Water Flowed Out Of A Tank At A Steady Rate. A Total Of $18 \frac{1}{2}$ Gallons Flowed Out Of The Tank In $4 \frac{1}{4}$ Hours. Which Expression Determines The Quantity Of Water Leaving The Tank Per Hour?A. $\frac{17}{4} +

Understanding the Problem: Calculating the Rate of Water Flow

When dealing with rates of change, it's essential to understand the concept of rate and how it's calculated. In this scenario, we're given the total quantity of water that flowed out of a tank and the total time it took for this to happen. Our goal is to determine the quantity of water leaving the tank per hour.

Calculating the Rate of Water Flow

To calculate the rate of water flow, we need to divide the total quantity of water that flowed out of the tank by the total time it took for this to happen. This can be represented mathematically as:

Rate = Total Quantity / Total Time

In this case, the total quantity of water that flowed out of the tank is $18 \frac{1}{2}$ gallons, and the total time it took for this to happen is $4 \frac{1}{4}$ hours.

Converting Mixed Numbers to Improper Fractions

Before we can perform the division, we need to convert the mixed numbers to improper fractions. To do this, we multiply the whole number part by the denominator and then add the numerator.

For the total quantity of water that flowed out of the tank, we have:

$18 \frac{1}{2} = \frac{(18 \times 2) + 1}{2} = \frac{36 + 1}{2} = \frac{37}{2}$ gallons

For the total time it took for this to happen, we have:

$4 \frac{1}{4} = \frac{(4 \times 4) + 1}{4} = \frac{16 + 1}{4} = \frac{17}{4}$ hours

Calculating the Rate of Water Flow

Now that we have the total quantity of water that flowed out of the tank and the total time it took for this to happen in improper fractions, we can perform the division to calculate the rate of water flow.

Rate = Total Quantity / Total Time = $\frac{37}{2}$ / $\frac{17}{4}$ = $\frac{37}{2}$ × $\frac{4}{17}$ = $\frac{37 \times 4}{2 \times 17}$ = $\frac{148}{34}$ = $\frac{74}{17}$ gallons per hour

Conclusion

In conclusion, the expression that determines the quantity of water leaving the tank per hour is $\frac{74}{17}$ gallons per hour.

Discussion

This problem requires us to understand the concept of rate and how it's calculated. We need to divide the total quantity of water that flowed out of the tank by the total time it took for this to happen. This can be represented mathematically as:

Rate = Total Quantity / Total Time

We also need to convert mixed numbers to improper fractions before performing the division. This is an essential skill in mathematics, as it allows us to perform calculations with fractions.

Real-World Applications

This problem has real-world applications in various fields, such as engineering, physics, and chemistry. For example, in engineering, we may need to calculate the rate of water flow in a pipe to determine the pressure drop or the flow rate of a pump. In physics, we may need to calculate the rate of water flow to determine the energy required to pump water through a pipe. In chemistry, we may need to calculate the rate of water flow to determine the concentration of a solution.

Tips and Tricks

When dealing with rates of change, it's essential to understand the concept of rate and how it's calculated. We need to divide the total quantity of water that flowed out of the tank by the total time it took for this to happen. This can be represented mathematically as:

Rate = Total Quantity / Total Time

We also need to convert mixed numbers to improper fractions before performing the division. This is an essential skill in mathematics, as it allows us to perform calculations with fractions.

Common Mistakes

When dealing with rates of change, it's essential to avoid common mistakes. One common mistake is to confuse the rate of change with the total quantity. For example, if we're given the total quantity of water that flowed out of the tank and the total time it took for this to happen, we may mistakenly calculate the rate of change as the total quantity divided by the total time. However, this is incorrect, as the rate of change is the total quantity divided by the total time.

