Problem 24 - Prealgebra

Question

A water tank is draining at a rate of 12 gallons every 8 minutes. If there are 234 gallons in the tank, when will it have just 138 gallons left?

Step-by-Step Solution

Answer

It will take 64 minutes to reach 138 gallons in the tank.

Step 1: Determine the volume of water to be drained

Calculate how much water needs to be removed to reach 138 gallons. Initial volume is 234 gallons and target volume is 138 gallons: \[ \text{Volume to drain} = 234 - 138 = 96 \text{ gallons} \]

Step 2: Calculate the rate of drainage

The tank drains at a rate of 12 gallons in 8 minutes. This rate means for every 8 minutes, 12 gallons are drained.

Step 3: Calculate total number of 12-gallon blocks to be drained

Since the tank drains in a set block of 12 gallons per 8 minutes, determine how many of these 12-gallon blocks make up the total 96 gallons that need to be drained. \[ \text{Number of blocks} = \frac{96}{12} = 8 \]

Step 4: Determine total time to drain the required amount

Since it takes 8 minutes to drain each 12-gallon block, calculate the total time needed for 8 blocks:\[ \text{Total time} = 8 \times 8 = 64 \text{ minutes} \]

Step 5: Conclusion on how to reach the target volume

It will take 64 minutes to drain 96 gallons from 234 gallons in the tank, leaving 138 gallons remaining.

Key Concepts

Rate of Change

Understanding the rate of change is crucial when dealing with problems that involve a continuous process over time. In this exercise, the water tank is draining at a constant rate. This means that every fixed period, a consistent amount of water is drained. Here, the rate is given as 12 gallons every 8 minutes.
To visualize this, think of it as a speedometer for water drainage. Every 8 minutes, imagine the tank losing exactly 12 gallons, just like a car travels a certain distance in a given time. Through this lens, we can easily predict and calculate future states of the water tank based on consistent changes over time, which leads us to the solution of the problem.

• This constant change is the essence of 'rate.'
• Understanding the concept of rate helps solve many real-world problems.

Volume Calculation

Volume calculation is about determining how much space is being used or needs to be emptied, as in this case. It is one of the first steps in solving a problem like this. Here, we start with an initial volume of water in the tank and identify how much needs to be drained to reach our desired level.
In simple terms, if the tank starts with 234 gallons and we want only 138 gallons left, we calculate the volume that must be removed:

• Subtract the target volume from the initial volume.
• This calculation gives us the volume of water that will be removed.

For this problem, we find that 234 gallons minus 138 gallons equals 96 gallons. Hence, 96 gallons need to be drained.
Mastering volume calculations assists in solving a wide range of practical problems, especially those involving filling or emptying a space.

Unit Conversion

Unit conversion plays a vital role when rates and measurements aren't in the same units. In this exercise, no conversion was necessary as both the volume of water drained and the time taken were already matched units.
However, understanding the concept is crucial. Often in real-world scenarios, you might need to convert units, such as converting minutes to hours for a better understanding of time-based tasks or using metric units for volume depending on the context.

• Always detect and convert differing units before proceeding to calculations.
• Use conversion factors, like 1 hour equals 60 minutes, to help translate measurements into the necessary units.

Getting familiar with unit conversion helps keep your calculations accurate and applicable.

Problem-Solving Steps

In tackling mathematical word problems, following a structured problem-solving approach makes it manageable and less intimidating. In this example, breaking down the problem into steps was essential.
First, calculate the desired volume to be drained. Then, use the known rate to determine how quickly that volume can be removed. From there, you can find how many times the process (in this instance, draining 12 gallons) needs to be repeated to achieve the target volume. Finally, calculate how long it will take in total.

• Identify what needs to be calculated first and break it into manageable steps.
• Check units and ensure consistency to maintain clear calculations.
• Proceed step-by-step, confirming each part before moving on to the next.

By following logical steps, word problems become straightforward puzzles rather than confusing challenges.