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Chapter 6: Problem 19

Write the ratio in lowest terms. \(8 \mathrm{m}\) to \(2 \mathrm{m}\)

Short Answer

Expert verified
4:1

Step by step solution

01

Write the ratio as a fraction

Express the given ratio in fraction form. Here, the ratio of 8 \text{m} to 2 \text{m} can be written as \( \dfrac{8 \text{m}}{2 \text{m}} \).
02

Cancel out the common units

The units 'm' (meters) in both the numerator and the denominator can be canceled out, simplifying the expression to \( \dfrac{8}{2} \).
03

Simplify the fraction

Divide both the numerator and the denominator by their greatest common divisor (GCD). For 8 and 2, the GCD is 2. So, \( \dfrac{8 \div 2}{2 \div 2} = \dfrac{4}{1} \).
04

Write the simplified ratio

The simplified ratio is then 4 to 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ratios
A ratio compares two quantities to show the relative size of one quantity to the other. Ratios can be written in three main forms: as a fraction, with a colon (e.g., 8:2), or using the word 'to' (e.g., 8 to 2). Ratios help us understand the relationship between different quantities. For example, in the exercise, the ratio 8 meters to 2 meters represents how much larger 8 meters is compared to 2 meters. To make ratios more understandable and easier to work with, we often simplify them.
Fractions
Fractions are a way to represent parts of a whole. They consist of a numerator (the top part) and a denominator (the bottom part). In our problem, we turned the ratio 8 meters to 2 meters into a fraction, giving us \(\frac{8 \text{m}}{2 \text{m}}\). Fractions help to visualize the division of quantities. When simplifying a ratio, converting it to a fraction allows us to perform easier arithmetic operations such as division. This conversion is the first step to simplification.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) is the largest number that evenly divides both the numerator and the denominator of a fraction. Finding the GCD is crucial in simplifying fractions because it helps reduce the fraction to its simplest form. In our example, the GCD of 8 and 2 is 2. Thus, we divide both 8 and 2 by 2, which simplifies \(\frac{8}{2}\) to \(\frac{4}{1}\). Calculating the GCD makes the numbers easier to work with and can reveal the simplest ratio between two quantities.
Unit Cancellation
Unit cancellation involves removing units that appear in both the numerator and the denominator of a fraction. This step simplifies the arithmetic in problems involving ratios with units. In our example, the units 'meters' appeared in both parts of the ratio 8 meters to 2 meters. By canceling out the 'meters,' we turned \(\frac{8 \text{m}}{2 \text{m}}\) into \(\frac{8}{2}\). Removing the units makes the mathematical process cleaner and focuses purely on the numerical relationship.

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