Let’s break down and solve each question clearly.

Question 3

32 L of water is poured into a water tank. If the tank is now $\frac{2}{3}$ full, how much water does the tank hold when full?

Let the capacity of the tank be $x$ liters.

From the problem, $\frac{2}{3} \times x = 32$.

Step 1: Solve for $x$: $x = \frac{32}{\frac{2}{3}} = 32 \times \frac{3}{2} = 48 \, \text{L}.$

Answer: The tank holds 48 liters when full.

Question 4

63 L of water was taken out of a water tank, and there is 65% of the water left. What is the original volume of the water tank?

Let the original volume of the tank be $x$ liters.

After 63 liters were removed, 65% of the tank remains: $0.65x = x - 63.$

Step 1: Solve for $x$: $0.65x = x - 63 \implies x - 0.65x = 63 \implies 0.35x = 63.$ $x = \frac{63}{0.35} = 180 \, \text{L}.$

Answer: The original volume of the tank is 180 liters.

Question 5

A tank is $\frac{1}{6}$ full. When 440 liters of water is added, the tank is now $\frac{5}{8}$ full. How much water will be in the tank when it is $\frac{1}{2}$ full?

Let the capacity of the tank be $x$ liters.

Step 1: Calculate the difference in fullness: $\frac{5}{8} - \frac{1}{6}.$ Finding a common denominator: $\frac{5}{8} = \frac{30}{48}, \quad \frac{1}{6} = \frac{8}{48}.$ $\frac{5}{8} - \frac{1}{6} = \frac{30}{48} - \frac{8}{48} = \frac{22}{48} = \frac{11}{24}.$

This means 440 liters corresponds to $\frac{11}{24}$ of the tank.

Step 2: Solve for the tank’s capacity $x$: $\frac{11}{24}x = 440 \implies x = \frac{440 \times 24}{11} = 960 \, \text{L}.$

Step 3: Calculate the amount of water when the tank is $\frac{1}{2}$ full: $\frac{1}{2}x = \frac{1}{2} \times 960 = 480 \, \text{L}.$

Answer: The tank will contain 480 liters when it is $\frac{1}{2}$ full.

Any questions or further clarifications?

Here are five related questions to expand your understanding:

1. How can we generalize the formula for tank capacity in similar fraction problems?
2. What are some real-world applications of these types of percentage or fraction problems?
3. How would the approach change if additional information about the tank were given?
4. Can you explain how to check for consistency in percentage-based problems?
5. How can unit conversion affect calculations in problems involving capacity?

Tip: Always double-check for proper alignment of fractions and percentages to ensure accuracy!