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Question

Difficulty: HardFraction Operations

A fuel tank on a generator is initially 78\frac{7}{8} full. After the generator consumes 1515 gallons of fuel, the tank is exactly 14\frac{1}{4} full. What is the total capacity of the fuel tank in gallons?

Answer: 24 gallons

Answer

The total capacity of the fuel tank is 24 gallons.
The correct answer is derived by finding the difference in the fraction of the tank's capacity (78−14=58\frac{7}{8} - \frac{1}{4} = \frac{5}{8}). Since this 58\frac{5}{8} portion corresponds to 1515 gallons, the total capacity is found by dividing 1515 by 58\frac{5}{8}, which yields 2424 gallons.

Step-by-Step Solution

1
Find a common denominator for the given fractions.
14=28\frac{1}{4} = \frac{2}{8}
To subtract the fractions, they must have the same denominator.
2
Calculate the fraction of the tank that was consumed by the generator.
78−28=58\frac{7}{8} - \frac{2}{8} = \frac{5}{8}
The difference between the initial and final fuel levels represents the exact amount consumed.
3
Set up an algebraic equation linking the fraction consumed to the actual volume in gallons.
58×Capacity=15\frac{5}{8} \times \text{Capacity} = 15
We know that 58\frac{5}{8} of the total capacity corresponds directly to the 1515 gallons used.
4
Solve the equation for the total capacity.
15×85=2415 \times \frac{8}{5} = 24
Multiplying by the reciprocal of the fraction isolates the capacity variable.

Key Concept

Applying fraction subtraction with unlike denominators and solving a missing-value equation using a reciprocal.
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Topics

Fraction Operations

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