Natural Science Using the same assumptions about octane ratings as in Example 1, solve the following problem.
A service station has 92-octane and 98-octane gasoline. How many liters of each gasoline should be mixed to provide 12 liters of 96-octane gasoline for a chemistry experiment?
Example 1
Business An oil company distributor needs to fill orders for 89-octane gas, but has only 87- and 93-octane gas on hand. How much of each type should be mixed together to produce 54,000 gallons of 89-octane gas?
Solution
Step 1 We must find how much 87-octane gas and how much 93-octane gas are needed for the 54,000 gallon mixture.
Step 2 Let x be the amount of 87-octane gas.
Step 3 Then 54,000 − x is the amount of 93-octane gas.
Step 4 The octane rating of a gasoline indicates that it has the same antiknock qualities as a standard fuel made of heptane and isooctane. The octane rating is the percentage of isooctane in the standard fuel.* So we assume that standard fuels are being mixed. We can summarize the relevant information in a chart.
Type of Gas | Quantity | % Isooctane | Amount of Isooctane |
87-octane | x | 87% | .87x |
93-octane | 54,000 − x | 93% | .93(54,000 − x) |
Mixture | 54,000 | 89% | .89(54,000) |
Step 5 The amount of isooctane satisfies this equation:
Step 6 To solve the equation, use the distributive property on the left side and multiply out the right side:
Step 7 So the distributor should mix 36.000 gallons of 87-octane gas with 54,000 − 36.000 = 18,000 gallons of 93-octane gas. Then the amount of isooctane in the mixture is
Hence, the octane rating of the mixture is , as required.
*This is one of several possible ways of determining octane ratings.
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