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Chapter 5: Problem 12

Simplify to lowest terms. $$\frac{-51}{85}$$

Short Answer

Expert verified
Question: Simplify the fraction $$\frac{-51}{85}$$ to its lowest terms. Answer: $$\frac{-3}{5}$$

Step by step solution

01

Find the GCD of 51 and 85

Let's start by finding the GCD of the two numbers 51 and 85. To do this, we can use the Euclidean Algorithm or prime factorization. In this case, we use prime factorization to find the GCD. 51 can be factored as 3 x 17, and 85 can be factored as 5 x 17. The GCD is the product of the common prime factors, hence GCD(51, 85) = 17.
02

Divide both numerator and denominator by GCD

Now that we have the GCD, we can divide both the numerator and denominator by 17. Our fraction becomes: $$\frac{-51}{85} = \frac{-1 \times 3 \times 17}{5 \times 17}$$ By dividing both the numerator and the denominator by 17, we get: $$\frac{-51}{85} = \frac{-1 \times 3}{5}$$ So the simplified fraction is: $$\frac{-51}{85} = \frac{-3}{5}$$

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Key Concepts

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

Greatest Common Divisor
The greatest common divisor, often abbreviated as GCD, is fundamental in simplifying fractions. It is the largest number that can evenly divide two or more numbers without leaving a remainder. Understanding how to find the GCD is crucial as it helps in reducing fractions to their simplest form.In our example, to simplify \( \frac{-51}{85} \), we first identify the GCD of 51 and 85.
  • We can determine the GCD by using either the Euclidean Algorithm or prime factorization.
  • By prime factorizing 51 and 85, we find the common factors shared by both numbers.
In this case, both 51 and 85 share the prime factor of 17, making it their GCD. By dividing the numerator and denominator of a fraction by the GCD, you effectively simplify it to the lowest terms.
Prime Factorization
Prime factorization involves breaking down a number into its smallest prime components. It helps us identify the shared factors between numbers, which is useful in computing the GCD.Consider the numbers in our fraction: - 51 can be expressed as \( 3 \times 17 \)- 85 can be broken into \( 5 \times 17 \)In this factorization, the number 17 is common. By recognizing common prime factors, you can determine the GCD efficiently. Prime factorization provides clarity and makes derivations methodical, ensuring that you're simplifying correctly each time.
Lowest Terms
When we talk about expressing a fraction in its lowest terms, we mean reducing it such that there are no common factors between the numerator and the denominator other than 1. This simplification makes the fraction easier to understand and work with.To achieve this, you:
  • First, identify the GCD of the numerator and the denominator.
  • Then, divide both parts of the fraction by this GCD.
In our example, the subtraction of 17 from both parts of \( \frac{-51}{85} \) results in \( \frac{-3}{5} \), a simplified fraction because no number other than 1 divides both 3 and 5. Understanding how to bring fractions to their lowest terms is essential for effective arithmetic operations and mathematical clarity.

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Most popular questions from this chapter

Add or subtract and write the resulting polynomial in descending order of degree. \(\left(5 y^{3}-\frac{4}{5} y^{2}+y-\frac{1}{6}\right)+\left(y^{3}+3 y^{2}-\frac{2}{3}\right)\)

Solve and check. $$-3 \frac{1}{2}+t=-4 \frac{1}{5}$$

Which unknown amount should be represented by the variable? Why? In a political poll, \(\frac{2}{5}\) of the respondents said that they would vote for a certain candidate. Of these respondents, \(\frac{1}{3}\) were men. What fraction of all respondents that voted for the candidate were women?

Solve. Write all answers in lowest terms. \(\frac{5}{6}\) of a company's employees live within a 15 -mile radius of the company. Of these employees, \(\frac{3}{4}\) live within a 10 -mile radius. What fraction of all employees live within a 10 -mile radius?

Add or subtract and write the resulting polynomial in descending order of degree. \(\left(2 x^{4}+\frac{3}{5} x^{2}-\frac{1}{8} x+1\right)-\left(6 x^{4}+\frac{1}{4} x^{2}-\frac{1}{2} x-\frac{1}{3}\right)\)
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