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Chapter 6: Problem 24

Write the ratio in lowest terms. \(75 \phi\) to \(100 \ \phi\)

Short Answer

Expert verified
The ratio in lowest terms is 3:4.

Step by step solution

01

Identify the Given Values

The ratio given is from 75 to 100. These represent the two numbers you are working with.
02

Write the Ratio as a Fraction

Express the ratio 75 to 100 as a fraction: \ \[ \frac{75}{100} \ \]
03

Find the Greatest Common Divisor (GCD)

Determine the GCD of 75 and 100. Factor 75: 1, 3, 5, 15, 25, 75 Factor 100: 1, 2, 4, 5, 10, 20, 25, 50, 100 The GCD of 75 and 100 is 25.
04

Simplify the Fraction

Divide both the numerator and the denominator by their GCD. \ \[ \frac{75 \div 25}{100 \div 25} = \frac{3}{4} \ \]
05

Write the Simplified Ratio

The simplified ratio is 3:4.

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Key Concepts

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

Understanding Ratios
A ratio is a way to compare two quantities. We often use ratios to show the relationship between two numbers. For example, in the exercise, we compared 75 to 100. Ratios can be written in several ways, such as using a colon (75:100) or as a fraction \(\frac{75}{100}\). Both methods tell us the same thing: how much one number relates to another. We often simplify ratios to make them easier to understand.
Fractions and Their Role in Ratios
Fractions are essential in simplifying ratios. They help us see the relationship between the numbers more clearly. When we wrote the ratio 75 to 100 as the fraction \(\frac{75}{100}\), it allowed us to simplify it easily. We can simplify a fraction by dividing the numerator and the denominator by their Greatest Common Divisor (GCD). For instance, dividing both 75 and 100 by their GCD of 25 gave us the simplified fraction \(\frac{3}{4}\), leading to the ratio 3:4.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) is the largest number that can divide two or more numbers without leaving a remainder. Finding the GCD is crucial for simplifying ratios and fractions. To find the GCD of two numbers like 75 and 100, we first identify their factors. The factors of 75 are:
  • 1
  • 3
  • 5
  • 15
  • 25
  • 75
The factors of 100 are:
  • 1
  • 2
  • 4
  • 5
  • 10
  • 20
  • 25
  • 50
  • 100
The common factors are 1, 5, and 25, with the greatest being 25. Once we find the GCD, we divide the numerator and denominator of our fraction by this number to simplify it. For 75 and 100, dividing both by 25 simplifies our fraction to \(\frac{3}{4}\), and thus the simplified ratio is 3:4.

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