SOLUTION: Please Help me work this out step by step... I Haven't done a mixture problem with so many variables before. I appreciate all the help I can get Ethonal and Toluene are two exampl

Question 955337: Please Help me work this out step by step... I Haven't done a mixture problem with so many variables before. I appreciate all the help I can get
Ethonal and Toluene are two examples of additives within your gasoline.
Type A gasoline is 5% Ethanol and 10% Toluene
Type B gasoline is 15% Ethanol and 8% Toluene
Type C gasoline is 10% Ethanol and 4% Toluene
A special Type D gasoline is 8.05% Ethanol and 6.84% Toluene. Precisely blending the correct quantities of Type A, B, and C together can create type D.
How much of each type should be blended together to create 100 gallons of Type D gasoline.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
let a = number of gallons of type A gasoline.
let b = number of gallons of type B gasoline.
let c = number of gallons of type C gasoline.
let d = number of gallons of type D gasoline.

D = 100 gallons.

you have 3 equations that need to be solved simultaneously.

they are:

a + b + c = 100

this equation tells you that the number of gallons of type A and type B and type C must be equal to the number of gallons of type D which is equal to 100 gallons.

.05a + .15b + .10c = .0805d = .0805 * 100 = 8.05

this equation tells you that the number of gallons of ethanol in type A and type B and type C gasoline must be equal to the number of gallons of ethanol in type D gasoline.

.10a + .08b + .04c = .0684d = .0684 * 100 = 6.84

this equation tells you that the number of gallons of toluene in type A and type B and type C gasoline must be equal to the number of gallons of toluene in type D gasoline.

the 3 equations are:

a + b + c = 100
.05a + .15b + .10c = 8.05
.10a + .08b + .04c = 6.84

when you solve these equations simultaneously, you will get:

a = 44
b = 5
c = 51

you will need 44 gallons of type A gasoline and 5 gallons of type B gasoline and 51 gallons of type C gasoline to make 100 gallons of type D gasoline.

solving this system of equations by elimination can be done as follows:

start with:

a + b + c = 100 (equation 1)
.05a + .15b + .10c = 8.05 (equation 2)
.10a + .08b + .04c = 6.84 (equation 3)

work with equations 1 and 2 to start.

multiply both sides of equation 1 by .05 to get:

.05a + .05b + .05c = 5 (equation 4)
.05a + .15b + .10c = 8.05 (equation 2)

subtract equation 4 from equation 2 to get:

.10b + .05c = 3.05 (equation 5)

now work with equation 1 and equation 3.

multiply both sides of equation 1 by .10 to get:

.10a + .10b + .10c = 10 (equation 6)
.10a + .08b + .04c = 6.84 (equation 3)

subtract equation 3 from equation 6 to get:

.02b + .06c = 3.16 (equation 7)

you now have 2 equations in 2 unknowns.

they are:

.10b + .05c = 3.05 (equation 5)
.02b + .06c = 3.16 (equation 7)

multiply both sides of equation 7 by 5 to get:

.10b + .05c = 3.05 (equation 5)
.10b + .30c + 15.8 (equation 8)

subtract equation 5 from equation 8 to get:

.25c = 12.75

solve for c to get:

c = 51

now that you have c = 51, you can go back to equation 5 or equation 8 and solve for b.

we'll use equation 5.

you get:

.10b + .05c = 3.05 (equation 5) becomes:
.10b + .05(51) = 3.05

solve for b to get:

.10b = 3.05 - .05(51) = .5

solve for b to get b = 5

you now have c = 51 and b = 5

go back to any of the original equations and solve for a.

we'll use equation 1.

you will get:

a + b + c = 100 (equation 1) becomes:

a + 5 + 51 = 100 which becomes:

a + 56 = 100

solve for a to get a = 44.

you now have:

a = 44
b = 5
c = 51

go back to your original equations and solve them using those values to see that all the equations are true.

they are, so the solution is correct.

RELATED QUESTIONS
It has been a long time since I've done this and need to help my son with his homework.... (answered by macston)

Could someone help me with this problem. I am lost. Step by step so I can understand what (answered by rfer)

I have a problem about factoring two variables I do not understand it. I need it worked... (answered by Alan3354)

Please help me solve this problem. I would appreciate step by step so that it can help... (answered by gonzo)

I have a problem I do not know how to work out. Could someone help me and show me step by (answered by stanbon,tara0066)

Could someone please help me with this problem. It says to graph the ellipse... (answered by ichudov)

We are doing 1 step equations at our adult education class and I do not understand these... (answered by ankor@dixie-net.com)

I have a problem 4a^2+18a+8/a+4 could you please work this out and show me step by... (answered by jim_thompson5910)

Please help me by writing this out step per step so I can understand and see how it is... (answered by mananth)