A businessman has $1000 income from bonds bearing 4% and 5%. If the amounts invested at 4% and 5% were interchanged, he would earn $50 more annually. Find the total amount invested by the businessman.
amount invested at 4% --- x
amount invested at 5% ---- y
.04x + .05y = 1000
4x + 5y = 100000 , #1
reversed:
.05x + .04y = 1050
5x + 4y = 105000 , #2
solve these two equations.
Well, well, well, it seems that the businessman is in quite a bond-ing situation! Let's see if I can help him out.
Let's call the amount invested at 4% "x" and the amount invested at 5% "y". So, initially, the businessman earns 4% of x and 5% of y in income.
According to the problem, if these amounts were interchanged, he would earn $50 more annually. So, the new income would be 5% of x and 4% of y.
Using some math magic, we can set up an equation:
0.04x + 0.05y = 0.05x + 0.04y + 50
Now, let's simplify this equation:
0.05x - 0.04x = 0.04y - 0.05y + 50
0.01x = -0.01y + 50
Now, let's move things around a bit:
0.01x + 0.01y = 50
x + y = 5000
Ta-da! We've got an equation that represents the total amount invested by the businessman. The sum of x and y is equal to 5000.
Now, all we have to do is assume that he invested a total of $5000. But remember, this is just an assumption. We don't have enough information to determine the exact amounts invested at each rate. So, the total amount invested by the businessman is $5000.
Well, I hope my clowning around helped shed some light on the situation. Just remember, laughter is the best investment!
Let the amount invested at 4% be x, and the amount invested at 5% be y.
According to the information given, the businessman earns $1000 from the bonds. Therefore, we can write the equation:
0.04x + 0.05y = 1000 -- Equation 1
If the amounts invested at 4% and 5% were interchanged, the businessman would earn $50 more annually. This can be expressed as:
0.05x + 0.04y = 1050 -- Equation 2
To solve the system of equations, we can multiply Equation 2 by 100 to eliminate the decimal points:
5x + 4y = 105000 -- Equation 3
Now, we can solve the system of equations by eliminating one variable:
Multiply Equation 1 by 4 and Equation 2 by 5:
0.16x + 0.20y = 4000 -- Equation 4
0.25x + 0.20y = 5250 -- Equation 5
Subtract Equation 4 from Equation 5:
0.25x - 0.16x = 5250 - 4000
0.09x = 1250
Solving for x:
x = 1250 / 0.09
x ≈ 13888.89
Next, substitute the value of x into Equation 1 to solve for y:
0.04(13888.89) + 0.05y = 1000
555.56 + 0.05y = 1000
0.05y = 444.44
y ≈ 8888.89
Therefore, the amount invested at 4% is approximately $13888.89, and the amount invested at 5% is approximately $8888.89.
The total amount invested by the businessman is:
Total amount = Amount invested at 4% + Amount invested at 5%
Total amount = $13888.89 + $8888.89
Total amount ≈ $22777.78
To solve this problem, let's assume that the amount invested at 4% is "x" dollars, and the amount invested at 5% is "y" dollars.
We know that the businessman has an income of $1000 from the investments. So we can set up the following equation:
0.04x + 0.05y = 1000 .....(equation 1)
Now, it is given that if the amounts invested at 4% and 5% were interchanged, the businessman would earn $50 more annually. Therefore, we can set up another equation using this information:
0.05x + 0.04y = 1000 + 50 .....(equation 2)
We now have a system of equations with two variables (x and y). We can solve this system of equations to find the values of x and y, which represent the amounts invested at 4% and 5%, respectively.
To solve this system of equations, we can use either the substitution method or the elimination method. In this case, let's use the elimination method:
Multiply equation 1 by 100 to get rid of the decimals:
4x + 5y = 100000 .....(equation 3)
Multiply equation 2 by 100 to get rid of the decimals:
5x + 4y = 100050 .....(equation 4)
Now, let's subtract equation 4 from equation 3 to eliminate the y variable:
(4x + 5y) - (5x + 4y) = 100000 - 100050
This simplifies to:
4x - 5x + 5y - 4y = -50
-x + y = -50 .....(equation 5)
Now, let's add equation 5 to equation 2 to eliminate the x variable:
(5x + 4y) + (-x + y) = 100050 + (-50)
This simplifies to:
4x + 5x + 4y + y = 100000
9x + 5y = 100000 .....(equation 6)
Now we have a new equation (equation 6) with only two variables (x and y). Let's solve this equation using the substitution method.
Rearrange equation 5 to solve for x:
-x + y = -50
x = y + 50 .....(equation 7)
Substitute equation 7 into equation 6:
9(y + 50) + 5y = 100000
Expand and simplify:
9y + 450 + 5y = 100000
14y + 450 = 100000
Subtract 450 from both sides:
14y = 99550
Divide by 14:
y = 7103.57
Now substitute the value of y back into equation 7 to find the value of x:
x = 7103.57 + 50
x = 7153.57
So, the amount invested at 4% is $7153.57 and the amount invested at 5% is $7103.57. To find the total amount invested by the businessman, we need to add these two amounts:
Total amount = Amount invested at 4% + Amount invested at 5%
Total amount = $7153.57 + $7103.57
Total amount = $14,257.14
Therefore, the total amount invested by the businessman is $14,257.14.