Answer
The answer is: 3 cubic yards of sand, 1.5 cubic yards of cement and 4.5 cubic yards of gravel.
Work Step by Step
We assign the variables:
x = volume of sand in cubic yards
y = volume of cement in cubic yards
z = volume of gravel in cubic yards
Step 1:
Find the equations that represent the problem.
We know that $x+y+z=9~yd^{3}$ because the sum of the elements of the concrete mixture has to equal the total volume of the concrete.
$x+y+z=9~yd^{3}$ -> Eq. 1
We can get the second equation from the wording in the problem.
From " twice as much sand as cement", we get,
$x=2y$ -> Eq. 2
If we solve for y, we get:
$y=\frac{x}{2}=0.5x$ -> Eq. 2
We can get the second equation from the wording in the problem.
From " twice as much sand as cement", we get,
$z=1.5x$ -> Eq. 3
Step 2:
Solve the system of equations. Using the substitution method,
-> Substitute Eq. 2 and Eq. 3 into Eq. 1
$x+0.5x+1.5x=9$
$3x=9$
$x=\frac{9}{3}$
$x=3yd^{3}$
-> Substitute the value for x into Eq. 2
$3=2y$
$y=\frac{3}{2}$
$y=1.5yd^{3}$
-> Substitute the value for x into Eq. 3
$z=1.5(3)$
$z=4.5$
Step 3:
The answer is 3 cubic yards of sand, 1.5 cubic yards of cement and 4.5 cubic yards of gravel.
