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Chapter 2: Problem 31

Determine whether each proportion is true or false. $$ \frac{5}{35}=\frac{8}{56} $$

Short Answer

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Step by step solution

01

Simplify the First Fraction

Simplify the fraction \( \frac{5}{35} \). Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 5. So, \( \frac{5}{35} = \frac{5 \div 5}{35 \div 5} = \frac{1}{7} \).
02

Simplify the Second Fraction

Simplify the fraction \( \frac{8}{56} \). Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 8. So, \( \frac{8}{56} = \frac{8 \div 8}{56 \div 8} = \frac{1}{7} \).
03

Compare Simplified Fractions

Now both fractions are simplified and can be compared. \( \frac{1}{7} = \frac{1}{7} \) which means the original proportion \( \frac{5}{35} = \frac{8}{56} \) is true.

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Key Concepts

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

Understanding Fractions
Fractions are a way to represent a part of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). The numerator shows how many equal parts we have, while the denominator shows how many equal parts make up a whole. For instance, in the fraction \( \frac{3}{4} \), 3 is the numerator, and 4 is the denominator. This fraction means we have 3 parts out of 4 equal parts.
Fractions are important in algebra because they allow us to represent and work with divisions that do not result in whole numbers. They are also used in proportions to show that two ratios are equivalent.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).
For example, consider the fraction \( \frac{8}{12} \). To simplify, we need to find the GCD of 8 and 12. The GCD is the largest number that divides both 8 and 12 without leaving a remainder. In this case, the GCD is 4.
So, we divide both the numerator and the denominator by 4:
\( \frac{8 \ \div 4}{12 \ \div 4} = \frac{2}{3} \)
The simplified form of \( \frac{8}{12} \) is \( \frac{2}{3} \).
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD), also known as the greatest common factor (GCF), is the largest positive integer that divides two or more integers without leaving a remainder. Finding the GCD is essential when simplifying fractions.
For instance, to find the GCD of 24 and 36, list the factors of each number:
  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

The common factors are 1, 2, 3, 4, 6, 12. The greatest of these is 12. So, the GCD of 24 and 36 is 12.
Using the GCD makes it easier to simplify fractions and work with proportions. In our example from the original exercise, we found the GCDs for \( \frac{5}{35} \) and \( \frac{8}{56} \) to simplify them.

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