Problem 34 - Algebra 1: Concepts and Skills

Question

Write the improper fraction as a mixed number. 34\. \(\frac{3}{2}$$ 35 . \frac{7}{4}$$ 36 . \frac{11}{3}$$ 37 . \frac{13}{6}\) 38\. \(\frac{16}{5}\) 39\. \(\frac{21}{9}\) 40\. \(\frac{18}{4}\) 41\. \(\frac{15}{7}\) 42\. \(\frac{30}{8}\) 43\. \(\frac{54}{12}\) 44\. \(\frac{84}{36}\) \(45 \cdot \frac{20}{3}\)

Step-by-Step Solution

Answer

The mixed numbers for the given fractions are: \(\frac{3}{2}=1 \frac{1}{2}, \frac{7}{4}=1 \frac{3}{4}, \frac{11}{3}=3 \frac{2}{3}, \frac{13}{6}=2 \frac{1}{6}, \frac{16}{5}=3 \frac{1}{5}, \frac{21}{9}=2 \frac{3}{9}, \frac{18}{4}=4 \frac{2}{4}, \frac{15}{7}=2 \frac{1}{7}, \frac{30}{8}=3 \frac{6}{8}, \frac{54}{12}=4 \frac{6}{12}, \frac{84}{36}=2 \frac{12}{36}, \frac{20}{3}=6 \frac{2}{3}\)

Step 1: Convert Improper Fraction to Mixed Number

To convert each improper fraction to a mixed number, first divide the numerator by the denominator to get the whole number. The remainder will be the numerator of the fractional part. The denominator will remain the same. Let's start doing this for each fraction.

Step 2: Convert Fraction to Mixed Number for \(\frac{3}{2}\)

Divide 3 by 2 we get 1 with a remainder of 1, thus the mixed number is \(1 \frac{1}{2}\).

Step 3: Convert Fraction to Mixed Number for \(\frac{7}{4}\)

Divide 7 by 4 we get 1 with a remainder of 3, thus the mixed number is \(1 \frac{3}{4}\).

Step 4: Follow the same steps for remaining fractions

Continue to apply the same procedure for all other improper fractions to get their corresponding mixed numbers.

Key Concepts

Improper Fraction

An improper fraction is a type of fraction where the numerator (the top number) is larger than or equal to the denominator (the bottom number). This means that the value of the fraction is equal to or greater than one. Unlike a proper fraction, where the numerator is less than the denominator, an improper fraction represents a whole plus a part of the whole.

For example, the fraction \( \frac{5}{3} \) can be viewed as 3 wholes (which makes 1) plus 2 additional parts. To visualize improper fractions, imagine slicing a pizza into 3 pieces (denominator) and having 5 slices (numerator) on your plate. You have more than one whole pizza—this is the essence of an improper fraction.

Mixed Number

A mixed number combines a whole number with a proper fraction, representing a sum of the whole parts and the fractional part. In essence, it's a more intuitive way to represent quantities that contain whole amounts plus a fraction of that amount. Mixed numbers are very practical, especially when dealing with measurements in everyday life, like cooking or measuring heights and distances.

Let's consider the example of \( \frac{7}{4} \). As a mixed number, it would be written as \( 1 \frac{3}{4} \), indicating 1 whole plus \( \frac{3}{4} \) of another whole. This form makes it clear how much more than the whole unit you have.

Numerator and Denominator

Understanding Numerators and Denominators

In every fraction, there are two key components: the numerator and the denominator. The numerator, positioned above the fraction bar, indicates how many parts of a whole are being considered. The denominator, below the fraction bar, shows into how many equal parts the whole is divided.

Grasping the roles of numerators and denominators is crucial for all operations with fractions. In improper fractions, the role of the numerator is interesting because it dictates how many whole pieces you have and how many extra pieces remain.

Division with Remainder

Breaking Down Division with Remainder

Division with remainder is a process where you divide two numbers and the result consists of a quotient and a remainder. The quotient is the number of times the divisor fits into the dividend, and the remainder is what's left over.

When converting an improper fraction to a mixed number, division with a remainder helps you find out how many whole parts you have and what fraction is left over. For instance, dividing 18 by 4 gives a quotient of 4 and a remainder of 2. So, \( \frac{18}{4} \) as a mixed number is \( 4 \frac{2}{4} \), which simplifies to \( 4 \frac{1}{2} \). This division is the backbone of understanding how to transform improper fractions into mixed numbers.