An executive invests $29,000, some at 6% and some at 4% annual interest. If he receives an annual return of $1,460, how much is invested at each rate?

Question

I am not sure how to set this up.

Answer 1

Here's an approach with fewer equations (and a different way of thinking about things), but it's basically the same math as Mark's below:

All of the $29,000 has earned at least 4%.  Some has earned an additional 2%.

If all of the $29,000 had earned 4%, that would mean earnings of $1,160. 

Since the total earnings are $1,460, the amount invested at the additional 2% yielded an extra $300.  How much would you have to invest at 2% to get $300?  That's $300 / 0.02 = $15,000.

Thus $15,000 was invested at 6%, with the remaining $14,000 at 4%.

Answer 2

Let x be the money he invested at 6% interest, and y be the money he invested at 4%.

Now we know that x+y = 29,000, because that is the total amount he invested.

Let's convert the percent annual interest to decimals: 6% = 0.06 and 4% = 0.04. To do this, divide 6% by 100%. The percent signs will cancel out and we'll get 0.06.

Then we have 0.06x + 0.04y = 1,460.

Now we have a system of linear equations.

x + y = 29,000

0.06x + 0.04y = 1,460

There are a couple ways to solve this, though I think the easiest is probably by substitution.

From the top equation, subtracting x from both sides: y = 29,000 - x.

Now substitute in for y in the second equation.

0.06x + 0.04(29,000-x) = 1460

0.06x + 1,160 - 0.04x = 1460

0.02x = 300

x = 15,000

Remember earlier, we solved for y in terms of x: y = 29,000-x.

So y = 29,000 - 15,000 = 14,000.

 

Thus, $15,000 was invested at 6% annual interest, and $14,000 was invested at 4% annual interest.

Answer 3

Start looking at the problem statement and try to find things that you can describe as a formula.

In this case, you can see that you will need two answers: an amount invested at each of two interest rates. Call them a and b. They will add up to the total amount invested: $29,000.

a + b = 29000

What else do we know? One piece is invested at 6%, the other at 4%. We can calculate the interest for each as: 

I_a = a * 0.04

I_b = b * 0.06

and the total interest will add up to $1460.

0.04 * a  + 0.06 * b = 1460

We now have two independent equations and two unknowns.  We can see that we don't really need the I_a and I_b equations.  I only included them to illustrate the thought process.  You could have gone directly to that final equation.  Here's the system of equations that we have to solve:

a + b = 29000

0.04 * a + 0.06 * b = 1460

Solve one of the equations for b in terms of a and use substitution:

b = 29000 - a

0.04 * a +0.06 * (29000 - a) = 1460

0.04 * a -0.06 * a + 0.06 * 29000 = 1460

-0.02 * a + 1740 = 1460

-0.02 * a = 1460 - 1740

-0.02 * a = -280

a = 14000

Use the first equation to get b:

a + b = 29000

14000 + b = 29000

b = 15000

Check your result:

0.04*14000 + 0.06*15000 = 560 + 900 = 1460

We're done!