Problem 50 Simplify. $$ \frac{75}{45} ... [FREE SOLUTION]

Chapter 1: Problem 50

Simplify. $$ \frac{75}{45} $$

Short Answer

Expert verified

$ \frac{5}{3} $

Step by step solution

Identify the Greatest Common Divisor (GCD)

Find the largest number that divides both the numerator (75) and the denominator (45) without leaving a remainder. The GCD of 75 and 45 can be found by factorizing each number: 75 = 3 * 5^2 and 45 = 3^2 * 5. The common factors are 3 and 5.

Divide Numerator and Denominator by the GCD

The greatest common divisor of 75 and 45 is 15. Divide the numerator and the denominator by this GCD. \[\frac{75 梅 15}{45 梅 15} = \frac{5}{3}\]

Simplified Fraction

The fraction simplified by dividing both the numerator and the denominator by their greatest common divisor (15) is $ \frac{5}{3} $. This process gives us the final simplified form.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Greatest Common Divisor

The Greatest Common Divisor (GCD) is essential when simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. Without the GCD, we wouldn't know the most accurate way to simplify a fraction.
To find the GCD of two numbers, follow these steps:

Factorize both numbers into their prime factors.
Identify the common factors.
Multiply the common factors to find the GCD.

For example, in the problem $ \frac{75}{45} $, we factorized 75 into $3 \times 5^2$ and 45 into $3^2 \times 5$. The common factors are 3 and 5, giving us a GCD of 15. Dividing both the numerator and the denominator by this GCD simplifies the fraction.

Numerator and Denominator

In a fraction, the numerator is the number on the top, and the denominator is the number on the bottom. The numerator represents how many parts we have, while the denominator shows how many parts make a whole.
For example, in $ \frac{75}{45} $, 75 is the numerator and 45 is the denominator. Understanding these positions helps us apply the correct steps in fraction simplification.
When we simplify a fraction, we divide both the numerator and the denominator by the same number to reduce it to its simplest form. Ensure you always divide both terms by their GCD to maintain the fraction's value.

Factorization

Factorization is breaking down a number into its prime factors. These are numbers that can only be divided by 1 and themselves. Factorization is crucial in finding the GCD for simplifying fractions.
For instance, to factorize 75, we find the numbers that multiply to give 75: 3 and $5^2$. Similarly, 45 factorizes into $3^2$ and 5. Identifying common factors, such as 3 and 5 in this example, allows us to determine the GCD.
Once we've factorized both the numerator and the denominator, we proceed to simplify the fraction by dividing both by their GCD.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Simplify. $$ \frac{75}{45} $$

Short Answer

Step by step solution

Identify the Greatest Common Divisor (GCD)

Divide Numerator and Denominator by the GCD

Simplified Fraction

Key Concepts

Greatest Common Divisor

Numerator and Denominator

Factorization

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Probability and Statistics

Pure Maths

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.