Problem 23 - Elementary Algebra Within Reach

Question

Find a ratio that compares the relative sizes of the quantities. (Use the same units of measurement for both quantities.) 75 centimeters to 2 meters

Step-by-Step Solution

Answer

The ratio comparing 75 centimeters to 2 meters is 3:8.

Step 1 - Convert meters to centimeters

First, convert the measurement from meters to centimeters. 1 meter is equal to 100 centimeters, thus 2 meters is equal to \(2 \times 100 = 200 \) centimeters.

Step 2 - Calculate the ratio

Now calculate the ratio by dividing the first quantity (75 cm) by the second quantity (200 cm). This gives \( \frac{75}{200} \).

Step 3 - Simplify the ratio

Simplify the ratio \( \frac{75}{200} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 25. This gives \( \frac{75 \div 25}{200 \div 25} = \frac{3}{8} \).

Key Concepts

Unit Conversion Basics

Whenever you compare two quantities using a ratio, it's crucial to ensure both quantities are in the same unit. This process is called unit conversion in mathematics. Let's explore how it works with an example:Suppose you want to compare 75 centimeters and 2 meters. Notice how these measurements are in different units—centimeters and meters. You need to convert one measurement to match the other for an accurate comparison.A simple way to convert units is by using a conversion factor. For example:

• 1 meter = 100 centimeters

To convert 2 meters into centimeters, multiply by the conversion factor: \(2 \times 100 = 200\) centimeters.Now, both quantities are in centimeters (75 cm and 200 cm), and you can proceed with finding the ratio.

The Art of Simplifying Ratios

Simplifying ratios is an essential skill in mathematics. It helps to express the ratio in the simplest form, making it easier to understand and interpret. Let's break down the process:Once you have comparable quantities (like 75 cm to 200 cm in our example), you can form a ratio, \( \frac{75}{200} \).To simplify, find the greatest common divisor (GCD) of the numerator and the denominator. For 75 and 200, the GCD is 25. You simplify the ratio by dividing both terms by their GCD:\[\frac{75 \div 25}{200 \div 25} = \frac{3}{8}\]Now the ratio is in its simplest form. This simple form, \(\frac{3}{8}\), tells us that for every 3 units of the first quantity, there are 8 units of the second quantity, presenting a clearer picture of their relation.

Problem Solving Strategies in Mathematics

Mathematics problem solving involves breaking down complex tasks into simpler, manageable steps. This strategy is valuable not only in simplifying ratios but in many math problems. To effectively solve problems:

• Identify what's being asked. This means understanding the goal of the problem.
• Ensure all quantities are comparable. Convert units if necessary, as shown in the unit conversion example above.
• Simplify expressions to their simplest form, as demonstrated in simplifying ratios.
• Check your work. Re-review each step to ensure accuracy and understanding.

Using these methods helps in organizing thoughts, reducing errors, and solving math problems more efficiently.