Calculating Water Flow Rate: A Step-by-Step Math Guide

Hey math enthusiasts! Let's dive into a cool problem involving water tanks and rates. We're going to break down how to calculate the rate at which water leaves a tank. This kind of problem pops up in all sorts of real-world scenarios, so understanding it is super useful. Let's get started, shall we?

The Problem: Water Flowing Out

Okay, so here’s the setup: We have a tank, and water is flowing out of it at a steady rate. This is the key – it means the flow is consistent; the same amount of water exits the tank every hour. We're told that a total of $18 \frac{1}{2}$ gallons flowed out of the tank in $4 \frac{1}{4}$ hours. Our mission, should we choose to accept it (and we do!), is to figure out how much water leaves the tank per hour. This is all about finding the flow rate.

To solve this, we need to know how much water was released and for how long. The problem gives us these vital pieces of information: the total volume of water and the time it took to drain. Our goal now is to determine the expression that correctly calculates the quantity of water leaving the tank each hour. Think of it like this: We want to find out how many gallons are lost in one single hour. This is where we'll use division to solve our math problem. We'll divide the total gallons of water that flowed out by the total number of hours.

Now, let's look at the given options to find out which expression will give us the correct answer for the rate of water flow. We're essentially looking for the correct way to express the division of the total gallons by the total hours to determine the amount of water lost per hour. This problem showcases a fundamental concept in mathematics: understanding rates. Rates are everywhere – speed, miles per hour, or even the price per item in the grocery store. This exercise is about translating a real-world problem into a mathematical expression and then solving it. This involves converting mixed numbers to improper fractions, understanding the concept of division in the context of rates, and finally, selecting the correct expression that represents the relationship between the total water volume and the total time. Alright, let's break it down further so we can solve this problem like pros. This will also sharpen our skills in interpreting word problems, a critical skill in all fields of life, not just mathematics.

Converting Mixed Numbers

Before we can do any calculations, let's get rid of those pesky mixed numbers and turn them into something easier to work with. Remember, a mixed number has a whole number and a fraction part. We need to convert them into improper fractions (where the numerator is larger than the denominator) to make the division easier.

So, let's convert $18 \frac{1}{2}$:

Multiply the whole number (18) by the denominator (2): $18 * 2 = 36$. Add the numerator (1) to the result: $36 + 1 = 37$. Put the result over the original denominator: $\frac{37}{2}$.

So, $18 \frac{1}{2} = \frac{37}{2}$.

Next, let's convert $4 \frac{1}{4}$:

Multiply the whole number (4) by the denominator (4): $4 * 4 = 16$. Add the numerator (1) to the result: $16 + 1 = 17$. Put the result over the original denominator: $\frac{17}{4}$.

So, $4 \frac{1}{4} = \frac{17}{4}$.

Now, we have the total gallons as $\frac{37}{2}$ and the time in hours as $\frac{17}{4}$. We've simplified the mixed numbers into improper fractions. This is a crucial step to correctly solve the problem. Converting these mixed numbers to improper fractions allows us to perform arithmetic operations more easily and accurately. This step ensures that we maintain mathematical integrity when calculating the water flow rate. This also makes the division operations more manageable. This also sets us up perfectly to perform the correct calculation in the next step. Remember, precision in these conversions is fundamental to arriving at the right answer.

Setting Up the Division

Now that we have the total gallons and the time in improper fractions, let's set up the division. We want to know how many gallons of water flow out per hour, so we need to divide the total gallons by the total hours. This gives us:

Flow rate = Total gallons / Total hours.

In our case:

Flow rate = $\frac{37}{2} \div \frac{17}{4}$.

This expression correctly represents the calculation needed to determine how much water leaves the tank each hour. Think about it: We're dividing the total amount of water that flowed out by the total time it took. This gives us the rate of flow—gallons per hour. Setting up the division is about expressing the relationship between the total volume and the duration. Because we want to find the rate per hour, the division order must be total gallons divided by total time. Understanding this is key to solving rate problems. This is because it directly addresses the question of how many gallons flow out of the tank for every hour. This is the cornerstone of how we arrive at the correct answer.

Evaluating the Options

Now, let's check the given options to see which one matches the expression we just derived. We're looking for an expression that represents $\frac{37}{2} \div \frac{17}{4}$.

The options are:

A. $\frac{17}{4} \div \frac{36}{2}$

B. (This option is missing, and we will find the correct answer ourselves.)

Let's analyze option A: $\frac{17}{4} \div \frac{36}{2}$. This expression does not match our derived expression of $\frac{37}{2} \div \frac{17}{4}$. It has the wrong numbers and the division order is incorrect.

So, we need to create the correct expression based on our work. The correct expression should be the total gallons ($\frac{37}{2}$) divided by the total time ($\frac{17}{4}$). None of the provided options are correct, so let us solve the problem ourselves.

Solving for the Flow Rate

We know that the flow rate is $\frac{37}{2} \div \frac{17}{4}$. To divide fractions, we multiply by the reciprocal of the second fraction (the one we're dividing by). The reciprocal of $\frac{17}{4}$ is $\frac{4}{17}$. So, our calculation becomes:

$\frac{37}{2} \times \frac{4}{17}$ = $\frac{37 * 4}{2 * 17}$ = $\frac{148}{34}$.

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

$\frac{148 \div 2}{34 \div 2} = \frac{74}{17}$.

So, the flow rate is $\frac{74}{17}$ gallons per hour. To convert this to a mixed number, we divide 74 by 17. 17 goes into 74 four times (4 * 17 = 68), with a remainder of 6. Therefore, $\frac{74}{17} = 4 \frac{6}{17}$. This means approximately 4 and 6/17 gallons of water leave the tank per hour.

In summary, we first converted the mixed numbers, set up the correct division problem and then solved it to get our answer. This final calculation provides us with the precise rate at which water is flowing out of the tank each hour. Converting the improper fraction back into a mixed number gives us a more intuitive understanding of the rate. Remember that the accuracy of our calculation depends on correctly setting up the division and accurately performing the arithmetic. So, the key to solving this problem lies in converting mixed numbers to improper fractions, setting up the division problem correctly, and then performing the calculations systematically.

Conclusion: The Final Answer

Since none of the answer options were correct, we had to do the calculation ourselves. Based on the steps we have completed, the flow rate is $\frac{74}{17}$ or $4 \frac{6}{17}$ gallons per hour.

Therefore, to determine the quantity of water leaving the tank per hour, you would need to calculate $\frac{37}{2} \div \frac{17}{4}$, which simplifies to $\frac{74}{17}$ or $4 \frac{6}{17}$ gallons per hour.

Great job sticking with me through this! You've successfully navigated a word problem involving rates. Keep practicing, and you'll become a math whiz in no time! Remember, these kinds of problems aren't just about getting the right answer; they're about thinking logically, breaking down complex situations, and building a strong foundation in math concepts. Keep up the excellent work, guys!