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Chapter 7: Problem 29

Write the simplified fractional form of each ratio or rate. 12 to 5.

Short Answer

Expert verified
The simplified form of the ratio 12 to 5 is \( \frac{12}{5} \).

Step by step solution

01

Understanding the Ratio

The given ratio is '12 to 5', which is normally written as the fraction \( \frac{12}{5} \). This fraction represents the relationship between the two numbers.
02

Simplifying the Fraction

In this step, we need to ensure the fraction \( \frac{12}{5} \) is in its simplest form.1. Check if the numerator (12) and the denominator (5) have any common factors other than 1.2. The factors of 12 are 1, 2, 3, 4, 6, 12.3. The factors of 5 are 1, 5.4. Since they don't share any common factors other than 1, the fraction \( \frac{12}{5} \) is already in its simplest form.

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Key Concepts

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

Understanding Ratios
A ratio represents a relationship between two numbers, showing how many times one value can contain another. For instance, if you have a ratio of 12 to 5, this means for every 12 of one quantity, there are 5 of another. Ratios can provide valuable insight into the relative sizes of two quantities and indicate a proportion.
Mathematically, ratios can be expressed as fractions, so the ratio of 12 to 5 is expressed as \( \frac{12}{5} \). Representing ratios as fractions aids in calculations and simplifications. This fractional form is crucial when comparing ratios or converting them into various formats, such as percentages or decimals.
Numerator and Denominator
In a fraction, the top number is called the numerator and the bottom number is called the denominator. Understanding these terms is key to simplifying ratios. For the fraction \( \frac{12}{5} \):
  • 12 is the numerator, indicating the number of parts considered.
  • 5 is the denominator, showing the total number of equal parts.
The numerator and denominator are essential in expressing the division of the described quantities. When we simplify fractions, we look for the greatest common factor between these two numbers. However, in \( \frac{12}{5} \), there are no common factors other than 1, so the fraction is already in its simplest form.
Finding Common Factors
To simplify a fraction, it's important to detect the common factors of its numerator and denominator. Common factors are integers that can evenly divide both the numerator and denominator without leaving a remainder.

For the fraction \( \frac{12}{5} \):
  • The factors of 12 are: 1, 2, 3, 4, 6, 12.
  • The factors of 5 are: 1, 5.
Since 12 and 5 share no common factors other than 1, the fraction cannot be simplified further. Identifying common factors helps to reduce fractions, making them easier to work with.
Remember, the greatest common factor is used to divide both the numerator and the denominator, leading to the simplest form of a fraction. If no greater common factor exists, the fraction is in its simplest form as is the case with \( \frac{12}{5} \).

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