Help!
Toshiro Hashimoto invested some money at 4.5% simple interest and $1000 more than four times the amount at 6%. His total annual income for one year from interest on the two investments was $801. How much did he invest at each rate?
X dollars @ 4.5 %
(4x + 1000) dollars @ 6%
0.04x + 0.06(4x + 1000) = 801
0.285x = 801
x = 801/0.285 = $ 2600 @ 4.5%
4x + 1000 = 4 * 2600 + 1000 = $ 11400
@ 6%
9
To solve this problem, let's break it down into steps:
Step 1: Assign variables:
Let x be the amount invested at 4.5% interest.
Then, the amount invested at 6% interest will be 4x + $1000.
Step 2: Calculate the annual income from each investment:
The income from the amount invested at 4.5% interest is given by the formula:
0.045x.
The income from the amount invested at 6% interest is given by the formula:
0.06(4x + $1000).
Step 3: Set up the equation:
According to the problem, the total annual income from both investments is $801, so we can set up the equation:
0.045x + 0.06(4x + $1000) = $801.
Step 4: Solve the equation:
Let's solve the equation step by step:
0.045x + 0.24x + 0.06($1000) = $801
0.045x + 0.24x + $60 = $801
0.285x = $801 - $60
0.285x = $741
Now, divide both sides of the equation by 0.285 to find the value of x:
x = $741 / 0.285
x ≈ $2600
Thus, Toshiro Hashimoto invested approximately $2600 at 4.5% interest.
To find the amount invested at 6%, substitute the value of x back into the equation:
4x + $1000 = 4($2600) + $1000
4x + $1000 = $10400 + $1000
4x + $1000 = $11400
Therefore, Toshiro Hashimoto invested $11400 at 6% interest.
To find out how much Toshiro Hashimoto invested at each rate, let's set up the equation using the information given.
Let:
- Amount invested at 4.5% = X dollars
- Amount invested at 6% = Y dollars
According to the problem, Toshiro invested $1000 more than four times the amount invested at 4.5% in the 6% investment. Mathematically, this can be expressed as:
Y = 4X + $1000 (Equation 1)
The total annual income from interest on the two investments is $801. We know that the interest earned is calculated using the formula:
Interest = Principal × Rate × Time
The interest earned from the investment at 4.5% can be calculated as:
Principal (X) × Rate (4.5%) × Time (1 year)
Similarly, the interest earned from the investment at 6% can be calculated as:
Principal (Y) × Rate (6%) × Time (1 year)
Since the total annual income is $801, we can form the equation:
0.045X + 0.06Y = $801 (Equation 2)
Now we have a system of two equations (Equations 1 and 2) with two variables (X and Y). We can solve this system of equations to find the values of X and Y.
One approach to solving this system is substitution:
1. Substitute the value of Y from Equation 1 into Equation 2:
0.045X + 0.06(4X + $1000) = $801
2. Simplify the equation:
0.045X + 0.24X + $60 = $801
0.285X = $801 - $60
0.285X = $741
3. Solve for X:
X = $741 / 0.285
X ≈ $2600
4. Substitute the value of X into Equation 1 to find Y:
Y = 4($2600) + $1000
Y = $10400 + $1000
Y = $11400
Therefore, Toshiro Hashimoto invested approximately $2600 at 4.5% and $11400 at 6%.