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Chapter 6: Problem 7

Determine whether the proportion is true or not true. $$\frac{45}{135}=\frac{3}{9}$$

Short Answer

Expert verified
The proportion is true as both fractions equivalent to \(\frac{1}{3}\) when simplified to their lowest terms.

Step by step solution

01

Simplify the fractions

Simplify each of the fractions to their lowest terms. This can be done by dividing the numerator and the denominator by their greatest common factor (GCF). The GCF of 45 and 135 is 45, so if we divide the numerator and the denominator of the first fraction by 45, we get: \(\frac{45}{135} = \frac{45 \div 45}{135 \div 45} = \frac{1}{3}\). The GCF of 3 and 9 is 3, so if we divide the numerator and the denominator of the second fraction by 3, we get: \(\frac{3}{9} = \frac{3 \div 3}{9 \div 3} = \frac{1}{3}\).
02

Compare the simplified fractions

Now that we have simplified both fractions, we can compare them. Since \(\frac{1}{3}\) is equal to \(\frac{1}{3}\), it means that \(\frac{45}{135}\) is proportional to \(\frac{3}{9}\).

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Key Concepts

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

Fractions
Fractions are a fundamental concept in mathematics used to represent parts of a whole. They consist of two numbers: a numerator on top and a denominator on the bottom. For example, in the fraction \(\frac{45}{135}\), 45 is the numerator, and 135 is the denominator.

Fractions can represent quantities more precisely than decimals, especially when dealing with exact values of parts.
  • The numerator reflects how many parts we have.
  • The denominator indicates the total number of equal parts.
Understanding fractions is crucial as they play an important role in various areas, including measurement, probability, and algebra.

This knowledge also supports concepts like division and multiplication of whole numbers and decimals.
Greatest Common Factor
The greatest common factor (GCF) is the largest number that divides two or more integers without leaving a remainder. Finding the GCF is essential when simplifying fractions to their lowest terms.

Take the numbers 45 and 135. The GCF of these numbers is 45 because it is the highest number that can divide both without leaving a remainder. Similarly, the GCF of 3 and 9 is 3.
  • To find the GCF, list the factors of each number.
  • Identify the largest factor shared by both numbers.
Using the GCF in equations helps simplify fractions, making them easier to compare or use in further calculations.
Simplification
Simplifying a fraction involves reducing it to its simplest form. This is done by dividing its numerator and denominator by their greatest common factor.

Let's simplify \(\frac{45}{135}\):
  • Find the GCF, which is 45.
  • Divide both the numerator and the denominator by 45.
  • The simplified form is \(\frac{1}{3}\).
The same process applies to \(\frac{3}{9}\), where the GCF is 3, resulting in \(\frac{1}{3}\).

Simplification reveals the true value of the fraction, removing any unnecessary complexity. It allows effortless comparison between different fractions by presenting them in their most straightforward form.
Mathematical Comparison
Comparing mathematical quantities often involves simplifying them to a common form. With fractions, this is normally done by reducing them to their lowest terms and observing if they are equal.

In the exercise, \(\frac{45}{135}\) simplifies to \(\frac{1}{3}\) and \(\frac{3}{9}\) also simplifies to \(\frac{1}{3}\). Since both fractions reduce to the same value, they are proportional, and the comparison shows they are equal.
  • Simplification helps identify equivalence between fractions.
  • This process is essential for solving proportion problems.
Grasping these comparison techniques is vital for understanding broader concepts in mathematics, such as solving equations and assessing ratios.

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Most popular questions from this chapter

Solve. Use the proportion method. What is \(87.4 \%\) of \(225 ?\)

Solve. Use the basic percent equation. Find \(10.7 \%\) of 485

In a survey conducted by Opinion Research Corp. for Lloyd's Barbeque Co., people were asked to name their favorite barbeque side dishes. \(38 \%\) named com on the cob, \(35 \%\) named cole slaw, \(11 \%\) named corn bread, and \(10 \%\) named fries. What percent of those surveyed named something other than corn on the cob, cole slaw, corn bread, or fries? (IMAGE CAN NOT COPY)

A building contractor estimates that 5 overhead lights are needed for every \(400 \mathrm{ft}^{2}\) of office space. Using this estimate, how many light fixtures are necessary for an office building of \(35,000 \mathrm{ft}^{2} ?\)

Write as a percent. Write the remainder in fractional form. $$\frac{3}{7}$$
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