A water tank is $\frac{3}{7}$ full. After adding $\frac{7}{5}$, it is $\frac{4}{5}$ full. What is its total capacity?
Answer (2)
Let C be the capacity of the tank.
Set up the equation: 7 3 C + 5 147 = 5 4 C .
Solve for C : 5 4 C − 7 3 C = 5 147 .
Find the capacity: C = 13 1029 .
Explanation
Problem Analysis Let's analyze the problem. We are given that a water tank is initially 7 3 full. After adding 5 7 of 21 to the tank, it becomes 5 4 full. We need to find the total capacity of the tank.
Defining Variables and Initial Setup Let C be the capacity of the tank. The initial amount of water in the tank is 7 3 C . The amount of water added is 5 7 × 21 = 5 147 . After adding the water, the tank is 5 4 full, so the amount of water is 5 4 C .
Setting up the Equation We can set up the equation: 7 3 C + 5 147 = 5 4 C
Isolating the Variable C Now, let's solve for C . First, we isolate the terms with C on one side of the equation: 5 4 C − 7 3 C = 5 147
Combining Terms with C To combine the terms with C , we need a common denominator for the fractions 5 4 and 7 3 . The least common denominator is 35. So we rewrite the fractions: 35 28 C − 35 15 C = 5 147 Combining the terms, we get: 35 13 C = 5 147
Solving for C To solve for C , we multiply both sides of the equation by 13 35 : C = 5 147 × 13 35 Simplifying, we have: C = 13 147 × 7 = 13 1029
Final Answer Therefore, the capacity of the tank is 13 1029 .
Examples
Understanding the capacity of tanks is crucial in many real-world applications, such as in water resource management, chemical engineering, and even in everyday scenarios like filling up a swimming pool. For instance, if you're managing a water reservoir, knowing the tank's capacity helps you plan for water distribution during dry seasons. In chemical plants, accurate tank capacity measurements are essential for mixing chemicals in the right proportions. Even at home, knowing the volume of your fish tank ensures you add the correct amount of water conditioner. The ability to calculate tank capacity using fractions and algebraic equations is a fundamental skill that bridges mathematics with practical problem-solving.
The total capacity of the water tank is 13 49 units after analyzing the water levels before and after the addition. By setting up an equation based on the fraction of the tank's capacity, we solved for the total capacity. This illustrates a practical application of fractions and algebra in real-world contexts.
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