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Mr. Jones has $22,000 to invest. He invests part at 5% and the rest at 9%. If he earns $1,540 in interest after 1 year, how much did he invest at each rate?

Respuesta :

frika

Answer:

$11,000 at 5%

$11,000 at 9%

Step-by-step explanation:

Mr. Jones has $22,000 to invest. Let $x be the amount he invested at 5% and then $(22,000-x) he invested at 9%.

Use formula:

[tex]I=P\cdot r\cdot t,[/tex]

where

I = interest

P = principal

r = rate (as decimal)

t = time

For 5%:

[tex]I_1=x\cdot 0.05\cdot 1\\ \\I_1=0.05x[/tex]

For 9%:

[tex]I_2=(22,000-x)\cdot 0.09\cdot 1\\ \\I_2=0.09(22,000-x)[/tex]

Since he earns $1,540 in interest, then

[tex]I_1+I_2=1,540[/tex]

So,

[tex]0.05x+0.09(22,000-x)=1,540\\ \\5x+9(22,000-x)=154,000\\ \\5x+198,000-9x=154,000\\ \\5x-9x=154,000-198,000\\ \\-4x=-44,000\\ \\4x=44,000\\ \\x=11,000\\ \\22,000-x=22,000-11,000=11,000[/tex]

TheBlueFox

Interest Formula ⇒ Amount Invested · Rate = Interest Earned

I would start this problem by setting up a chart based on

the formula Amount Invested · Rate = Interest Earned.

The chart is attached below.

Down the left side of the chart, we have the the 2

different types of investments that Mr. Jones makes,

the 5% investment and the 9% investment.

For the amount invested column, we know that Mr. Jones invested

$22,000, but we don't know how much he invested at each rate.

So we use x and 22,000 - x to represent

the amount he invested at each rate.

For the rate column, we translate our

two percentages to decimal form.

So 5% becomes .05 and 9% becomes .09.

Based on our formula, we can multiply the first two columns

in our chart together to fill out the third column.

So we have x(.05) which is .05x and

22,000 - x(.09) or .09(22,000 - x).

Reading through the last part of the problem, we know that Mr. Jones

earned a total of $1,540 in interest for the year.

So if we add up the two amounts of interest that he

earned at each rate, we will get a total of 1540.

So we can setup an equation that reads

.05x + .09(22,000 - x) = 1540.

Solving from here, we first get rid of the decimals by multiplying each side of the equation by 100 to get 5x + 9(22,000 - x) = 154,000.

Now distribute the 9 through the parentheses.

This gives us 5x + 198,000 - 9x = 154,000.

We can simplify on the left side

to get -4x + 198,000 = 154,000.

Now subtract 198,000 from

both sides  to get -4x = -44,000.

Then divide both sides by -4 to find that x = 11,000.

So we know that Mr.Jones invested 11,000 at 5% interest and the amount he invested at 9% is represented by 22,000 - x which is 22,000 - 11,000 or $11,000.

So Mr. Jones invested $11,000 at 5%

and he invested $11,000 at 9%

Ver imagen TheBlueFox

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