after winning $280,000 in the lottery, maurika deided to place the money in three different investments : a certificate of deposit paying 4% , a money market certificate paying 5% , and some Aa bonds paying 7%. After 1 year she earned $15,400 in interest. find how much was invested at each rate if $20,000 more was invested at 7% than at 5%
Well, well, Maurika struck it rich in the lottery! Let's figure this out.
Let's say Maurika invested x dollars at 4%, y dollars at 5%, and y + 20000 dollars at 7%.
We're told that after 1 year, she earned $15400 in interest.
The interest earned from the certificate of deposit at 4% would be x * 0.04.
The interest earned from the money market certificate at 5% would be y * 0.05.
The interest earned from the Aa bonds at 7% would be (y + 20000) * 0.07.
Now we can set up an equation:
x * 0.04 + y * 0.05 + (y + 20000) * 0.07 = 15400
Simplifying, we have:
0.04x + 0.05y + 0.07y + 1400 = 15400
Combining like terms, we get:
0.04x + 0.12y = 14000
But we also know that $20000 more was invested at 7% than at 5%, so we have:
y + 20000 = y
Simplifying, we have:
20000 = y
Substituting this value back into the equation, we get:
0.04x + 0.12(20000) = 14000
Simplifying further, we have:
0.04x + 2400 = 14000
Subtracting 2400 from both sides, we get:
0.04x = 11600
Dividing both sides by 0.04, we find:
x = 290000
So, Maurika invested $290,000 at 4%, $20,000 at 5%, and $220,000 at 7%.
That's a lot of money to juggle, but hey, it's a good problem to have!
Let's assume the amount invested at 5% is x.
According to the given information, the amount invested at 7% is $20,000 more than the amount invested at 5%, which means the amount invested at 7% is x + $20,000.
The total amount invested is the sum of the amounts invested at each rate, so it is given by x + (x + $20,000) + $280,000 (the principle amount).
To calculate the interest earned, we need to multiply the amount invested at each rate by the respective interest rate and then add them together:
(4% of x) + (5% of (x + $20,000)) + (7% of $280,000) = $15,400
Now let's solve the equation step by step:
0.04x + 0.05(x + $20,000) + 0.07($280,000) = $15,400
0.04x + 0.05x + 0.05($20,000) + 0.07($280,000) = $15,400
0.04x + 0.05x + $1,000 + $19,600 + $19,600 = $15,400
0.09x + $40,200 = $15,400
0.09x = $15,400 - $40,200
0.09x = -$24,800
x = -$24,800 / 0.09
x ≈ -$275,555.56
Since the investment amount cannot be negative, there seems to be an error in the given information or calculation. Please double-check the numbers provided.
To find the amount invested at each rate, let's use algebraic equations to represent the given information.
Let's assume that Maurika invested x dollars at 4%, y dollars at 5%, and z dollars at 7%.
According to the information given, she earned $15,400 in interest in one year. We can use this information to set up the equation:
0.04x + 0.05y + 0.07z = 15,400 ----- (equation 1)
Additionally, it is stated that $20,000 more was invested at 7% than at 5%. So we can set up the second equation:
z = y + 20,000 ----- (equation 2)
Now we have a system of equations to solve: equation 1 and equation 2.
Let's solve equation 2 for y:
y = z - 20,000
Substitute this value of y in equation 1:
0.04x + 0.05(z-20,000) + 0.07z = 15,400
Now, solve this equation to find the value of z.
Invested $X at 5%.
Invested x+20,000 at 7%.
Invested 280,000-(x + (x+20,000)) = 280,000 - 2x - 20,000 = -2x + 260,000
at 4%.
In each case, I = P*r*t.
I = x*0.05*1 + (x+20,000)*0.07*1 + (-2x+260,000)*0.04*1 = 15,400.
0.05x + 0.07x+1400 + (-0.08x)+10,400 = 15,400,
0.12x - 0.08x + 11,800 = 15,400,
X = $3600 at 5%.
x + 20,000 = 3600 + 20,000 = $23,600 at 7%.
-2x + 260,000 = -7200 + 260,000 = $252,800 at 4%.