical interest word problem.
Sam invests $6,000 in two bank accounts. One of the accounts pays 8% interest per year, and the other account pays 10% interest per year. If the total interest earned on the investments is $560 after one year, how much money was invested in each account?
What are we supposed to find in this problem? The last
part of the problem gives us the answer to this question. We are trying
to find the amount of money that was invested in each account.
Let’s define a variable. We don’t know how much money was invested in either account, so we’ll make a decision to use an x to represent the amount of money invested in the first account (the one that earns 8% per year). What do we call the amount of money invested in the second account? We could give it a different variable name, like y, but in this chapter we’re going to limit our problems to a single variable.
So picture this: Sam carries a stack of $6,000 to the first bank and says, "I’d like to deposit x dollars in an account that pays 8% interest per year." The teller then takes away the x dollars from his $6,000, and Sam deposits the remaining money in other bank. Remember what the words "take away" meant to you when you were an elementary-school math student? Subtract, right?? So "take away x dollars from $6,000" can be written symbolically as 6,000-x. That’s how much money is invested in the second bank.
In interest problems, it is often useful to use a chart
to organize the information in the problem. We’ll label the top of the
chart with the two things we’re looking for (the money invested at 8% and
the money invested at 10%), and we’ll label the side of the chart with
the two pieces of information that the problem provides (the principal
and the interest earned). Since the problem gives us information about
total principal and total interest, we’ll also include a column for totals.
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So far, we know the following:
6,000 – x dollars were invested at 10%
the total principal invested was $6,000
the total interest earned was $560
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To fill in the interest boxes, remember that in one year’s
time I=PR. So in each interest box, we will multiply the amount of principal
invested in that account by the interest rate earned on that money. Now
our chart looks like this:
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.08x + .10(6,000 – x) = 560
To solve the equation, we’ll first multiply both sides of the equation by 100 to clear the decimals away. Remember, multiplying by 100 will shift one decimal point in each term to the right by two places.
100[.08x + .10(6,000 – x)] = 100[560]
8x + 10(6,000 – x) = 56,000
8x + 60,000 – 10x = 56,000
-2x + 60,000 = 56,000
-2x = -4,000
x = 2,000
Since x represents the amount of money invested in the
first account, we can say that $2,000 was invested at 8%.
We need to finish the problem. We were asked how much money was invested in each account, and since we don’t yet know how much money was invested at 10% we only have half of our answer. According to our chart, the amount of money invested in the second account is equal to 6,000- x, so
6,000 – x = 6,000 – 2,000 = 4,000
$4,000 was invested at 10%
Before we leave this problem, let’s check our answer with
the words of the problem:
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| $6,000 was invested in all | 2,000 + 4,000 = 6,000 |
| $560 was earned in interest | Interest earned at 8%:
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Our answers check. We’re done!