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Chapter 6: Problem 11

Determine whether each proportion is true or false. $$ \frac{15}{9}=\frac{5}{3} $$

Short Answer

Expert verified
The proportion \( \frac{15}{9} = \frac{5}{3} \) is true.

Step by step solution

01

Understand the Proportions

We are asked to determine if the two ratios \( \frac{15}{9} \) and \( \frac{5}{3} \) are equivalent. When two fractions are equal, they form a true proportion. Our task is to verify this.
02

Simplify the First Fraction

Let's simplify the first fraction \( \frac{15}{9} \). Find the greatest common divisor (GCD) of 15 and 9, which is 3. Divide both the numerator and the denominator by 3. \[ \frac{15 \div 3}{9 \div 3} = \frac{5}{3} \]
03

Compare the Simplified Fractions

Now that the first fraction \( \frac{15}{9} \) simplifies to \( \frac{5}{3} \), compare it with the second fraction \( \frac{5}{3} \). Since they are the same, the original proportion 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 parts of a whole. A fraction consists of two parts: the numerator and the denominator. The numerator is the top number and shows how many parts of a certain size you have. The denominator, the bottom number, tells you how many of those parts make up a whole.
For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator. This fraction tells you that you have 3 parts out of 4 equal parts that make a whole. Understanding the role of both the numerator and denominator is essential when working with fractions because it helps you determine the size of the part you're dealing with relative to the whole.
  • Fractions express ratios or comparisons between two quantities.
  • They can show division since \( \frac{3}{4} \) can be interpreted as 3 divided by 4.
  • In various scenarios, such as measuring ingredients for a recipe or representing data, fractions are incredibly useful.
Concept of Equivalent Fractions
Equivalent fractions are different fractions that represent the same part of a whole. You can create equivalent fractions by multiplying or dividing both the numerator and denominator of a fraction by the same number. This doesn't change the value of the fraction; you're just resizing it.
For instance, the fractions \( \frac{1}{2} \) and \( \frac{2}{4} \) are equivalent because when you multiply both the numerator and the denominator of \( \frac{1}{2} \) by 2, you get \( \frac{2}{4} \).
  • Equivalent fractions have a similar ratio between the numerator and denominator.
  • To determine if two fractions are equivalent, you can simplify them to see if they reduce to the same fraction.
  • This concept is fundamental in determining if two proportions are true, as in the exercise with \( \frac{15}{9} = \frac{5}{3} \).
Greatest Common Divisor (GCD)
The Greatest Common Divisor, often abbreviated as GCD, is the largest number that can evenly divide both the numerator and the denominator of a fraction without leaving a remainder. Using the GCD is a vital step in simplifying fractions.
For example, if we have the fraction \( \frac{15}{9} \), the GCD of 15 and 9 is 3. You find the GCD by identifying all the divisors of both numbers and selecting the largest shared one.
  • Using the GCD helps in reducing fractions to their simplest form by dividing both the numerator and denominator by this number.
  • Finding the GCD can involve prime factorization or using methods such as the Euclidean algorithm.
  • Once simplified, if two fractions are equal, they confirm a true proportion, just like \( \frac{15}{9} \) simplifies to \( \frac{5}{3} \).

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Most popular questions from this chapter

Find each unit rate. In the United States in \(2016,\) the total number of students enrolled in public schools was 50,400,000 . There are 98,300 public schools. Write a unit rate in students per school. Round to the nearest whole. (Source: National Center for Education Statistics)

A grocer will refuse a shipment of tomatoes if the ratio of bruised tomatoes to the total batch is at least 1 to \(10 .\) A sample is found to contain 3 bruised tomatoes and 33 good tomatoes. Determine whether the shipment should be refused.

For each proportion, solve for the variable. $$ \frac{z}{1150}=\frac{588}{483} $$

Describe a situation in which writing a proportion might solve a problem related to driving a car.

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