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How Do You Solve a More Complicated Interest Problem?

Elana invested a total of $6400. Part of her investment was at 6% interest, and part of her investment was at 8.5% interest. If she made a total of $1035 in interest over the last 30 months, how much money did she invest at each rate?

Summary

  1. m represents the amount of money invested at 6% interest
  2. n represents the amount of money invested at 8.5% interest
  3. We have two equations with two unknowns, so we can solve for m and n
  4. We have m in terms of n, so we can plug it into the second equation to find n
  5. Simplify to find that n=1200
  6. Plug 1200 in for n in the first equation to find that m=5200

Notes

    1. m will be one of our variables
    2. m will represent the amount of money Elana invested at a 6% interest rate
    3. n will be our other variable
    4. n will represent the amount of money Elana invested at an 8.5% interest rate
    1. Remember, our variables m and n were both parts of her investment
    2. So if we add them together, we'll get the total amount she invested, $6400
    1. m represents the amount of money Elana invested at a 6% interest rate
    2. n represents the amount of money Elana invested at an 8.5% interest rate
    1. m represents the amount of money Elana invested at a 6% interest rate
    2. n represents the amount of money Elana invested at an 8.5% interest rate
    3. Add m and n to get Elana's total investment, which we know is $6400
    1. The formula for simple interest is I=prt, where:
    2. I is the amount of interest earned
    3. p is the amount of money invested
    4. r is the interest rate
    5. t is time in years
    1. The formula we have is the formula for interest when there is only one interest rate
    2. But we can still use it, we just have to adapt it a little to fit our problem
    1. ITOT represents the total amount of interest earned
    1. We need to find the amount of interest she earned at each rate separately, then add them together
    2. So one part of our formula will be the interest earned at a 6% rate
    3. And we'll add that to the interest earned at an 8.5% rate
    4. This will give us the total amount of interest earned
    1. pr1t will be the amount of interest earned at a 6% rate
    2. p is the amount she invested at 6%
    3. r1 is the first interest rate, 6%
    4. t, time, will be the same for both parts, because she invested them both at the same time
    1. pr2t will be the amount of interest earned at an 8.5% rate
    2. p is the amount she invested at 8.5%
    3. r2 is the second interest rate, 8.5%
    4. t, time, will be the same for both parts, because she invested them both at the same time
    1. This will be the adapted formula we can use to solve our problem
    2. Notice how it is similar to the formula for simple interest, but now we're adding two things on the right hand side
    3. We're adding the amounts of interest earned at each separate interest rate
    1. ITOT is 1035, the total interest earned
    1. Plug 1035 in for ITOT
    1. Remember, m represents the amount of money Elana invested at a 6% interest rate
    2. So p, the amount of money invested in the first part, will be equal to m
    1. Remember, m represents the amount of money Elana invested at a 6% interest rate
    2. So p, the amount of money invested in the first part, will be equal to m
    3. r1 is the first interest rate, 6%
    4. But we can't multiply with percents, so we need to convert to a decimal
    5. Drop the percent sign, move the decimal point two places to the left, and we get 0.06
    1. Remember, t represents time in years
    2. Since our problem gives us time in months, we'll need to convert it to years so we can use it in our formula
    1. Remember, t represents time in years
    2. Since our problem gives us time in months, we'll need to convert it to years so we can use it in our formula
    1. To convert months to years:
    2. Divide the number of months, 30, by the number of months in a year, 12
    3. 30 divided by 12 is 2.5
    4. So t will be 2.5 years
    1. So the interest earned at a 6% rate can be represented by pr1t
    2. p=m, our variable that represents the amount of money earned at 6%
    3. r1=0.06, which is 6%, the interest rate, converted to a decimal
    4. t=2.5, the number of years the investment earned interest
    1. Remember, n represents the amount of money Elana invested at an 8.5% interest rate
    2. So p, the amount of money invested in the second part, will be equal to n
    1. The interest earned at an 8.5% rate can be represented by pr2t
    2. p=n, our variable that represents the amount of money earned at 8.5%
    3. r2=0.085, which is 8.5%, the interest rate, converted to a decimal
