Althea invested $10,000 for one year, part at 7% and part at 6 1/4%. If she earned a total interest of
$656.50, how much was invested at each rate?
x = amount at 7%
y = amount at 6.25%
x + y = 10,000 or x = 10,000 - y
.07x + .0625y = 656.50
You can solve by substitution:
.07(10,000 - y) +.0625y = 656.50
Let's assume Althea invested x dollars at 7% and y dollars at 6 1/4%.
The interest earned from the investment at 7% is given by:
0.07x
The interest earned from the investment at 6 1/4% is given by:
0.0625y
According to the information given, the total interest earned is $656.50. So, we have the equation:
0.07x + 0.0625y = 656.50
We also know that the total amount invested is $10,000. So, we have the equation:
x + y = 10,000
We can use these two equations to solve for x and y. Let's multiply the second equation by 0.07 to eliminate x:
0.07x + 0.07y = 700
Now, subtract this equation from the first equation:
0.07x + 0.0625y - (0.07x + 0.07y) = 656.50 - 700
Simplifying the equation:
0.07x - 0.07x + 0.0625y - 0.07y = -43.50
Combining like terms:
-0.0075y = -43.50
Dividing both sides by -0.0075:
y = (-43.50)/(-0.0075)
y ≈ 5800
Substituting the value of y back into the second equation:
x + 5800 = 10,000
Simplifying the equation:
x = 10,000 - 5800
x ≈ 4200
Therefore, Althea invested approximately $4,200 at 7% and $5,800 at 6 1/4%.
To find out how much Althea invested at each rate, let's set up equations based on the given information.
Let x be the amount invested at 7% and y be the amount invested at 6 1/4%.
We know that Althea invested a total of $10,000, so we have:
x + y = 10000
The interest earned on the amount invested at 7% can be calculated using the formula: Interest = Principal * Rate * Time.
In this case, the interest earned on the amount invested at 7% is 7% of x.
The interest earned on the amount invested at 6 1/4% can be calculated using the same formula. However, it's important to convert the percentage to decimal form. 6 1/4% is equal to 6.25%. So, the interest earned on the amount invested at 6 1/4% is 6.25% of y.
The total interest earned is given as $656.50, so we have:
0.07x + 0.0625y = 656.50
Now, we can solve this system of equations to find the values of x and y.
Method 1: Substitution
1. Solve the first equation for x: x = 10000 - y.
2. Substitute this expression for x in the second equation:
0.07(10000 - y) + 0.0625y = 656.50
Multiply and simplify:
700 - 0.07y + 0.0625y = 656.50
-0.0075y = -43.50
Divide both sides by -0.0075:
y = 5800.
3. Substitute the value of y into the first equation to solve for x:
x + 5800 = 10000
x = 4200.
Therefore, Althea invested $4200 at 7% and $5800 at 6 1/4%.
Method 2: Elimination
1. Multiply the first equation by 0.07 to eliminate x in the second equation:
0.07(x + y) = 0.07(10000)
0.07x + 0.07y = 700
2. Multiply the second equation by 100 to eliminate decimals:
7x + 6.25y = 65650
3. Subtract the equation obtained in step 1 from the equation obtained in step 2:
7x + 6.25y - 0.07x - 0.07y = 65650 - 700
6.93x + 6.18y = 64950
4. Solve this new equation:
6.93x + 6.18y = 64950
Multiply the equation by 20 to eliminate decimals:
139x + 123.6y = 1299000
5. Multiply the first equation by -139 to eliminate x in the second equation:
-139x - 139y = -1390000
6. Add the two equations together:
-139x - 139y + 139x + 123.6y = -1390000 + 1299000
-15.4y = -91000
Divide both sides by -15.4:
y ≈ 5912.34.
7. Substitute the value of y into the first equation to solve for x:
x + 5912.34 = 10000
x ≈ 4087.66.
Therefore, Althea invested approximately $4087.66 at 7% and approximately $5912.34 at 6 1/4%.