A bank lent $1.2 million for the development of three new products, with one loan each at 6%, 7%, and 8%. The amount lent at 8% was equal to the sum of the amounts lent at the other two rates, and the bank's annual income from the loans was $88,000. How much was lent to each rate?
To solve this problem, we can use a system of equations.
Let's denote the amount lent at 6% as x, the amount lent at 7% as y, and the amount lent at 8% as z.
We are given three pieces of information:
1. The amount lent at 8% was equal to the sum of the amounts lent at the other two rates: z = x + y.
2. The bank's annual income from the loans was $88,000. We can calculate the income from each loan as follows:
- Income from the loan at 6% is 0.06x.
- Income from the loan at 7% is 0.07y.
- Income from the loan at 8% is 0.08z.
So, we have the equation: 0.06x + 0.07y + 0.08z = $88,000.
Now, let's solve this system of equations.
From the first equation, we can rewrite z in terms of x and y: z = x + y.
Substituting this into the second equation, we get: 0.06x + 0.07y + 0.08(x + y) = $88,000.
Expanding and simplifying, we have: 0.06x + 0.07y + 0.08x + 0.08y = $88,000.
Combining like terms, we get: 0.14x + 0.15y = $88,000.
Dividing both sides by 0.01, we have: 14x + 15y = $8,800,000. (equation A)
Now, using the first equation (z = x + y), we can substitute this into equation A:
14x + 15y = $8,800,000 becomes 14x + 15(x + y) = $8,800,000.
Expanding and simplifying, we get: 14x + 15x + 15y = $8,800,000.
Combining like terms, we have: 29x + 15y = $8,800,000.
Now, we have a system of two linear equations:
Equation A: 14x + 15y = $8,800,000.
Equation B: 29x + 15y = $8,800,000.
We can solve this system of equations using substitution or elimination. I'll use elimination.
Subtracting equation A from equation B, we eliminate the y term:
(29x + 15y) - (14x + 15y) = $8,800,000 - $8,800,000.
This simplifies to: 15x = 0.
Dividing by 15, we find: x = 0.
Now, substituting this value of x back into equation A, we get:
14(0) + 15y = $8,800,000.
This simplifies to: 15y = $8,800,000.
Dividing by 15, we find: y = $586,666.67.
Now, we can substitute the values of x and y into the equation z = x + y:
z = 0 + $586,666.67.
Therefore, z = $586,666.67.
To summarize, the amount lent at 6% is $0, the amount lent at 7% is $586,666.67, and the amount lent at 8% is also $586,666.67.