Problem 5 - Algebra 2 and Trigonometry

Question

Write each ratio in simplest form. \(3 : 18\)

Step-by-Step Solution

Answer

The ratio \(3 : 18\) in simplest form is \(1 : 6\).

Step 1: Understanding the Ratio

A ratio is a comparison of two numbers or quantities. Here, the ratio is given as \(3 : 18\), which can also be expressed as the fraction \(\frac{3}{18}\).

Step 2: Determine the Greatest Common Divisor (GCD)

To simplify the ratio, we need to find the greatest common divisor of 3 and 18. The divisors of 3 are \(1, 3\). The divisors of 18 are \(1, 2, 3, 6, 9, 18\). The greatest common divisor is \(3\).

Step 3: Simplifying the Fraction

We divide both the numerator and the denominator of the fraction \(\frac{3}{18}\) by their greatest common divisor, which is \(3\). So, \(\frac{3 \div 3}{18 \div 3} = \frac{1}{6}\).

Step 4: Writing the Simplified Ratio

The simplified form of the ratio \(3 : 18\) is represented by the fraction \(\frac{1}{6}\) or simply the ratio \(1 : 6\).

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental concept in mathematics that helps to simplify numbers neatly. It stands for the largest positive integer that divides two or more numbers without leaving a remainder. In the context of ratios, knowing how to find the GCD allows us to break down ratios into their simplest form.
To find the GCD of two numbers like 3 and 18, we need to list out the divisors of each. The divisors of 3 are 1 and 3. For 18, the divisors include 1, 2, 3, 6, 9, and 18. Among these divisors, 3 is the largest number common to both lists, making it the GCD.
Finding the GCD helps tremendously when simplifying ratios and fractions, ensuring we can reduce them to the simplest values possible while maintaining their underlying relationship. This skill is also beneficial in many practical situations, such as cooking measurements or solving problems involving proportions.

Fraction Simplification

Fraction simplification involves reducing fractions to their lowest terms, which is an important skill everyone should master. This process makes fractions easier to understand and compare. To simplify a fraction, divide both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD).
Let's return to the fraction \(rac{3}{18}\). We already know from our GCD calculation that 3 is the GCD for 3 and 18. We divide both the numerator and the denominator by this number: \[ \frac{3 \div 3}{18 \div 3} = \frac{1}{6}. \]
This results in \(\frac{1}{6}\), which is the fraction in its simplest form. By simplifying the fraction, we maintain the same value and simplify the comparison. Fraction simplification helps in various mathematical applications, from solving equations to interpreting data more effectively.

Ratio Concepts

Ratios are an essential concept that appear in mathematics and day-to-day life. They compare two quantities to show their relative sizes. Understanding how to simplify ratios makes them easier to work with, especially when trying to compare different ratios or understand proportional relationships.
Converting a ratio like 3:18 into a simpler form is straightforward. The first step is expressing the ratio as a fraction (\(\frac{3}{18}\)), after which we simplify it using the GCD method described earlier.
By dividing both parts of the ratio by the GCD, we transform 3:18 into 1:6. This simplified ratio represents the same relationship but in a more digestible manner. Grasping ratio concepts is extremely beneficial when solving problems related to rate, scale, probability, and many everyday situations where a proportionate comparison is involved.