Problem 5 - Prealgebra

Question

Express each of the following rates as a ratio with the given units. Gallons/Second The flow of water from a water faucet can fill a 3 -gallon container in 15 seconds. Give the ratio of gallons to seconds as a rate in gallons per second.

Step-by-Step Solution

Answer

The rate is \( \frac{1}{5} \) gallons per second.

Step 1: Identify Given Values

We need to express the rate of water flow from a faucet in gallons per second. The problem states that a water faucet can fill a 3-gallon container in 15 seconds.

Step 2: Set Up the Ratio

To find the rate in gallons per second, we must express the given values as a fraction, with gallons as the numerator and seconds as the denominator. So, we write the ratio as \( \frac{3 \text{ gallons}}{15 \text{ seconds}} \).

Step 3: Simplify the Fraction

The fraction \( \frac{3}{15} \) can be simplified. Both the numerator (3) and the denominator (15) can be divided by their greatest common divisor, which is 3. So, \( \frac{3}{15} = \frac{3 \div 3}{15 \div 3} = \frac{1}{5} \).

Step 4: State the Rate

The simplified fraction \( \frac{1}{5} \) represents the rate of water flow in gallons per second. Hence, the water faucet releases water at a rate of \( \frac{1}{5} \) gallons per second.

Key Concepts

Simplifying fractions

Simplifying fractions is an essential skill in math that helps us express numbers in their simplest form. This makes them easier to work with and understand. Imagine you have a fraction like \( \frac{3}{15} \). To simplify it, you need to find the greatest common divisor (GCD) of the numerator and the denominator.

In this case, the GCD of 3 and 15 is 3. You then divide both the numerator and the denominator by this number. So, \( \frac{3}{15} \) becomes \( \frac{3 \div 3}{15 \div 3} = \frac{1}{5} \). This process reduces the fraction to its simplest form.

• Simplifying helps in easier computation.
• It reduces the terms to their basic form.
• Makes comparison between fractions more straightforward.

Simplifying is especially useful when dealing with rates, as it lets us see the core relationship between two measurements or quantities.

Rates

A rate is a specific type of ratio where two quantities with different units are compared. In real life, rates are everywhere—like speeds in miles per hour or prices in dollars per item. Understanding rates helps in comparing different quantities efficiently.

To express a rate, we often use the format of a fraction, comparing two different units. For example, in our exercise, the rate of water flow was described as gallons per second. We start with the ratio \( \frac{3 \text{ gallons}}{15 \text{ seconds}} \) and simplify it to find the rate in its simplest form—\( \frac{1}{5} \text{ gallons per second} \).

• Rates tell us how one quantity changes with respect to another.
• They are used in various fields such as physics, finance, and daily transactions.
• Rates allow us to understand the efficiency or speed of processes.

Understanding the concept of rates is crucial for solving real-world problems efficiently.

Units of measurement

Units of measurement are essential in expressing quantities and comparing them. They give meaning to numbers, allowing us to understand size, amount, or duration. In math, units help us measure things like distance, time, volume, and weight.

When we talk about rates, it's important to pay attention to the units involved. For example, in our exercise, the units were gallons and seconds. Understanding how these units interact helps us grasp the actual quantity being measured. In our case, the number of gallons per second tells us about the water flow rate.

• Consistency in units is critical when performing math calculations.
• Mismatched units can lead to incorrect interpretations or results.
• Units ensure clarity and precision in communication.

Whether working with inches or liters, appropriately using and understanding units is fundamental in math and science.