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Chapter 1: Problem 29

Simplify each fraction by reducing it to its lowest terms. $$\frac{10}{16}$$

Short Answer

Expert verified
The fraction \(\frac{10}{16}\) simplified to its lowest terms is \(\frac{5}{8}\).

Step by step solution

01

Identify the greatest common divisor

Identify the largest integer that can divide both 10 and 16 without a remainder. In this case, the greatest common divisor is 2.
02

Divide numerator and denominator by the greatest common divisor

Now, divide both the numerator (10) and the denominator (16) by 2. This gives \(\frac{10}{2}\) for the numerator and \(\frac{16}{2}\) for the denominator.
03

Simplify the fraction

Simplifying both \(\frac{10}{2}\) and \(\frac{16}{2}\) yields \(\frac{5}{8}\). This is the fraction \(\frac{10}{16}\) in its simplest form.

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Key Concepts

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

Greatest Common Divisor (GCD)
Understanding the Greatest Common Divisor (GCD) is vital when simplifying fractions. The GCD of two numbers is the largest integer that divides both numbers without leaving a remainder.

For instance, let's look at the numbers 10 and 16. The divisors of 10 are 1, 2, 5, and 10, while the divisors of 16 are 1, 2, 4, 8, and 16. The common divisors are 1 and 2, hence the GCD is 2. This is the largest number that fits into both 10 and 16 equally.

Finding the GCD is the first crucial step in reducing fractions because it shows us by how much we can 'scale down' both the numerator and the denominator, ensuring the fraction stays equivalent but becomes simpler.
Numerator and Denominator
Every fraction consists of two parts: the numerator and the denominator. The numerator, situated at the top, tells us how many parts we have. The denominator, at the bottom, tells us into how many parts the whole is divided.

In the fraction \(\frac{10}{16}\), 10 is the numerator, meaning 10 parts, and 16 is the denominator, indicating that the whole is divided into 16 equal parts. When simplifying, the goal is to reduce both of these numbers equally to get the simplest form of the fraction that still represents the same proportion of the whole.
Reducing Fractions to Lowest Terms
Reducing fractions to their lowest terms means making the numerator and the denominator as small as possible without changing the value of the fraction. This is achieved by dividing both the numerator and the denominator by their GCD.

Let's revisit our example with \(\frac{10}{16}\). We already identified the GCD as 2. So, by dividing both 10 and 16 by 2, we get \(\frac{5}{8}\). Now, neither 5 nor 8 can be divided by the same number other than 1, so \(\frac{5}{8}\) is the fraction in its lowest terms. This simpler fraction is easier to work with and visually more understandable, showing us the true ratio between the numerator and the denominator.

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Most popular questions from this chapter

Explain how to multiply two real numbers. Provide examples with your explanation.

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The quotient of \(-25\) and the sum of \(-21\) and 16

In Exercises \(139-142\), write an algebraic expression for the given English phrase. The value, in cents, of \(x\) nickels

From here on, each exercise set will contain three review exercises. It is essential to review previously covered topics to improve your understanding of the topics and to help maintain your mastery of the material. If you are not certain how to solve a review exercise, turn to the section and the worked example given in parentheses at the end of each exercise. Consider the set $$\\{-6,-\pi, 0,0, \overline{7}, \sqrt{3}, \sqrt{4}\\}$$ List all numbers from the set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. (Section 1.3 Example 5 )

Without using a number line, describe how to add two numbers with different signs. Give an example.
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