A petroleum company has two different sources of crude oil. The first source provides crude oil that is 25% hydrocarbons, and the second one provides crude oil that is 45% hydrocarbons. In order to obtain 180 gallons of crude oil that is 40% hydrocarbons, how many gallons of crude oil must be used from each of the two sources?
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Let's call the first source A, and the second source B. The volume constraint is A + B = 180
Now let's look at our concentration constraint. The first source is 25% HC's so 1 gallon from the first source contains 0.25*1 = 0.25 gal HC's. So the HC's from the first source are A*0.25. The second source and the final product may be handled in the same manner to give our concentration constraint:
A*0.25 + B*0.45 = 180*0.4 = 72
We now have two equations with two unknowns:
A + B = 180
A*0.25 + B*0.45 = 72
Solving the first eqn for A gives us A = 180 - B. Substituting this into the second eqn gives us (180 - B)*0.25 + B*45 = 72. Solving for B gives us B = 135. Clearly, A = 45
Now let's look at our concentration constraint. The first source is 25% HC's so 1 gallon from the first source contains 0.25*1 = 0.25 gal HC's. So the HC's from the first source are A*0.25. The second source and the final product may be handled in the same manner to give our concentration constraint:
A*0.25 + B*0.45 = 180*0.4 = 72
We now have two equations with two unknowns:
A + B = 180
A*0.25 + B*0.45 = 72
Solving the first eqn for A gives us A = 180 - B. Substituting this into the second eqn gives us (180 - B)*0.25 + B*45 = 72. Solving for B gives us B = 135. Clearly, A = 45
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Let x be number of gallons of 0.45 crude.
Then 180 - x is the # gal of 0.25 crude.
1. (0.45)x=# gal's of pure hc's from the 0.45 crude
2. (0.25)(180-x) = # gal's of pure hc's from the 0.25 crude
3. (0.40)(180) = # gal's of pure hydrocarbons from the desired mixture of the two, being 0.40 pure
.45x + 45 -.25x = 72 (equation 1, above, plus equation 2 equals equation 3).
0.2x = 27
x= 135
180-x=45
One needs 135 gallons of the 45% oil, and 45 gallons of the 25% stuff.
Pretty Slick, huh?
(Pun intended)!
Then 180 - x is the # gal of 0.25 crude.
1. (0.45)x=# gal's of pure hc's from the 0.45 crude
2. (0.25)(180-x) = # gal's of pure hc's from the 0.25 crude
3. (0.40)(180) = # gal's of pure hydrocarbons from the desired mixture of the two, being 0.40 pure
.45x + 45 -.25x = 72 (equation 1, above, plus equation 2 equals equation 3).
0.2x = 27
x= 135
180-x=45
One needs 135 gallons of the 45% oil, and 45 gallons of the 25% stuff.
Pretty Slick, huh?
(Pun intended)!
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28 gallons