# You invested $4000, part at 5% and the remainder at 9% annual interest. At the end of the year, the total interest from these investments was $311. How much was invested at each rate?

##### 2 Answers

#### Explanation:

Let the part invested at

So we can write

or

Multiplying both sides of

We get

Subtracting

We get

or

or

or

So plugging the value

we get

or

or

Improved my method by cutting out s step.

$2775 was invested at 9%

$1225 was invested at 5%

#### Explanation:

Suppose all the money was invested at 5% then the income would be

Suppose all the money was invested at 9% then the income would be

Consider this transition of total interest received by varying the amount deposited in each account.

This can be modelled by modelling just one account. If all of the money is in the 9% account then none is in the 5% account. If all the money is in the 5% account then there is none in the 9% account. So one account infers directly how much is in the other as the funds available is fixed at $4000

The result is a straight line graph where the gradient is the change in interest depending on how much is in each account.

The equation of this graph will be:

We are told that the target interest is $311.

Set

Dropping the $ sign for now

Subtract 200 from both sides

Multiply both sides by 25

Thus the amount of the principle sum in the 5% account is:

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