Answer:
The quantity of fuel bought at $4 is 40.84 gallons
The quantity of fuel bought at $3 is 116.699 gallons
The quantity of fuel bought at $5 is 17.505 gallons
Step-by-step explanation:
Here we have
Gas used on East coast = X gallons
Gas consumed in the mid-US = Y gallons
Gas consumed in the West coast = Z gallons
Where Y = 2×(X + Z)..........(1)
4·X + 3·Y + 5·Z = 515.........(2)
X + Y + Z = 150..................(3)
Therefore substituting for Y in equation (3) we have
X + 2×(X + Z) + Z = 150
3·X + 3·Z = 150
X = (150 - 3·Z)/3 = 50 - Z
Therefore, equation (2) becomes;
4·(50 - Z) + 3·(2×((50 - Z) + Z)) + 5·Z = 515
200 - 4·Z + 300 -6·Z + 6·Z + 5·Z = 515
500 + 2·Z = 515
Z = 15
X = 50 - Z = 50 - 15 = 35
Y = 2×(X + Z) = 2×(35 + 15) = 2 × 50 = 100
(b) Whereby the bought 174 gallons based on the same ratio as in part (a), we have;
Ratio in part (a) is given as follows;
Proportion of Z gallons = 15/150 = 1/10
Proportion of X gallons = 35/150 = 7/30
Proportion of Y gallons = 100/150 = 2/3
Therefore the proportion of 174 gallons that was consumed in the East coast, X gallons = (7/30)×174 = 203/5 gallons = 40.6 gallons
The proportion of 174 gallons that was consumed in the mid-US, Y gallons = (2/3)×174 = 116 gallons
The proportion of 174 gallons that was consumed in the West coast, Z gallons = (1/10)×174 = 87/5 gallons = 17.4 gallons
Based on price of per gallon, we have
The proportion of $515 that was consumed in the East coast on X gallons = $4 ×35 = $140
Proportion = 140/515= 28/103
The proportion of $515 that was consumed in the mid-US on Y gallons = $3 ×100= $300
Proportion = 300/515= 60/103
The proportion of $515 that was consumed in the West coast on Z gallons = $5 ×15= $75
Proportion = 75/515= 15/103
Therefore, the quantity of fuel bought at $4 = ((28/103) × 601)/4 = 40.84 gallons
The quantity of fuel bought at $3 = ((60/103) × 601)/3 = 116.699 gallons
The quantity of fuel bought at $5 = ((15/103) × 601)/5 = 17.505 gallons.
