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Jo invested $63,000. A portion earning a simple interest rate of 5% per year and the rest earning a rate of 7% per year. After one year the total interest

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2 Answers

In order to solve this problem, one must construct a system of equations. Let x represent the amount of money invested at 7% and y the amount invested at 5%. In order to find the amount of interest, we construct this equation:

3,890 = .07x + .05y

Remember that 7% is .07 when expressed as a decimal (and similarly for 5%). 3,890 is the amount of interest earned in one year. .07x is 7% of x and .05y is 5% of y, which are the amounts of interest each investment earned in one year. We add them together to get the total interest earned.

Now we need to express what x and y are in a separate equation. We know that x + y must be $63,000 because x and y are parts of the total investment, which was $63,000. That equation looks like this:

63,000 = x + y

From here, we go about solving our system of equations. First, let's find a value for x:

63,000 = x + y

63,000 - y = x

Now that we have a value for x, we can substitute it into the other equation and solve for y:

3,890 = .07x + .05y

3,890 = .07(63,000 - y) + .05y

3,890 = (4,410 - .07y) + .05y

3,890 = 4,410 - .02y

-520 = -.02y

26,000 = y

At this point we know that $26,000 was invested at 5%. If we take 5% of $26,000 we get $1,390 in interest for this portion of the total investment. Since $1,390 is less than the total interest earned of $3,890 we can move on relatively confident that we are on the right track.

The final step to solving this problem is to follow the same pattern, but to solve for y in the first step and substitute it in to find the value of x.

Solve for y:

63,000 = x + y

63,000 - x = y

Now substitute that in and find the value of x:

3,890 = .07x + .05(63,000 - x)

3,890 = .07x + (3,150 - .05x)

3,890 = .02x + 3,150

740 = .02x

37,000 = x

We find that $37,000 was invested at 7%. This investment yields $2,590 in interest. Remember, the problem isn't done until the answer is checked. For a system of equations, one must check both equations.

First the equation of investments:

63,000 = x + y

63,000 = 37,000 + 23,000

63,000 = 63,000

Then the equation of interest:

3,890 = .07x + .05y

3,890 = .07(37,000) + .05(26,000)

3,890 = 2,590 + 1,300

3,890 = 3,890

Both equations check out and we are finished!

Comments

You're very welcome!

Comment

Hi Jamey,

This problem can be solved by using a system of equations, or by writing one equation with each unknown value written in terms of x. I will explain the latter.

We are trying to determine how much money Jo invested at each rate. Let's use x to represent the amount the invested at 5%. To avoid using a second variable to represent the value invested at 7%, we can represent it in terms of x. In other words, we know this value is the remainder of the $63,000 after a portion has been invested at 5%. We can express this value as 63,000 - x. 

Because x (amount invested at 5%) + 6300 -x (amount invested at 7%) = $63,000

Now that we have a way to express both unknowns, we can write the equation. 

The annual interest earned from each investment is the product of the principal (initial amount) and the interest rate, 5% and 7%, respectively. Let's represent the percents as decimals. Percent means "per 100," so 5% is 5 per 100, or 5/100. Five hundredths is 0.05. 7% is 7 per 100, or 7/100. Seven hundredths is 0.07.

Our equation can be written as (0.05)(x) + (0.07)(63,000 - x) = 3890

Translated, five percent of our first investment plus seven percent of our second investment yields $3,890 in interest. 

Let's simplify this equation. 0.05 multiplied by x is 0.05x. 

0.07 needs to be distributed (multiplied) by each of the terms inside the parentheses. That results in 4,410 and 0.07x.

Our equation now reads .05x + 4410 - .07x = 3890

We can combine the like terms of .05x and -.07x. They are like terms because they have the same variable and the same degree (exponent). We simply need to add the coefficients: .05 - .07 = -0.02

Our equation simplified reads -0.02x + 4410 = 3890

To solve for x, we need to isolate the variable. This is done by using inverse operations to "undo" what has happened to the x. 

X is being multiplied by -0.02 and added by 4410. 

To "undo" adding 4410, we need to subtract 4410 for both sides. We are subtracting from both sides because it is an equality. If we don't perform the operations on both sides, then we are not treating each side equally and it ceases to be an equality. 

-0.02x + 4410 - 4410 = 3890 - 4410  = 

-0.02x = -520

X is also being multiplied by -0.02. To "undo" the multiplication we need to divide, or multiply by its multiplicative inverse (better known as a reciprocal)

-0.02 written as a fraction is -2/100. It's reciprocal is 100/-2. Simplified is -50. So to "undo" multiplying by -0.02, we can simply multiply by 50. This has to be done on both sides. 

(-2/100)(-100/2)(x) = 520 (100/-2) = 

x = 26,000.

Therefore $26,000 is the amount invested at 5%

To find the amount invested at 7%, we simply take the difference from the initial $63,000. $63,000 - $26000 = $37,000. So, $37,000 was invested at 7%

To check your work, evaluate the equation with these values: Find 5% of $26,000 and 7% of $63,0000. The sum should be $3,890.

26,000 * 0.05 = 1300

37,000 * 0.07 = 2590

1300 + 2590 = 3,890

I hope this helped you, Jamey!

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