amount of money loaned at 9.4%.

Question

A bank loaned out $9900, part of it at a rate of 9.4% per year and the rest of it at a rate of 7.2% per year. The total amount of interest owed to the bank at the end of one year was $863.3. Find the amount of money that the bank loaned out at 9.4%.

Answer 1

Hello Ansaf, let's see if we can solve this.

As I read it, this is a Simple Interest problem. This means that we will use the formula for Simple Interest which is as follows:

I=P*r*t

I = Interest ($)

P = Initial Value, or the Principal value. ($)

r = The rate at which you're building interest. This value is the decimal form of the percentage.

t = The time you spend building interest in years.

In this problem, you can get the solution by treating your problem as a summation of two different Simple Interest formulas.

I_T=P₁*r₁*t+P₂*r₂*t

Note: Since both interests are occurring over the same time interval, we only need 1 time variable.

Now let's look at the problem.

We can see one set of interest has a rate of 9.4%, or 0.094. The other has a rate of 7.2%, or 0.072. Let's use these values as our r₁ and r₂ respectively. We can see this happens over the course of only 1 year so t can be 1. We also know that the total interest at the end of the year is $863.30 so let's make this our total interest value, I_T.

To recap:

r₁=0.094

r₂=0.072

t=1

I_T=863.30

This takes care of everything but two variables. The general rule is 1 equation per 1 unknown. This means we need a formula to relate the two initial amounts. Since we know the bank loaned a total amount of $9900, we can say that both initial amounts will sum to this number. Thus: P₁+P₂=9900 Since we are solving for the initial amount that gets lent at the rate of 9.4%(P₁), let's rewrite this formula as this so we don't have to solve for it later:

P₂=9900-P₁

Now that we have collected our variables, we can plug into the original formula we made.

I_T=P₁*r₁*t+P₂*r₂*t

863.30=P₁*0.094*1+(9900-P₁)*0.072*1

Solving this we get:

863.30=P₁*0.094+(9900-P₁)*0.072

863.30=0.094P₁+712.8-0.072P₁

863.30=0.022P₁+712.8

150.51=0.022P₁

Being mindful to get our units, we get the answer:

P₁=$75250/11, or approximately $6840.91