    4. t=2.5, the number of years the investment earned interest
    1. 1035 was the total amount of interest earned
    1. Simplify by multiplying the numbers we have
    1. Now the right hand side just has the two variables we defined at the beginning
    2. m represents the amount of money Elana invested at a 6% interest rate
    3. n represents the amount of money Elana invested at an 8.5% interest rate
    1. Our unknowns are the variables m and n
    2. m represents the amount of money Elana invested at a 6% interest rate
    3. n represents the amount of money Elana invested at an 8.5% interest rate
    4. Our first equation is m+n=6400
    5. Our second equation is 1035=m(0.15)+n(0.2125)
    1. When we have two equations with two unknowns, we call that a system of linear equations
    1. m is the first variable we will solve for
    2. m represents the amount of money Elana invested at a 6% interest rate
    1. m represents the amount of money Elana invested at a 6% interest rate
    2. n represents the amount of money Elana invested at an 8.5% interest rate
    3. We want to get m by itself on one side
    4. Since subtraction is the opposite of addition, we can subtract n from both sides to move it to the right hand side
    1. m represents the amount of money Elana invested at a 6% interest rate
    2. n represents the amount of money Elana invested at an 8.5% interest rate
    3. Now we have m by itself on the left hand side and all the n terms on the right hand side
    4. So we have m in terms of n
    1. m represents the amount of money Elana invested at a 6% interest rate
    2. n represents the amount of money Elana invested at an 8.5% interest rate
    3. Since we found that m=6400-n, we can substitute this value for n in our second equation
    1. m represents the amount of money Elana invested at a 6% interest rate
    2. n represents the amount of money Elana invested at an 8.5% interest rate
    3. Since we found that m=6400-n, we can substitute this value for n in our second equation
    1. m represents the amount of money Elana invested at a 6% interest rate
    2. n represents the amount of money Elana invested at an 8.5% interest rate
    3. Since we found that m=6400-n, we can substitute this value for n in our second equation
    1. m represents the amount of money Elana invested at a 6% interest rate
    2. n represents the amount of money Elana invested at an 8.5% interest rate
    3. Since we found that m=6400-n, we can substitute this value for n in our second equation
    4. So instead of m(0.15), we will have (6400-n)(0.15)
    1. n represents the amount of money Elana invested at an 8.5% interest rate
    2. Multiply 0.15 by each term in the parentheses
    1. Like terms are terms that have the same variables raised to the same power
    2. In our problem, we have two terms that have n raised to the first power, -0.15n and 0.2125n
    3. We can combine these to simplify our equation
    1. Take -0.15n+0.2125n
    2. Add the coefficients and keep the variable the same
    1. n represents the amount of money Elana invested at an 8.5% interest rate
    2. Now our second equation is fully simplified, so we can solve
    1. n represents the amount of money Elana invested at an 8.5% interest rate
    1. We want to get n by itself on one side
    2. Subtraction is the opposite of addition
    3. So we can get rid of the 960 on the right hand side by subtracting it from both sides
    4. 1035-960 gives us 75 on the left hand side
    5. And the 960's cancel on the right hand side to leave us with 0.0624n
    1. We want to get n by itself on one side
    2. Division is the opposite of multiplication
    3. So we can get rid of the 0.0624 on the right hand side by dividing it from both sides
    4. 75/0.0624 gives us 1200 on the left hand side
    5. The 0.0624's cancel on the right hand side to leave us with n
    1. n represents the amount of money Elana invested at an 8.5% interest rate
    2. We have solved for n!
    3. n equals $1200, the amount Elana originally invested at an 8.5% rate
    1. But we're not done yet
    2. We still need to find m, the amount she invested at a 6% rate
    3. Since now we have a value for n, 1200, we can plug it into our first equation to find m
    1. m represents the amount of money Elana invested at a 6% interest rate
    2. We can plug the value we found for n, 1200, into our first equation to solve for m
    1. m represents the amount of money Elana invested at a 6% interest rate
    2. We got 6400-n when we solved for m in terms of n
    1. We know that n=1200
    2. So we can plug it into m=6400-n to find m
    1. Plugging 1200 in for n gives us m=6400-1200
    2. 6400-1200=5200
    3. So m=5200, the amount of money Elana invested at a 6% rate