$13,138 is invested, part at 9% and the rest at 7% If the interest earned from the amount invested at 9% exceeds the interest earned from the amount invested at 7% by $537.46, how much is invested at each rate?
| Item (Loan and %) |
Principal (Amount Invested in $) |
Rate (as a decimal) |
Time (in years) |
Total Interest (in $) |
| 7% loan |
x |
.07 |
1 (assumed) |
.07x |
| 9% loan |
$13138 - x |
.09 |
1 (assumed) |
.09($13138 – x) |
| Total |
$13138 |
------------------------ |
|
.07x + .09(13138 – x) |
We will have to assume this is a Simple Interest problem because there is nothing in the problem stating that it is compound interest. Therefore, with Simple Interest, the Total Interest = Principal * Rate * Time.
There is also nothing that states the amount of time the loan is for; therefore, you can assume the loan is for 1 year. The chart is provided only to assist one with coming up with the two equations. Since the total amount invested is $13,138, then you can call one of the amounts x, since you do not know what that is, and the other amount will be the total amount invested minus x. Therefore, if you call the Amount Invested at 7% x, then the amount invested at 9% must be $13,138 – x.
Next, you can multiple across using the formula. Therefore, the Total Interest for the 7% loan would be: x * .07 *1 = .07x. This is because 7% as a decimal is .07 because to change from a percent to a decimal, you can divide the number given by 100, or you can move the decimal point back two decimal places.
By the same formula, the Total Interest for the 9% loan would be: ($13,138 – x) * .09 * 1 = .09($13,138 – x) = $1,182. 42 - .09x.
Now, you need to concentrate separately on the Total Interest earned for each loan. In this case, make sure to use a separate variable other than x. In this case, I will use “y”. Since the interest earned from the amount invested at 9% exceeds the interest earned from the amount invested at 7%, I will call the interest earned from the 7% loan the “y.” Therefore, since the total amount of interest earned at 7% is .07x, and this is also “y”, now we know that y = .07x. Since the interest earned from the 9% loan exceeds the interest from the 7% loan by $537.45, this means that the total interest from the 9% loan is also equal to .07x + $537.45 because exceeds implies addition. Therefore, by the use of substitution, .07x + $537.45 = $1,182.42 - .09x. We can now use this equation to solve for x. Since .07x + $537.45 = $1,182.42 - .09x, we can move the x variable to the left hand side by doing the opposite to both sides of the equation. The opposite of -.09x is +.09x. Therefore, we can add .09x to both sides of the equation. Therefore, .07x + .09x + $537.45 = $1,182.42 - .09x + .09x. This simplifies to a new equation, by collecting like terms. .07x + .09x = .16x and -.09x + .09x = 0. Therefore, we now have .16x + $537.45 = $1,182.42. Now, we need to get the number on the other side of the equation from the variable term. The opposite of +$537.45 is - $537.45. Therefore, we can subtract $537.45 from both sides of the equation, and now .16x + $537.45 - $537.45 = $1,182.42 - $537.45. Since $537.45 - $537.45 = 0 and $1,182.42 - $537.45 = $644.97, we now have a new equation of .16x = $644.97. Once you are down to one term on each side of the equation, then you can divide. The opposite of multiplying x by .16 is dividing x by .16. Thus, you can now divide both sides of the equation by .16. Therefore, .16x / .16 = $644.97 /.16. This simplifies to x = $4031.06.
Therefore, since x was the amount invested at 7%, the amount invested at 7% was $4,031.06. To obtain the amount invested at 9%, simply subtract $13,138 – x, which means $13,138 - $4,031.06 = $9,106.94. Therefore, the amount invested at 9% was $9,106.94.
To check your answer, you can multiply .07 * $4,031.06, which is $282.17. You can then multiply .09 * $9,106.94, which is $819.62. Then, check to see if when you subtract the two amounts, you obtain the $537.45 that interest from the 9% loan exceeded the interest from the 7% loan by. $819.62 - $282.17 = $537.45. Therefore, your answers are correct and you have solved the problem.
