Question

An actor invested some money at 6​% simple​ interest, and ​$48 comma 000 more than three times the amount at 7 %. The total annual interest earned from the investment was ​$37 comma 110. How much did he invest at each​ amount?

He invested ​$__
nothing at 6​% and ​__
nothing at 7​%

Answer 1

He invested $125,000 at 6% and $423,000 at 7%.

Explanation:

Let
`x` be the amount of money invested at 6%. Since the amount of money invested at 7% is 3 times the amount of money invested at 6% plus $48,000,
`3x+48,000` is the amount invested at 7%.

We know that both investments yield $37,110 in interest, so the sum of the money invested at 6% and the money invested at 7% is $37,110; in other words:

6%
`x`+7%
`(3x+48,000)` = 37,110

We need to convert both interest rates to decimals, so we are going to divide both rates by 100%

6%
`x`+7%
`(3x+48,000)` = 37,110

`(6)/(100) x+(7)/(100) (3x+48,000)=37,110`

`0.06x+0.07(3x+48,000)=37,110`

Now we can solve our equation, step-by-step, to find the amount of each investment.

Step 1. Distribute 0.07 to 3x and 48,000

`0.06x+0.07*3x+0.07*48,000=37,110`

`0.06x+0.21x+3,360=37,110`

Step 2. Combine like terms

`0.27x+3,360=37,110`

Step 3. Subtract 3,360 from both sides of the equation

`0.27x+3,360-3,360=37,110-3360`

`0.27x=35,750`

Step 4. Divide both sides of the equation by 0.27

`(0.27x)/(0.27x) =(33,750)/(0.27)`

`x=125,000`

We now know that he invested $125,000 at 6%. Since the amount invested at 7% is
`3x+48,000`, we just need to replace x with 125,000 to find it:

Amount invested at 7% =
`3(125,000)+48,000`

Amount invested at 7% =
`375,000+48,000`

Amount invested at 7% =
`423,000`

We can conclude that the actor invested $125,000 at 6% and $423,000 at 7%.