Problem 43 - Prealgebra

Question

Isabel is putting water into a 20 -gallon fish tank using a 2-quart pitcher. How many pitchers of water will she need to fill the tank?

Step-by-Step Solution

Answer

Isabel needs 40 pitchers of water to fill the tank.

Step 1: Understand the Units Involved

The fish tank capacity is given in gallons, while the pitcher volume is given in quarts. There are 4 quarts in a gallon. We need to account for this conversion when calculating how many pitchers a tank can hold.

Step 2: Convert Tank Capacity to Quarts

Convert the capacity of the fish tank from gallons to quarts. Since 1 gallon is equal to 4 quarts, we have:\[ 20 \text{ gallons} \times 4 \text{ quarts/gallon} = 80 \text{ quarts} \]So, the fish tank can hold 80 quarts of water.

Step 3: Calculate the Number of Pitchers Needed

Now that we know the fish tank holds 80 quarts, we need to determine how many 2-quart pitchers are required to reach this amount.Using the formula:\[ \frac{\text{Total quarts in tank}}{\text{Volume of pitcher in quarts}} = \frac{80}{2} = 40 \]Thus, Isabel will need 40 pitchers of water to fill the tank.

Key Concepts

Volume Measurement

In many everyday scenarios, we deal with the concept of volume measurement. Volume is essentially the amount of space that an object or a substance occupies. When we think about filling objects like tanks or containers, volume gives us a quantitative grasp of how much liquid they can hold. For instance, Isabel's fish tank has a volume of 20 gallons. In mathematical terms, it's important to grasp the basic units of measurement, such as liters, milliliters, gallons, and quarts. Understanding these units helps us perform precise calculations and efficiently plan our usage of resources.

Capacity Conversion

Capacity conversion is a crucial skill when dealing with problems that involve volume. In this exercise, we have two different units: gallons and quarts. To find how many pitchers Isabel needs, first, we must convert everything to a single unit. In this case, we convert gallons to quarts because it aligns with the pitcher’s measurement.

• 1 gallon equals 4 quarts

To convert the 20-gallon tank to quarts, simply multiply it by 4:\[20 \times 4 = 80 \text{ quarts}\]By this conversion process, we translate the fish tank's volume into the same unit as the pitcher, which enables easier calculations. Conversion helps in aligning measurements and preparing for accurate arithmetic operations.

Problem Solving

Problem-solving is about breaking down a task into manageable steps. For Isabel's situation, we achieve this by understanding the problem, executing conversions, and making calculations.First, we interpret the task: Isabel wants to know how many times she will use a 2-quart pitcher to fill a 20-gallon tank. Start by converting the gallons to quarts, as already discussed. Next, calculate how many pitchers are necessary by dividing the total quarts by the pitcher’s volume:\[\frac{80}{2} = 40\]This division tells us that 40 pitchers of water are required. Breaking problems into clear, smaller parts makes solving them more straightforward and manageable.

Mathematical Reasoning

Mathematical reasoning is the process of making logical inferences based on given information. It involves understanding the relationship between numbers and quantities. In Isabel’s problem: 1. Recognize that the number of quarts a tank can hold is dependent on its gallon capacity. 2. Realize that converting units is necessary to make accurate calculations. 3. Recognize that dividing the capacity of the tank by the volume of each pitcher yields the number of pitchers needed. These steps show how mathematical reasoning connects understanding with action, leading to accurate conclusions. It not only involves performing arithmetic operations but also applying concepts in a logical sequence to solve real-world problems.