$45,000 are to be divided among three shares in such a way that triple of one share and double of the second and 4 times that 3rd share are equal. Find the share of each.
If the shares are x,y,z, then
3x = 2y = 4z
also, x+y+z = 45000
see what you can do with that
Solution
Let's assume that the first share is x, the second share is y, and the third share is z.
According to the given information:
Triple of one share, which is 3x, is equal to the double of the second share, which is 2y.
Also, four times the third share, which is 4z, is equal to the double of the second share, which is 2y.
So we have the following equations:
Equation 1: 3x = 2y
Equation 2: 4z = 2y
We also know that the sum of all three shares is $45,000:
Equation 3: x + y + z = $45,000
Now let's solve these equations to find the values of x, y, and z.
From Equation 1, we can express x in terms of y:
x = (2y) / 3
Substituting this value of x in Equation 3:
(2y) / 3 + y + z = $45,000
Simplifying:
2y + 3y + 3z = $135,000
5y + 3z = $135,000
From Equation 2, we can express z in terms of y:
z = (2y) / 4
z = y / 2
Substituting this value of z in the previous equation:
5y + 3(y / 2) = $135,000
Simplifying:
10y + 3y = 2 * $135,000
13y = 2 * $135,000
13y = $270,000
y = $270,000 / 13
Calculating y:
y ≈ $20,769.23
Substituting this value of y in Equation 1:
3x = 2 * $20,769.23
3x = $41,538.46
x = $41,538.46 / 3
Calculating x:
x ≈ $13,846.15
Finally, substituting y = $20,769.23 in Equation 2 to find z:
4z = 2 * $20,769.23
z = (2 * $20,769.23) / 4
Calculating z:
z ≈ $10,384.62
So, the three shares are approximately:
First share: $13,846.15
Second share: $20,769.23
Third share: $10,384.62
To solve this problem, we need to set up a system of equations based on the given information.
Let's assume the shares are represented by x, y, and z.
We are given that the triple of one share is equal to the double of the second share, and four times the third share. Mathematically, this can be expressed as:
3x = 2y = 4z
We also know that the sum of all the shares is $45,000. Therefore:
x + y + z = $45,000
Now we have a system of equations:
3x = 2y = 4z
x + y + z = $45,000
To solve this system, we can use the method of substitution.
First, let's solve the equation 3x = 2y for x:
3x = 2y
x = (2y) / 3
We can substitute (2y) / 3 for x in the second equation:
(2y) / 3 + y + z = $45,000
Now we can simplify this equation:
(2y + 3y) / 3 + z = $45,000
(5y + 3z) / 3 = $45,000
To solve for y, we can cross-multiply:
5y + 3z = $45,000 * 3
5y + 3z = $135,000
Since we have two equations with two variables, we can solve for y and z simultaneously.
Next, let's solve the equation 2y = 4z for z:
2y = 4z
z = (2y) / 4
z = y / 2
Now we can substitute y / 2 for z in the equation 5y + 3z = $135,000:
5y + 3(y / 2) = $135,000
5y + (3/2)y = $135,000
(10/2)y + (3/2)y = $135,000
(13/2)y = $135,000
To solve for y, we can multiply both sides of the equation by 2/13:
y = ($135,000) * (2/13)
y = $20,769.23 (approx.)
Now that we have y, we can substitute this value into the equation z = y / 2:
z = ($20,769.23) / 2
z = $10,384.62 (approx.)
Finally, we can use these values to find x by substituting into the equation x + y + z = $45,000:
x + ($20,769.23) + ($10,384.62) = $45,000
x + $31,153.85 = $45,000
x = $45,000 - $31,153.85
x = $13,846.15 (approx.)
Therefore, the share of each person is approximately:
Person 1 (x): $13,846.15
Person 2 (y): $20,769.23
Person 3 (z): $10,384.62