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Chapter 3: Problem 32

Convert to fraction notation. $$ 205 \frac{3}{14} $$

Short Answer

Expert verified
\( \frac{2873}{14} \)

Step by step solution

01

Identify the Mixed Number Components

The mixed number is given as \( 205 \frac{3}{14} \). Here, 205 is the whole number part, and \( \frac{3}{14} \) is the fractional part.
02

Convert the Whole Number to a Fraction

Convert the whole number 205 to a fraction. This is done by expressing it as \( \frac{205}{1} \).
03

Convert the Mixed Number to an Improper Fraction

To convert the mixed number \( 205 \frac{3}{14} \) to an improper fraction, multiply the whole number by the denominator of the fractional part and add the numerator of the fractional part. Thus, we compute: \[ 205 \times 14 + 3 = 2870 + 3 = 2873 \] So, the numerator of the improper fraction is 2873, and the denominator remains 14.
04

Write the Final Fraction

Combine the results from the previous step to form the improper fraction: \( \frac{2873}{14} \).

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Key Concepts

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

Mixed Number Conversion
A mixed number consists of a whole number and a fraction, like the one given in your exercise: 205 \( \frac{3}{14} \). To convert a mixed number to an improper fraction, you follow a few straightforward steps. Start by identifying the components of the mixed number. Here, 205 is the whole number and \( \frac{3}{14} \) is the fraction.
Next, convert the whole number to a fraction. This typically means placing it over 1, resulting in \( \frac{205}{1} \). Once this is done, you need to handle the mixed number conversion by calculating a new numerator. Multiply the whole number by the denominator of the fractional part and add the numerator of the fractional part:
Fraction Notation
Understanding fraction notation is crucial for working with mixed numbers and improper fractions. Fractions consist of two parts: a numerator (top number) and a denominator (bottom number). For example, in \( \frac{3}{14} \), 3 is the numerator, and 14 is the denominator. The denominator tells you how many equal parts make up a whole, and the numerator tells you how many of those parts are being considered. Mixed numbers combine whole numbers and fractions. For instance, 205 \( \frac{3}{14} \) combines the whole number 205 with the fraction \( \frac{3}{14} \). Breaking down these parts makes it easier to convert between different types of fractions.
Improper Fractions
An improper fraction has a numerator larger than its denominator. This means the fraction represents a value greater than 1. To convert a mixed number like 205 \( \frac{3}{14} \) to an improper fraction, you perform a series of multiplication and addition steps:
  • Multiply the whole number by the denominator: \( 205 \times 14 = 2870 \)
  • Add the result to the numerator: \( 2870 + 3 = 2873 \)
Now, you have your new numerator, 2873. The denominator remains the same as in the fractional part, which is 14. Therefore, the improper fraction is \( \frac{2873}{14} \). This fraction is called 'improper' because it鈥檚 larger than 1, a key feature distinguishing it from proper fractions, where the numerator is less than the denominator.

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