Examples of Setting Up & Solving "Investment" Word Problems
Purplemath
"Investment" word problems, using the simple-interest formula, I = Prt, pretty much all work the same one of two ways: Either the exercise is just one application of the formula, or else the investment is split in some manner, so you'll be applying the formula more than once.
If you set up your investment word problems so everything is labeled and well-organized, they should all work out fairly easily. Just take your time and do things in an orderly fashion.
In the following exercises, I've done the set-up, but not the complete solutions. (Click on the links for solution info.)
- An investment of $3,000 is made at an annual simple interest rate of 5%. How much additional money must be invested at an annual simple interest rate of 9% so that the total annual interest earned is 7.5% of the total investment?
First, pick a variable to stand for the unknown amount of additional monies to be invested; I'll use x.
Then fill in the P, r, and t columns with the given values.
| I | P | r | t | |
|---|---|---|---|---|
| first | 3,000 | 0.05 | 1 | |
| additional | x | 0.09 | 1 | |
| total | 3,000 + x | 0.075 | 1 |
Then multiply across all three rows (from the right to the left) in order to fill in the I column. In the "first" row, simplify the product, since it's just numerical (that is, there is no variable).
| I | P | r | t | |
|---|---|---|---|---|
| first | (3,000)(0.05) = 150 | 3,000 | 0.05 | 1 |
| additional | 0.09 x | x | 0.09 | 1 |
| total | (3,000 + x)(0.075) | 3,000 + x | 0.075 | 1 |
The total interest earned will be the sum of the interest from each of the two investments, so add down the I column to get the following equation:
150 + 0.09x = (3,000 + x)(0.075)
To find the solution, solve for the value of x.
"But, wait!", I hear you cry; "I thought you said that there is *never* a ‘total’ rate because you can't add rates. So why is there a 0.075 for r in the ‘total’ column here?"
Good question. But the "total" here isn't the sum of the two other rates. Instead, in this particular exercise, the 7.5% interest rate in the "total" row is actually the equivalent rate. You're figuring out how much to invest in each of the two accounts so that, once you're done with everything, you will have earned on the two accounts what would be the equivalent of 7.5% on the entire amount invested.
The probability that you will have a number to put in the "total" box of the "interest" column is low, but it is not zero.
- A total of $6,000 is invested into two simple interest accounts. The annual simple interest rate on one account is 9%; on the second account, the annual simple interest rate is 6%. How much should be invested in each account so that both accounts earn the same amount of annual interest?
You need to figure out how much should be invested in each account. Use the variable x to stand for the amount invested in the 9% account. Then there will be 6,000 − x left to invest in the 6% account. (You can use the variable to stand for the amount invested in the 6% account, and then "the rest" goes into the 9% account — there is no rule that says you have to assign the variable in the way I did. But make sure, either way, that you clearly label your variable with its definition, so you don't lose track of its meaning.)
Use the information from the exercise statement to fill in the P, r, and t columns:
| I | P | r | t | |
|---|---|---|---|---|
| 9% acct. | x | 0.09 | 1 | |
| 6% acct. | 6,000 − x | 0.06 | 1 | |
| total | — | 6,000 | — | — |
Then multiply to the left to fill in the I column.
| I | P | r | t | |
|---|---|---|---|---|
| 9% acct. | 0.09x | x | 0.09 | 1 |
| 6% acct. | (6,000 − x)(0.06) | 6,000 − x | 0.06 | 1 |
| total | — | 6,000 | — | — |
In this exercise, there is no "total" interest; the two interest payments are instead meant to be equal. There is no "total" time, because both investments are made over the same time period. And there is *never* a "total" interest rate. So, in this particular exercise, it just so happens that you don't actually need the "total" row at all. This is not unusual, and should not (on its own) cause you to doubt your work.
From the interest column, use the fact that the two yields are supposed to be equal, setting the two interest expressions equal to each other to get your equation.
Then solve for the value of x (being the amount invested at 9% interest), and back-solve (by subtracting this amount from the total of $6,000) to find the value invested in the 6% account.
Note: This exercise's set-up used that "how much is left" construction, mentioned earlier.
- An investor deposited an amount of money into a high-yield mutual fund that returns a 9% annual simple interest rate. A second deposit, $2,500 more than the first, was placed in a certificate of deposit (that is, in a cd) that returns a 5% annual simple interest rate. The total interest earned on both investments for one year was $475. How much money was deposited in the mutual fund?
The amount invested in the cd is defined in terms of the amount invested in the mutual fund, so let x stand for the amount invested in the mutual fund.
Fill in the P, r, and t columns:
| I | P | r | t | |
|---|---|---|---|---|
| mutual fund | x | 0.09 | 1 | |
| cd | x + 2,500 | 0.05 | 1 | |
| total | 475 | 2x + 2,500 | — | — |
Then multiply to the left to fill in the I column's boxes for each of the two investment instruments.
| I | P | r | t | |
|---|---|---|---|---|
| mutual fund | 0.09x | x | 0.09 | 1 |
| cd | (x + 2,500)(0.05) | x + 2,500 | 0.05 | 1 |
| total | 475 | 2x + 2,500 | — | — |
Use the fact that the two interest payments are meant to sum to the given total value, and add down the I column to get your equation.
Then solve for the value of x.
In this problem, there is no total for the "rate" or "time" columns. Hence, the dashes in those boxes.
- The manager of a mutual fund placed 30% of the fund's available cash in a 6% simple interest account, 25% in 8% corporate bonds, and the remainder in a money market fund that earns 7.5% annual simple interest. The total annual interest from the investments was $35,875. What was the total amount invested?
In this exercise, the investment amount is split into three parts: one part is the 30% of the total investment that was put into the 6% account; another part is the 25% put into the 8% bonds; the third part is whatever is left after the first two parts are invested.
| I | P | r | t | |
|---|---|---|---|---|
| 6% acct. | 0.06 | 1 | ||
| 8% bonds | 0.08 | 1 | ||
| 7.5% fund | 0.75 | 1 | ||
| total | 35,875 | — | — |
Let x stand for the total amount invested, and then create expressions for the amount invested in each of these two accounts.
total invested: x
invested at 6%: 0.3x
invested at 8%: 0.25x
Then the amount that is left to invest in the 7.5% fund is represented by subtracting the amounts already dealt with from the total, which is 100% of x:
1x − 0.30x − 0.25x = 0.45x
Use this information to fill in the "principal" column.
| I | P | r | t | |
|---|---|---|---|---|
| 6% acct. | 0.30x | 0.06 | 1 | |
| 8% bonds | 0.25x | 0.08 | 1 | |
| 7.5% fund | 0.45x | 0.75 | 1 | |
| total | 35,875 | 1x | — | — |
Then multiply to the left to fill in the "interest" column.
| I | P | r | t | |
|---|---|---|---|---|
| 6% acct. | (0.30x)(0.06) = 0.018x | 0.30x | 0.06 | 1 |
| 8% bonds | (0.25x)(0.08) = 0.02x | 0.25x | 0.08 | 1 |
| 7.5% fund | (0.45x)(0.075) | 0.45x | 0.75 | 1 |
| total | 35,875 | 1x | — | — |
The total interest is the sum of what was earned on all three acounts, so add down the "interest" column to get your equation.
Then solve for the value of x, and interpret this value in the given context.
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