A bank loaned out $27,500, part of it at the rate of 5% annual interest, and the rest at 14% annual interest. The total interest earned for both loans was $3,130.00. How much was loaned at each rate?

Question

How much was loaned at 5%?

How much was loaned at 14%?

Answer 1

So let us write down what we know. There are two parts to this loan:

x, y

Step 1 - Write down the equation for overall principal amount loaned out by the bank:

The equation for overall initial principal is this:

x + y = 27500

Step 2 -

Now let us write the equation for overall interest earned by the bank. One part of the loan is at 5% and another at 14%.

The equation for overall interest will be this:

x(5%) + y(14%) = 3130

(reason - The reason we have the above as the interest equation is this. The question says one part of the loan is at 5% and another at 14%. In our equation is x is one part of the loan and we can charge it 5% by multiplying it and y is another part and we can charge it 14% by multiplying that. Remember, simple interest is basically Principal Amt X interest rate %.)

Step 3 -

We have two equations and two variables. Because of this fact, we can solve for one of the variables (y) and substitute it into the second equation. Let us take a look at what this means:

We said that in step 1:

x + y = 27500

From this, we can solve for y.

y = 27500-x

As you can see, this y is just the overall principal minus x (one part of the loan).

Step 4 -

Let us take the solved equation from step 3 and substitute it into the interest equation from step 2.

This is the equation from step 2 ---> x(5%) + y(14%) = 3130

When we substitute 27500-x for y, it will now look like this:

x(5%) + (27500-x)(14%) = 3130

Step 5 -

Now that we have an equation with just one variable, we can solve the equation and find x:

(Note, recall that % means divided by 100 so 5% and 14% are just .05 and .14 respectively.)

x(5%) + (27500-x)(14%) = 3130

x(0.05) + (0.14*27500) - (0.14*x) = 3130

Rewrite the terms so the x are next to each other and you can see what is happening:

(.05x-.14x) + (.14*27500) = 3130

-.09x +3850 = 3130

Subtract 3850 from both sides:

-.09x + 3850 - 3850 = 3130 - 3850

Now, it will become:

-.09x = -720

x = 720/0.09

x = 8000

Step 6 -

Great, now we have x. Let us solve for y:

In step 1, made this equation:

x + y = 27500

We know what x is, so let us substitute it in:

8000 + y = 27500

Subtract 8000 from both sides:

y+8000-8000=27500-8000

y = 19500

So, x is 8000 and y is 19500. When we made these equations, we said x was at 5% and y was at 14%.

Therefore, the answer to the question is this:

8000 dollars was loaned at 5% and 19500 was loaned at 14%.

Answer 2

Assume simple interest since there was no mention of compound interest.

Also assume the time period was one year.

Use i = Prt

Let x be the amount invested at 5% interest and 27500-x be the amount invested at 14%.

0.05x + 0.14(27500-x) = 3130.00

0.05x+3850-0.14x=3150

0.09x=700

x = 700/0.09 = $7777.78

27500-7777.78 = $19722.22

$7777.78 was invested at 5% interest

$19722.52 was invested at 14% interest