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Chapter 4: Problem 2

Simplify the fraction. $$-\frac{104}{36}$$

Short Answer

Expert verified
-26/9

Step by step solution

01

- Identify the common factors

First, identify the greatest common divisor (GCD) of the numerator (-104) and the denominator (36). The GCD of 104 and 36 is 4.
02

- Divide both numerator and denominator by the GCD

Next, divide both the numerator and the denominator by the GCD. \[ \frac{-104}{36} = \frac{-104 \text{ 梅 } 4}{36 \text{ 梅 } 4} \ = \frac{-26}{9} \]
03

- Simplified fraction

The fraction \(\frac{-26}{9}\) is in its simplest form. Thus, \(\frac{-104}{36}\) simplifies to \(\frac{-26}{9}\).

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Key Concepts

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

Understanding the Greatest Common Divisor (GCD)
The greatest common divisor (GCD) is a crucial concept in simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator.

Here's how to find the GCD:
  • List the factors of both numbers.
  • Identify the largest factor that appears in both lists.
For example, for 104 and 36:
  • Factors of 104: 1, 2, 4, 8, 13, 26, 52, 104
  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The largest common factor is 4, so the GCD of 104 and 36 is 4.
Understanding and using the GCD makes it easier to simplify fractions by reducing both the numerator and denominator by the same factor.
What is the Numerator?
The numerator is the top part of a fraction. It tells you how many parts of a whole you have. In our example, the numerator is -104.

When dealing with negative numerators, keep in mind:
  • It indicates a negative value.
  • Division by the GCD should account for the sign.
For instance, dividing -104 by the GCD of 4, gives us -26. So, the simplified numerator is -26.
What is the Denominator?
The denominator is the bottom part of a fraction. It shows into how many parts the whole is divided. In our example, the denominator is 36.

Remember these points about denominators:
  • The denominator must never be zero.
  • Division by the GCD helps simplify the fraction while keeping the denominator positive.
In our exercise, dividing 36 by the GCD of 4, results in 9. Hence, the simplified denominator is 9.
The Process of Fraction Simplification
Fraction simplification makes fractions easier to work with. Here's a quick guide:
  • Find the greatest common divisor (GCD) of the numerator and denominator.
  • Divide both the numerator and the denominator by this GCD.
  • The result is a simplified fraction.
Looking at our example:
  • Original fraction: \(-\frac{104}{36}\)
  • GCD: 4
  • Divided numbers: \(\frac{-104 \text{ 梅 } 4}{36 \text{ 梅 } 4} = -\frac{26}{9}\)
  • Simplified fraction: -\frac{26}{9}
This final fraction is easier to understand and work with in further calculations.

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Most popular questions from this chapter

Convert the improper fraction to a mixed number. $$-\frac{115}{7}$$

Simplify the fraction to lowest terms. Write the answer as a fraction or a whole number. $$\frac{77}{110}$$

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Jonathan and Jared both sold candy bars for a fundraiser. Jonathan sold 25 of his 35 candy bars, and Jared sold 24 of his 28 candy bars. a. What fractional part of his total number of candy bars did each person sell? b. Which person sold the greater fractional part?

Which expressions are equivalent to \(-\frac{4}{5} ?\) \(\begin{array}{lll}\text { a. } \frac{-4}{5} & \text { b. } \frac{-4}{-5} & \text { c. } \frac{4}{-5} & \text { d. }-\frac{5}{4}\end{array}\)
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