A woman has money in two accounts. One account pays 7% annual interest, whereas the other pays 15% annual interest. If she has $1,500 more invested at 15% than she does at 7% and her total interest for a year is $1,545, how much does she have in each account?
Please show the steps!
simply solve ...
.07x + .15(x+1500) = 1545
where x is the amount invested at 7%
Let's assume the amount invested at 7% is x dollars. Therefore, the amount invested at 15% will be (x + $1,500) dollars.
The interest earned on the amount invested at 7% is given by 0.07x dollars.
The interest earned on the amount invested at 15% is given by 0.15(x + $1,500) dollars.
According to the problem, the total interest earned for a year is $1,545. Hence, we can write the equation:
0.07x + 0.15(x + $1,500) = $1,545
Now, let's solve the equation step by step:
Step 1: Distribute 0.15 to the terms inside the parentheses:
0.07x + 0.15x + 0.15($1,500) = $1,545
Step 2: Simplify:
0.07x + 0.15x + $225 = $1,545
Step 3: Combine like terms:
0.22x + $225 = $1,545
Step 4: Subtract $225 from both sides of the equation:
0.22x = $1,545 - $225
0.22x = $1,320
Step 5: Divide both sides of the equation by 0.22 to isolate x:
x = $1,320 / 0.22
x ≈ $6,000
So the amount invested at 7% is approximately $6,000.
And the amount invested at 15% is x + $1,500 = $6,000 + $1,500 = $7,500.
Therefore, the woman has $6,000 invested at 7% and $7,500 invested at 15%.
To solve this problem, let's break it down into smaller steps.
Step 1: Define variables
Let x be the amount invested at 7% interest.
Then, the amount invested at 15% interest would be (x + $1,500), as it is $1,500 more than the other account.
Step 2: Calculate the interest earned
The interest earned from the account at 7% interest is 0.07x.
The interest earned from the account at 15% interest is 0.15(x + $1,500).
Step 3: Set up the equation
The total interest earned is the sum of the interest from both accounts, which is $1,545.
So, we can write the equation as:
0.07x + 0.15(x + $1,500) = $1,545.
Step 4: Solve the equation
First, distribute 0.15 to (x + $1,500):
0.07x + 0.15x + 0.15($1,500) = $1,545.
Simplify the equation:
0.07x + 0.15x + $225 = $1,545.
Combine like terms:
0.22x + $225 = $1,545.
Next, isolate x on one side of the equation:
0.22x = $1,545 - $225.
Simplify the right side:
0.22x = $1,320.
Now, solve for x:
x = $1,320 / 0.22.
x ≈ $6,000.
Step 5: Find the amounts in each account
The amount invested at 7% interest is $6,000.
The amount invested at 15% interest is (x + $1,500) ≈ ($6,000 + $1,500) ≈ $7,500.
Therefore, the woman has $6,000 in the account with 7% interest and $7,500 in the account with 15% interest.