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Chapter 3: Problem 1

Write the ratio as a fraction in simplest form. 36 to 9

Short Answer

Expert verified
The ratio 36 to 9 in simplest form is 4.

Step by step solution

01

Understand the Ratio

A ratio represents a comparison between two or more quantities. Here, we have a ratio of 36 to 9. This can be written as \(\frac{36}{9}\).
02

Simplify the Fraction

To simplify the fraction, divide both the numerator and denominator by their greatest common divisor (GCD). The GCD of 36 and 9 is 9. So, \(\frac{36}{9} = \frac{36 \div 9}{9 \div 9} = \frac{4}{1}\).
03

Write the Final Answer

The ratio of 36 to 9 in its simplest form as a fraction is \(\frac{4}{1}\) or simply 4.

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Key Concepts

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

Understanding the Greatest Common Divisor (GCD)
The greatest common divisor (GCD) is a key concept in simplifying fractions and ratios. It is the largest positive integer that divides two or more numbers without leaving a remainder. For example, in our exercise, we need to simplify the ratio 36 to 9. To do this, we first find the GCD of 36 and 9.
  • To find the GCD, list the factors of each number.
  • The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
  • The factors of 9 are 1, 3, and 9.
  • Notice that 9 is the largest number that appears in both lists.
So, the GCD of 36 and 9 is 9. Knowing the GCD allows us to simplify fractions by dividing both the numerator and the denominator by this common factor. This step is crucial in making ratios simpler and more comprehensible.
Breaking Down Fractions
A fraction represents a part of a whole or, in the context of ratios, a part-to-part comparison between two quantities. In the exercise, the ratio 36 to 9 can be written as the fraction \( \frac{36}{9} \). This fraction means that for every 9 units of the second quantity, there are 36 units of the first quantity.
Understanding fractions involves looking at its two main components:
  • The numerator, which is the top number, indicating the number of parts being considered. Here, it is 36.
  • The denominator, which is the bottom number, showing the total number of equal parts. Here, it is 9.
By understanding fractions in this way, we make the process of manipulating and simplifying them more intuitive. This sets the stage for effective ratio simplification.
Simplifying Ratios: The Process
Ratio simplification is a method used to express the relationship between two numbers in the simplest form. Once we have a ratio expressed as a fraction, like \( \frac{36}{9} \), the goal is to simplify it using the greatest common divisor (GCD).
Here's how you simplify:
  • Identify the GCD of both the numerator and the denominator. As earlier identified, the GCD is 9.
  • Divide both the numerator and the denominator by this GCD to simplify the fraction.
  • For our example: \( \frac{36 \div 9}{9 \div 9} = \frac{4}{1} \).
After simplification, the fraction \( \frac{4}{1} \) shows the most simplified form of the original ratio, which effectively communicates the same relationship as the original larger numbers, but in a cleaner and more understandable manner.

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