Interest rates math

Question

a total of $5000 is invested part at 5% simple interest and part at 6% if the total annual return from the two investments is $270 how much is invested at each rate

Answer 1

A simple interest rate formula is:

I = PRT

We can simply represent the two investments as:

I₁ = P₁(0.05)(1) → I₁ = 0.05P₁

I₂ = P₂(0.06)(1) → I₂ = 0.06P₂

We also know:

I₁ + I₂ = 270

If we add the first two equations from above, we get:

I₁ + I₂ = 0.05P₁ + 0.06P₂

this becomes:

First equation: 270 = 0.05P₁ + 0.06P₂

We also know:

Second equation: 5000 = P₁ + P₂

Now that we have our two equations, we can solve for P1 and P2.

First step is to multiply our second equation by 0.05 so:

First: 270 = 0.05P₁ + 0.06P₂

Second: 250 = 0.05P₁ + 0.05P₂

We can then subtract the second equation from the first to eliminate P1:

20 = 0.01P₂ → P₂ = 2000

Lastly, we can recall that originally 5000 = P₁ + P₂

Solving for P₁ gives us P₁ = 3000.

Overall, we initially invested $3000 for our 5% interest and $2000 for our 6% interest.

Answer 2

.05(X) + .06(5000-X) = 270

.05X-.06X = 270- 300

.01X =30

X = 30/.01 = $3,000 invested at 5%

5000-X = $2,000 invested at 6%

270/5000 = 5.4%

5 is within .4 of 5.4

6 is within .6 of 5.4

.6/.4 = 3/2

3,000/2,000 = 3/2

you immediately know you need more at 5% than 6% since 5.4% is closer to 5%. You need exactly 3/2 or 1.5 times as much. $2,000 x 1.5 = $3,000