Problem 2 - Prealgebra

Question

Solve each of the following word problems by translating the statement into a proportion. Be sure to show the proportion used in each case. [Examples \(1-4]\) Distance An airplane flies \(1,260\) miles in 3 hours. How far will it fly in 5 hours?

Step-by-Step Solution

Answer

The airplane will fly 2,100 miles in 5 hours.

Step 1: Understand the Problem

We need to find out how far the airplane will fly in 5 hours, given that it flies 1,260 miles in 3 hours.

Step 2: Translate the Word Problem into a Proportion

Set up the proportion by comparing the distance flown to the time taken: \( \frac{1260\text{ miles}}{3\text{ hours}} = \frac{x\text{ miles}}{5\text{ hours}} \), where \( x \) represents the distance flown in 5 hours.

Step 3: Solve the Proportion

To solve for \( x \), cross-multiply the terms of the proportion: \( 3x = 1260 \times 5 \).

Step 4: Perform the Calculation

Calculate \( 1260 \times 5 = 6300 \). Then divide both sides by 3 to find \( x \): \( x = \frac{6300}{3} = 2100 \).

Step 5: Interpret the Result

The airplane will fly 2,100 miles in 5 hours based on our proportion.

Key Concepts

Understanding Word Problems

Word problems in math can sometimes feel intimidating, but they become manageable when broken down into simpler parts. The first step is to thoroughly understand the problem statement. This typically involves identifying what is being asked and determining the information you already have. For instance, in our example, we know the speed of the airplane in miles per hour based on a given distance and time. We need to find how far it would go in a different time span. Once you've gathered and understood the details, the next step is to formulate these into a mathematical expression, often a proportion, to proceed with the solution.
One strategy is to:

• Identify the quantities involved.
• Decipher the relationships between these quantities.
• Translate these insights into equations or proportions.

Breaking down the word problem into smaller tasks can significantly simplify the problem-solving process.

Solving Distance Problems

Distance problems typically involve the relationship between distance, speed, and time – a fundamental concept in physics and daily life applications. In such problems, distance traveled can often be determined by multiplying the speed (how fast something is moving) by the time spent moving.
For example, using the formula:

• Distance = Speed × Time

In the provided exercise example, the problem gives the distance the airplane flew in a certain timeframe and asks for the distance at a different time frame. It's crucial to maintain consistency in units (miles, hours, etc.) to ensure accurate calculations. Structuring information effectively allows setting up a proportion, which will assist in finding the unknown quantities, like our distance flown in 5 hours.
Understanding these relationships converts complex distance problems into well-structured equations that are less daunting.

Utilizing Cross-Multiplication

Cross-multiplication is a valuable tool when dealing with proportions because it allows us to find unknowns efficiently. A proportion is essentially an equation that states that two ratios are equivalent. When expressed in the form of fractions \[\frac{a}{b} = \frac{c}{d}\]where a and d are terms from one side and b and c are from the other, cross-multiplication involves multiplying across the equals sign:

\(a \cdot d = b \cdot c\)

This clears the proportion of fractions and makes solving for the variable straightforward. In our example, we use this method to equate the known distance-to-time ratio with the unknown one, allowing us to solve for the missing distance.

• Multiply diagonally across the proportion.
• Set the products equal to each other.
• Solve the resulting equation for the unknown variable.

Cross-multiplication not only simplifies solving proportions but also enhances computational accuracy and reduces calculation steps.