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Brainliest to correct!! Please help :)

Katie invested a total of $8,000, part at 3% simple interest and part at 4% simple interest. At the end of 1 year, the investments had earned $293 interest. How much was invested at each rate?

Katie invested $__ at a rate of 3% and $__ at a rate of 4%

Sagot :

Answers:  2700 and 5300 in that order

Katie invested $2700 at a rate of 3% and $5300 at a rate of 4%

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Explanation:

  • x = amount invested at a rate of 3%
  • y = amount invested at a rate of 4%

Those two amounts must add to 8000 because this is how much she invests total. So x+y = 8000 is the first equation to set up.

The formula we'll be using is the simple interest formula, which is,

i = p*r*t

where p is the amount deposited or principal, r is the interest rate, and t is the number of years. Luckily, we don't have to worry about the time value t because t = 1 so it effectively goes away. We just need to worry about p and r.

The variable p will be tied directly to the values of x and y, as these are deposit amounts. The r values are decimal forms of the percentages (eg: 3% converts to 0.03)

In short, Katie will earn

  • 0.03x dollars from the first account (3% account)
  • 0.04y dollars from the second account (4% account)

In total, she earns 0.03x+0.04y dollars combined. This is set equal to 293 as this is the stated interest she earns over the entire year.

So another equation to set up is 0.03x+0.04y = 293

This only works for simple interest. Compound interest may likely be a different story.

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To summarize so far, we found these two equations:

  • x+y = 8000
  • 0.03x+0.04y = 293

We have a number of options on how to solve this system. I'll use substitution.

Solve the first equation for y

x+y = 8000

y = 8000-x

Then we'll plug this into the other equation to solve for x like so,

0.03x+0.04y = 293

0.03x+0.04( y ) = 293

0.03x+0.04( 8000-x ) = 293

0.03x+0.04(8000)+0.04(-x) = 293

0.03x+320-0.04x = 293

-0.01x+320 = 293

-0.01x = 293-320

-0.01x = -27

x = -27/(-0.01)

x = 2700

This tells us that Katie invested $2700 at 3%

Use this x value to find y

y = 8000-x

y = 8000-2700

y = 5300

She also invested $5300 at 4%

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As a quick check,

  • x+y = 2700+5300 = 8000

so that works out

Also,

  • 0.03*x = 0.03*2700 = 81
  • 0.04*y = 0.04*5300 = 212
  • 0.03x+0.04y = 81+212 = 293

which also works out. The answers are fully confirmed.

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