Problem 1 - 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 A woman drives her car 235 miles in 5 hours. At this rate how far will she travel in 7 hours?

Step-by-Step Solution

Answer

She will travel 329 miles in 7 hours.

Step 1: Understand the Problem

We need to find out how far the woman will travel in 7 hours if she maintains a constant speed. We are given that she traveled 235 miles in 5 hours.

Step 2: Define the Proportion

The distance traveled is directly proportional to time, meaning we can set up the proportion \( \frac{235}{5} = \frac{x}{7} \), where \( x \) is the distance traveled in 7 hours.

Step 3: Solve the Proportion

Cross-multiply to solve for \( x \): \( 235 \times 7 = 5 \times x \). This gives us \( 1645 = 5x \).

Step 4: Calculate the Result

Divide both sides of the equation by 5 to isolate \( x \): \( x = \frac{1645}{5} \). Thus, \( x = 329 \).

Step 5: Write the Final Answer

The woman will travel 329 miles in 7 hours at the same speed.

Key Concepts

Distance Rate Problems

Distance rate problems like the one described here are common types of word problems in math. They deal with the relationship between distance, rate (or speed), and time, and are crucial for understanding how these three factors interact.

In this particular exercise, you're asked to determine how far someone can travel given a constant speed over a certain period. The essential equation connecting these is:

• Distance = Rate × Time

This equation is the foundation of solving any distance rate problem. It tells us that if you know two of these variables, you can easily find the third. Here, you know the rate (distance per time, calculated as 235 miles in 5 hours) and the time (7 hours). Therefore, you can calculate the distance the woman will travel.

By setting up the proportion, \[\frac{235\text{ miles}}{5\text{ hours}} = \frac{x\text{ miles}}{7\text{ hours}},\]you essentially use this foundational formula to find your unknown variable \(x\).

Cross-Multiplication

Cross-multiplication is a mathematical technique that's particularly useful in solving proportions. When you have an equation of the form \(\frac{a}{b} = \frac{c}{d}\), cross-multiplication allows you to eliminate the fractions, making it easier to solve for the unknown variable.

Here's how it works:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Set this product equal to the product of the denominator of the first fraction and the numerator of the second fraction.

For example, in this exercise, you start with the proportion:\[\frac{235}{5} = \frac{x}{7}.\]Next, you cross-multiply:\[235 \times 7 = 5 \times x.\]This simplifies your equation to\[1645 = 5x.\]Now, solving for \(x\) becomes a straightforward process, as you only need to divide both sides by 5.

Word Problem Translation

Word problems can often seem daunting at first glance, but breaking them down into smaller, manageable components can simplify the process significantly. Translating word problems into mathematical equations or expressions is a skill that can be developed over time.

Here’s a simple approach to tackle word problems like the one given:

• Carefully read the problem to understand what's being asked.
• Identify the known and unknown variables.
• Translate the words into a math equation, often using known formulas, like Distance = Rate × Time in distance problems.

In this problem, you start by identifying that you know the total distance (235 miles) and time (5 hours) based on the woman's previous trip. You are tasked to find out how far she will go in 7 hours, assuming the rate remains constant.

By translating these pieces of information into a proportion, you've effectively moved from a word problem to something more tangible and solvable:\[\frac{235}{5} = \frac{x}{7}.\]This approach empowers you to tackle complex word problems by systematically decoding their elements and transforming them into solvable math problems.