Question

Express each of the following improper fractions as mixed fractions: (a) \(\frac{20}{3}\) (b) \(\frac{32}{7}\) (c) \(\frac{60}{9}\) (d) \(\frac{102}{50}\) (e) \(\frac{120}{11}\)

Short answer

Question: Convert the following improper fractions to mixed fractions: (a) \(\frac{20}{3}\) (b) \(\frac{32}{7}\) (c) \(\frac{60}{9}\) (d) \(\frac{102}{50}\) (e) \(\frac{120}{11}\) Answer: (a) \(6\frac{2}{3}\) (b) \(4\frac{4}{7}\) (c) \(6\frac{2}{3}\) (d) \(2\frac{1}{25}\) (e) \(10\frac{10}{11}\)

Step-by-step solution

Identify the improper fraction

An improper fraction is a fraction in which the numerator is greater than the denominator.

Divide the numerator by the denominator

For each given improper fraction, perform the division to find the quotient as the whole number part and the remainder as a new numerator over the original denominator.

(a) Convert \(\frac{20}{3}\)

Divide 20 by 3 to get the quotient (6) and remainder (2). So, the mixed fraction is \(6\frac{2}{3}\).

(b) Convert \(\frac{32}{7}\)

Divide 32 by 7 to get the quotient (4) and remainder (4). So, the mixed fraction is \(4\frac{4}{7}\).

(c) Convert \(\frac{60}{9}\)

First simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD) which is 3. So, \(\frac{60}{9} = \frac{20}{3}\), now divide 20 by 3 to get the quotient (6) and remainder (2). So, the mixed fraction is \(6\frac{2}{3}\).

(d) Convert \(\frac{102}{50}\)

First simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD) which is 2. So, \(\frac{102}{50} = \frac{51}{25}\), now divide 51 by 25 to get the quotient (2) and remainder (1). So, the mixed fraction is \(2\frac{1}{25}\).

(e) Convert \(\frac{120}{11}\)

Divide 120 by 11 to get the quotient (10) and remainder (10). So, the mixed fraction is \(10\frac{10}{11}\).

Key concepts

Mixed Fractions

A mixed fraction is a combination of a whole number and a proper fraction. Mixed fractions are an excellent way to express quantities in a more understandable form compared to improper fractions. Converting an improper fraction to a mixed fraction involves recognizing the whole number part and a fractional part, improving clarity.

• The whole number comes from dividing the numerator by the denominator fully.
• The remainder from this division becomes the new numerator for the proper fraction part.
• The denominator remains the same as that of the original improper fraction.

For example, when converting the improper fraction \(\frac{20}{3}\), divide 20 by 3 to get 6 as a quotient (the whole number) and 2 as the remainder (the new numerator), resulting in the mixed fraction \(6\frac{2}{3}\). Understanding this helps in simplifying and visualizing math problems easily.

Numerator and Denominator

Numerator and denominator are key elements of any fraction. A fraction essentially represents a part of a whole and uses two numbers to show this:

• Numerator: The top part of a fraction indicating how many parts we have.
• Denominator: The bottom part of a fraction indicating the total number of equal parts that make up a whole.

In an improper fraction, the numerator is larger than the denominator, meaning we have more parts than the whole. This difference is crucial as it requires conversions into mixed fractions to express the quantity correctly. For \(\frac{32}{7}\), the 32 (numerator) tells us the parts we have, while 7 (denominator) tells us how many parts make a whole.

Fraction Simplification

Fraction simplification is an important step to make fractions easier to understand and work with. Simplification is done by reducing fractions to their simplest form:

• Identify the greatest common divisor (GCD) for the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

This results in a smaller, equivalent fraction that is easier to interpret. Take the fraction \(\frac{60}{9}\). Here, divide both 60 and 9 by 3 (GCD = 3) to get \(\frac{20}{3}\). This simplifies the process of converting it to a mixed fraction. Simplification often precedes further operations with fractions to ensure all calculations are as straightforward as possible.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that can divide two numbers without a remainder. Finding the GCD is a helpful tool in simplifying fractions. For example, to simplify an improper fraction \(\frac{102}{50}\):

• List the factors of both numbers.
• Identify the largest factor they share.

In this case, the GCD is 2, as it divides both 102 and 50, simplifying the fraction to \(\frac{51}{25}\). It is an essential technique for fraction simplification, providing a neat and clean fraction for further mathematical operations. Understanding how to find the GCD and applying it will strengthen your skills in managing fractions efficiently.