Conclusion

In conclusion, the expression that determines the quantity of water leaving the tank per hour is $\frac{74}{17}$ gallons per hour. This problem requires us to understand the concept of rate and how it's calculated. We need to divide the total quantity of water that flowed out of the tank by the total time it took for this to happen. This can be represented mathematically as:

Rate = Total Quantity / Total Time

We also need to convert mixed numbers to improper fractions before performing the division. This is an essential skill in mathematics, as it allows us to perform calculations with fractions.
Q&A: Calculating the Rate of Water Flow

In the previous article, we discussed how to calculate the rate of water flow from a tank. Here are some frequently asked questions and answers to help you better understand the concept:

Q: What is the rate of water flow?

A: The rate of water flow is the quantity of water that flows out of a tank per unit of time. In this case, we're calculating the rate of water flow in gallons per hour.

Q: How do I calculate the rate of water flow?

A: To calculate the rate of water flow, you need to divide the total quantity of water that flowed out of the tank by the total time it took for this to happen. This can be represented mathematically as:

Rate = Total Quantity / Total Time

Q: What if the total quantity of water that flowed out of the tank is given as a mixed number?

A: If the total quantity of water that flowed out of the tank is given as a mixed number, you need to convert it to an improper fraction before performing the division. To do this, you multiply the whole number part by the denominator and then add the numerator.

Q: What if the total time it took for the water to flow out of the tank is given as a mixed number?

A: If the total time it took for the water to flow out of the tank is given as a mixed number, you need to convert it to an improper fraction before performing the division. To do this, you multiply the whole number part by the denominator and then add the numerator.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you multiply the whole number part by the denominator and then add the numerator. For example, if you have the mixed number $4 \frac{1}{4}$, you would multiply the whole number part (4) by the denominator (4) and then add the numerator (1) to get $\frac{17}{4}$.

Q: What if I have a fraction with a denominator that is not a multiple of the numerator?

A: If you have a fraction with a denominator that is not a multiple of the numerator, you need to find the least common multiple (LCM) of the denominator and the numerator. The LCM is the smallest number that both the denominator and the numerator can divide into evenly.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you need to list the multiples of each number and find the smallest number that appears in both lists. For example, if you have the numbers 4 and 6, the multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, and the multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60. The smallest number that appears in both lists is 12, so the LCM of 4 and 6 is 12.

Q: What if I have a fraction with a denominator that is a multiple of the numerator?

A: If you have a fraction with a denominator that is a multiple of the numerator, you can simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that both the numerator and the denominator can divide into evenly.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you need to list the factors of each number and find the largest number that appears in both lists. For example, if you have the numbers 12 and 18, the factors of 12 are 1, 2, 3, 4, 6, 12, and the factors of 18 are 1, 2, 3, 6, 9, 18. The largest number that appears in both lists is 6, so the GCD of 12 and 18 is 6.

Q: What if I have a fraction with a numerator that is a multiple of the denominator?

A: If you have a fraction with a numerator that is a multiple of the denominator, you can simplify the fraction by dividing the numerator and the denominator by their GCD.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide the numerator and the denominator by their GCD. For example, if you have the fraction $\frac{12}{18}$, you can simplify it by dividing the numerator and the denominator by their GCD, which is 6. This gives you the simplified fraction $\frac{2}{3}$.

Q: What if I have a fraction with a numerator that is not a multiple of the denominator?

A: If you have a fraction with a numerator that is not a multiple of the denominator, you need to find the LCM of the denominator and the numerator. The LCM is the smallest number that both the denominator and the numerator can divide into evenly.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you need to list the multiples of each number and find the smallest number that appears in both lists. For example, if you have the numbers 4 and 6, the multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, and the multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60. The smallest number that appears in both lists is 12, so the LCM of 4 and 6 is 12.

Conclusion

In conclusion, calculating the rate of water flow from a tank requires understanding the concept of rate and how it's calculated. We need to divide the total quantity of water that flowed out of the tank by the total time it took for this to happen. This can be represented mathematically as:

Rate = Total Quantity / Total Time

We also need to convert mixed numbers to improper fractions before performing the division. This is an essential skill in mathematics, as it allows us to perform calculations with fractions.