Confusing algebra

Question

Lubricating oil worth 28 cents/quart is to be mixed with oil worth 33 cents/quart to make up 45 quarts of a mixture to sell at 30 cents/quart. What volume of each grade should be taken

Answer 1

Let x = the number of quarts of lubricant

Let y = the number of quarts of oil

 

Using these variables, we can write equations that represent the total number of quarts and total cost per quart of mixture.

 

x + y = 45               eq1

 

28x + 33y = 30 * 45             eq2

 

we multiply the right side of eq2 by total quarts in mixture to maintain equal units.   

 

28x + 33y = 1350              eq2

 

 

We have a system of equations.  The bold equations will be the equations to work with.  To solve this system, we use the substitution method.

 

Substitute eq1 into eq2 so that eq2 is in terms of x.

 

28x + 33(45 - x) = 1350

 

 

Solve for x from this equation.

 

28x + 1485 - 33x = 1350

 

-5x = -135

 

   x = 27

 

 

Substitute this value of x into eq1 to solve for y.

 

y = 45 - 27

 

y = 18

 

 

Number of quarts in lubricant is 27.

Number of quarts in oil is 18.

 

We are not done yet.  Now, we have to convert quarts to volume.

 

Use the conversion factor:

 

1 quart = 57.75 cubic inches

 

 

So we multiply each quart by 57.75

 

Answer 2

Let x = # quarts at 28 cents per quart

y = # quarts at 33 cents per quart

 

We want a mixture of oil that totals 45 quarts at 30 cents per quart.  

45 x $0.30 = $13.50

 

We want to set up a system of 2 equations and 2 unknowns.

 

So      x + y = 45

and  .28x + .33y = 13.50

 

Let's use substitution to solve, letting y = 45 - x

 

Substituting this in the 2nd equation,

 

.28 x + .33 (45-x) = 13.50

.28x + 14.85 - .33x = 13.50

-.05x = -1.35

x = 27

 

Now we can use the first equation to solve for y.  y = 45 - x = 45-27 = 18

 

We should use these two values in the 2nd equation to check our work.

 

.28 (27) + .33 (18) = 7.56 + 5.94 = 13.50

 

So we need 27 quarts of the 28-cent oil and 18 quarts of the 33-cent oil.

 

Hope this helps.

 

Answer 3

0.28x+0.33(45-x)=0.3(45)

0.28x+14.85-0.33x=13.5

-0.05x+14.85=13.5

-0.05x=-14.85+13.5

-0.05x=-1.35

0.05x=1.35

5x=135

x=135/5

x=27 quarts of the $0.28 oil

45-27=18 quarts of the $0.33 oil