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Chapter 3: Problem 38

Convert to a mixed numeral. $$ \frac{30}{9} $$

Short Answer

Expert verified
3\(\frac{1}{3}\)

Step by step solution

01

Divide the numerator by the denominator

First, divide 30 by 9 to determine how many whole numbers fit into the fraction. Using long division, 30 divided by 9 is 3 with a remainder of 3.
02

Write the mixed numeral

Since 30 divided by 9 equals 3 with a remainder of 3, the mixed numeral is written with the whole number 3 and the fraction part as the remainder over the original denominator. So, the mixed numeral is: \[ 3 \frac{3}{9} \]
03

Simplify the fraction

The fraction \( \frac{3}{9} \) can be simplified. Both 3 and 9 are divisible by their greatest common divisor, which is 3. Dividing the numerator and the denominator by 3, the fraction simplifies to:\[ \frac{3 \div 3}{9 \div 3} = \frac{1}{3} \]Thus, the simplified mixed numeral is:\[ 3 \frac{1}{3} \]

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Key Concepts

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

long division
Long division is a method for dividing large numbers. It helps break down the problem into smaller, more manageable parts. When you have a fraction like \(\frac{30}{9}\), the numerator (30) is divided by the denominator (9). Here are the steps:

  • Divide: Determine how many times 9 fits into 30. The answer is 3, because 9 times 3 is 27.
  • Multiply: Multiply the quotient (3) by the divisor (9). This gives you 27.
  • Subtract: Subtract 27 from the original numerator (30). This leaves a remainder of 3.

By using long division, we've determined that 30 divided by 9 is 3 with a remainder of 3. Now, we can use this information to write the fraction as a mixed number.
simplifying fractions
Simplifying fractions helps express them in their simplest form. When a fraction is simplified, the numerator and denominator are both divided by their greatest common divisor (GCD). For example, let's look at \(\frac{3}{9}\):
  • Identify the GCD: The greatest common divisor of 3 and 9 is 3.
  • Divide: Divide both the numerator and denominator by their GCD. This gives us \(\frac{3 \div 3}{9 \div 3} = \frac{1}{3}\).

By carrying out these steps, we simplify \(\frac{3}{9}\) to \(\frac{1}{3}\). This process makes the fraction easier to understand and work with in equations.
greatest common divisor
The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator without leaving a remainder. Finding the GCD is crucial for simplifying fractions. Let's see how to find the GCD of 3 and 9:

  • List the factors of both numbers:
    - Factors of 3: 1, 3
    - Factors of 9: 1, 3, 9
  • Identify the largest common factor: In this case, both numbers share 1 and 3 as factors, but the largest is 3.

Thus, the GCD of 3 and 9 is 3. Using the GCD, we can simplify the fraction \(\frac{3}{9}\) to \(\frac{1}{3}\), and put our mixed number in its simplest form \(3 \frac{1}{3}\). Understanding the GCD is vital for fraction operations and simplification.

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Most popular questions from this chapter

Find the average of \(10 \frac{2}{3}\) and \(24 \frac{5}{6}\).

Simplify. $$ \frac{\frac{3}{8}}{\frac{11}{8}} $$

Simplify. $$ \frac{\frac{5}{18}}{1 \frac{2}{3}} $$

Simplify. $$ \frac{\frac{5}{9}-\frac{1}{6}}{\frac{2}{3}} $$

Simplify. $$ 5 \cdot 2^{2} \div 10 \quad[1.9 \mathrm{c}] $$
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