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Chapter 1: Problem 27

Simplify by factorization and cancellation:\(\frac{21}{49}\)

Short Answer

Expert verified
\(\frac{3}{7}\)

Step by step solution

01

Identify Common Factors

To simplify the fraction \(\frac{21}{49}\), begin by identifying the greatest common factor (GCF) of the numerator 21 and the denominator 49. Both numbers are divisible by 7.
02

Divide the Numerator by the GCF

Divide the numerator 21 by the GCF, which is 7. This results in \(\frac{21}{7} = 3\).
03

Divide the Denominator by the GCF

Next, divide the denominator 49 by the GCF, which is 7. This results in \(\frac{49}{7} = 7\).
04

Write the Simplified Fraction

Combine the results from Steps 2 and 3 to rewrite the fraction. The simplified fraction is \(\frac{3}{7}\).

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor, or GCF, is the largest number that can divide two numbers without leaving any remainder. It's like finding the biggest common denominator among a set of numbers. When simplifying fractions, identifying the GCF is a crucial first step.
  • Consider the numbers 21 and 49 in our example.
  • Both numbers are divisible by 7.
  • Thus, 7 is the GCF.
Recognizing the GCF helps us reduce a fraction to its simplest form. It's done by dividing both the numerator and the denominator by their GCF. Using the GCF ensures that we don't miss any common factors, leading to the most simplified version of the fraction.
Fraction Simplification
Fraction simplification is the process of reducing fractions to their simplest form. This is done when neither the numerator nor the denominator has any common factors left, except for 1. The goal is to express the fraction in the simplest way possible, making mathematical operations easier.
  • Step 1: Find the GCF of both the numerator and denominator.
  • Step 2: Divide both the numerator and denominator by the GCF.
Let's simplify the fraction \(\frac{21}{49}\):
  • Numerator: 21 divided by 7 gives 3.
  • Denominator: 49 divided by 7 gives 7.
Now, the fraction becomes \(\frac{3}{7}\). Simplifying fractions not only makes equations simpler but also helps in gaining a deeper understanding of the relationship between numbers.
Mathematics Education
In mathematics education, learning how to simplify fractions is a vital skill. It is part of a deeper understanding of number relationships, mathematics operations, and problem-solving. Simplifying fractions enhances critical thinking and analytical skills.
  • It helps students engage with more complex mathematical concepts.
  • By practicing fraction simplification, students start to see the hidden patterns in numbers.
  • This aids in developing a confidence in manipulating numbers and fractions.
Great math education allows students to make connections between various mathematical concepts, fostering a love and understanding of numbers that extends beyond the classroom.

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Most popular questions from this chapter

Express as decimal fractions:\(\frac{5}{32}\)

Evaluate:\(4 \div 2 \times 7-2\)

Simplify by factorization and cancellation:\(\frac{63}{294}\)

Use the rules of rounding to give each of the following to.8, \(7,6,5,4,3,2\) and 1 significant figures:\(\sqrt{2}=1.414213562373\)

Factorize in prime numbers:810
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