Question

A bank lent $900,000 for the development of two new products with one loan at 5% and the other at 6%. The amount lent at 6% is $300,000 more than the amount lent at 5%. What is the annual income from these loans?

Answer 1

Let's denote the amount lent at 5% as \( x \). According to the problem, the amount lent at 6% is \( x + 300,000 \).

We know that the total amount lent is $900,000, so we can set up the following equation:

\[
x + (x + 300,000) = 900,000
\]

Simplifying this equation:

\[
2x + 300,000 = 900,000
\]

Subtracting 300,000 from both sides gives:

\[
2x = 600,000
\]

Dividing both sides by 2:

\[
x = 300,000
\]

This means the amount lent at 5% is $300,000. Now, we can find the amount lent at 6%:

\[
x + 300,000 = 300,000 + 300,000 = 600,000
\]

So, the amounts lent are:
- At 5%: $300,000
- At 6%: $600,000

Now, we calculate the annual income from these loans:

1. Income from the 5% loan:

\[
\text{Income from 5% loan} = 0.05 \times 300,000 = 15,000
\]

2. Income from the 6% loan:

\[
\text{Income from 6% loan} = 0.06 \times 600,000 = 36,000
\]

Now, adding both incomes together gives the total annual income:

\[
\text{Total Annual Income} = 15,000 + 36,000 = 51,000
\]

Therefore, the annual income from these loans is \(\boxed{51,000}\).