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?

Question

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?

Answer 1

if the amounts are x,y,z for 6,7,8% respectively (in thousands), then we have

x+y+z = 1200
z = x+y
.06x + .07y + .08z = 88

x,y,z = 200,400,600

Answer 2

Let's assume the amount lent at 6% is x.

Therefore, the amount lent at 7% is also x, and the amount lent at 8% is 2x (since it is equal to the sum of the amounts lent at the other two rates).

The total income from the loans is the sum of the incomes at each rate:
0.06x + 0.07x + 0.08(2x) = 88000

Now, let's simplify and solve for x:
0.06x + 0.07x + 0.16x = 88000
0.29x = 88000
x = 88000 / 0.29
x ≈ 303,448.28

So, the amount lent at 6% is approximately $303,448.28.
The amount lent at 7% is also approximately $303,448.28.
And the amount lent at 8% is approximately 2 * $303,448.28 = $606,896.56.

Answer 3

To find out how much was lent at each interest rate, let's assign variables to the unknowns. Let:

- x = amount lent at 6%
- y = amount lent at 7%
- z = amount lent at 8%

From the given information, we can determine the following relationships:

1) The total amount lent is $1.2 million:
x + y + z = 1,200,000

2) The amount lent at 8% is equal to the sum of the amounts lent at the other two rates:
z = x + y

3) The bank's annual income from the loans is $88,000, which is the sum of the interest earned from each loan:

0.06x + 0.07y + 0.08z = 88,000

These three equations form a system of equations that we can solve simultaneously to find the values of x, y, and z.

First, let's simplify equation 3 by substituting z with x + y (from equation 2):

0.06x + 0.07y + 0.08(x + y) = 88,000

Next, we can simplify the equation by distributing the 0.08:

0.06x + 0.07y + 0.08x + 0.08y = 88,000

Combine like terms:

0.14x + 0.15y = 88,000

Now we have a system of two equations (from equations 1 and 3) with two unknowns:

x + y + z = 1,200,000 ---(equation 1)

0.14x + 0.15y = 88,000 ---(equation 3)

By solving this system of equations, we can find the values of x and y, and then calculate z.