Solving a Gas Tank Capacity Problem Using Algebra
Understanding the capacity of a gas tank can be crucial for various applications, from vehicle maintenance to large industrial systems. In this article, we will explore how to solve a specific gas tank problem using algebraic methods. The problem at hand is: when 20 liters are added to a gas tank which is 1/4 full, the tank is then 2/3 full. The objective is to determine the capacity of the tank in liters. We will solve this problem using multiple methods, including algebraic equations and proportional reasoning.
Algebraic Method
Let's denote the total capacity of the tank as (T) liters. According to the given conditions:
When 20 liters are added to a 1/4 full tank, the tank becomes 2/3 full.
This can be expressed algebraically as:
$$frac{1}{4}T 20 frac{2}{3}T$$
Subtracting (frac{1}{4}T) from both sides:
$$20 frac{2}{3}T - frac{1}{4}T$$
Combining the fractions on the right hand side:
$$20 frac{8}{12}T - frac{3}{12}T$$ or $$20 frac{5}{12}T$$
Multiplying both sides by (frac{12}{5}) to solve for (T):
$$T 20 times frac{12}{5} 48 text{ liters}$$
Proportional Reasoning
To approach the problem using proportional reasoning, we first identify the difference in tank fullness between 1/2 full and 2/3 full:
The difference between 1/2 full and 2/3 full is 1/6 of the tank because 2/3 4/6 less 1/2 3/6 is 1/6.
Since 15 liters is 1/6 of the tank, the capacity of the tank can be calculated as:
$$15 text{ liters} frac{1}{6}T$$
Multiplying both sides by 6:
$$T 15 times 6 90 text{ liters}$$
Multiplicative Reasoning
Another method involves recognizing that the difference in the fractions 2/3 and 1/4 is equal to the difference in the tank's fullness when 20 liters are added. This difference is 5/12 of the tank's capacity. Therefore:
20 liters frac{5}{12}T
Solving for (T):
$$T 20 times frac{12}{5} 48 text{ liters}$$
Practical Implications
Understanding the tank capacity can be crucial for several reasons. For instance, if the fuel gauge is factually accurate in reality, the tank should hold the calculated capacity. However, in practice, the tank capacity is often more than the displayed figure due to factors like the fuel gauge's inaccuracy and the space taken up by the fuel pump and hoses.
Different methods to solve for the gas tank capacity allow us to validate and cross-check the results, ensuring accuracy in our calculations.
Conclusion
By applying algebraic and proportional reasoning, we have determined that the gas tank capacity is 48 liters when 20 liters are added to a tank that is 1/4 full, making the tank 2/3 full. This problem illustrates the effectiveness of algebraic and proportional methods in solving practical problems.