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Improper Fractions(Definition, Conversions & Examples)

improper fraction

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Fractions represent the parts of a whole or collection of objects. A fraction has two parts. The number on the top of the line is called the numerator and the one at the bottom of the line is called the denominator.

Based on the relative values of numerator and denominator, a fraction can be either a proper fraction or an improper fraction. In this article, we’ll discuss improper fractions.

What is an Improper Fraction?

An improper fraction is one of the two main types of fractions. The other one is a proper fraction.

An improper fraction is a type of fraction in which the numerator is greater than or equal to the denominator. For that reason, these fractions are also called ‘top-heavy’ fractions. 

For example, $ \frac {7}{3}$ is an improper fraction as in this fraction $7$ (numerator) is greater than $3$ (denominator).

Note: The top (numerator) is heavier (greater than) the bottom (denominator).

Operations on improper fractions like addition & subtraction or multiplication & division are easy as compared to mixed fractions. These arithmetic operations on improper fractions are the same as that with proper fractions.

Note: Improper fractions are always greater than or equal to $1$.

Improper Fractions and Mixed Fractions

As mentioned above improper fractions are greater than or equal to $1$. Mixed fractions are also the fractions that are greater than or equal to $1$. So, improper fractions and mixed fractions are related to each other.

A mixed fraction is written as a combination of a whole number and a proper fraction. For example a mixed fraction $4 \frac {3}{4}$ is same as improper fraction $\frac {19}{4}$. Compared to improper fractions, mixed fractions are easier to interpret and compare. 

Note: 

  • Any improper fraction can be written as a mixed fraction.
  • Any mixed fraction can be written as an improper fraction.
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Converting Improper Fraction To Mixed Fraction

An improper fraction can be converted to a mixed fraction by dividing the numerator by the denominator. After conversion, the denominator of a mixed fraction will always be the same as that of the original improper fraction.

For example, mixed fraction of $\frac {27}{4}$ is $6 \frac {3}{4}$.

Steps to Convert Improper Fraction to Mixed Fraction

The following steps are used to convert an improper fraction to a mixed fraction.

Step 1– Divide the numerator with the denominator.

Step 2– Note down the values of quotient and remainder.

Step 3– Arrange these values of the quotient, remainder, and divisor in the following manner to express a fraction as a mixed number:

improper fraction
Improper fraction format
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Examples

Ex 1: Convert $\frac{17}{4}$ into mixed fraction

improper fraction

$\frac{17}{4} = 4 \frac {1}{4}$

Ex 2: Convert $\frac{59}{8}$ into mixed fraction

improper fraction

$\frac{59}{8} = 7 \frac {3}{8}$

Note: Improper fraction and mixed fraction have the same denominator.

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Converting Mixed Fraction To Improper Fraction

As we can convert an improper fraction to a mixed fraction, we can also perform the reverse process. That is we can convert a mixed fraction to an improper fraction.

Steps to Convert Mixed Fraction To Improper Fraction

The steps involved in converting a mixed fraction to an improper fraction are

Step 1: Multiply the denominator of the mixed fraction with the whole number part.

Step 2: Add the numerator to the product obtained from step 1.

Step 3: Write the improper fraction with the sum obtained from step 2 in the numerator/denominator form.

Examples

Ex 1: $8 \frac {2}{3}$ 

The whole part is $8$

The fraction part is $\frac {2}{3}$, where the numerator is $2$ and the denominator is $3$.

Multiplying the denominator with the whole part and adding the numerator: $3 \times 8 + 2 = 26$

Therefore, the improper fraction of $8 \frac {2}{3}$ is $\frac {26}{3}$

Ex 2: $11 \frac {5}{7}$

The whole part is $11$

The fraction part is $\frac {5}{7}$ where the numerator is $5$ and the denominator is $7$.

The improper fraction of $11 \frac{5}{7} = \frac{11 \times 7 + 5}{7} = \frac {82}{7}$.

Improper Fractions and Decimal Numbers

Improper fractions can be converted into numbers with a decimal point and vice-versa. Since an improper fraction is greater than or equal to $1$, the whole part of the decimal number will also be greater than or equal to $1$.

Converting Improper Fraction to Decimal Number

Improper fractions can be converted to decimals easily by dividing the numerator with the denominator. 

An improper fraction is converted to 

  • Whole number when the numerator is completely divisible by the denominator (Remainder is zero ($0$))
  • A number with a decimal part when the numerator is not divisible by the denominator (Remainder is not zero ($0$))

Steps to Convert Improper Fraction to Decimal Number

The steps involved in converting an improper fraction to a decimal number are

Step 1: Divide the numerator by the denominator

Step 2: The quotient is the decimal representation of the improper fraction

Axiom, Postulate & Theorem

Examples

Ex 1: Convert $\frac {15}{2}$ to decimal number

improper fraction

$\frac {15}{2} = 7.5$

Ex 2: Convert $\frac {21}{4}$ to decimal number

improper fraction

$\frac {21}{4} = 5.25$

Converting Decimal Number to Improper Fraction

Decimal numbers can be converted to improper fractions by using 10, 100, 1000, … depending on the number of decimal places.

Steps to Convert Decimal Number to Improper Fraction

The steps involved in converting a decimal number to an improper fraction are

Step 1: Count the number of digits after the decimal point in the decimal number

Step 2: Write either 10, 100, or 1000, … depending on the number of digits after the decimal point in the denominator

Step 3: Write the number after removing the decimal point in the numerator

Examples

Ex 1: $7.5$ to improper fraction

The number of digits after the decimal point is $1$.

Since the number of digits after the decimal point is $1$, therefore denominator is $10$.

The number after removing the decimal point is $75$.

Therefore, $7.5 = \frac {75}{10} = \frac {15}{2}$

Ex 2: $5.25$ to improper fraction

The number of digits after the decimal point is $2$.

Since the number of digits after the decimal point is $2$, therefore denominator is $100$.

The number after removing the decimal point is $525$.

Therefore, $5.25 = \frac {525}{100} = \frac {21}{4}$

Arithmetic Operations With Improper Fractions

The four basic arithmetic operations are 

  • Addition
  • Subtraction
  • Multiplication
  • Division

To perform addition or subtraction, the improper fractions must be the like fractions. If the fractions are not like fractions, the first step is to convert them into like fractions.

For multiplication or division, it is not necessary for an improper fraction to be a like fraction. Multiplication or division can be performed on like as well as unlike fractions.

Conclusion

Improper fractions are the ones where the numerator is greater than or equal to the denominator. Any improper fraction can be converted to a mixed fraction or vice-versa. The process of arithmetic operations on improper fractions is the same as that of proper fractions.

Practice Problems

  1. Which of the following are improper fractions?
    1. $\frac {2}{3}$
    2. $\frac{17}{11}$
    3. $\frac{5}{5}$
    4. $\frac{1}{9}$
    5. $\frac{23}{12}$
  2. Convert the following improper fractions to mixed fractions
    1. $\frac{7}{3}$
    2. $\frac{17}{9}$
    3. $\frac{23}{13}$
    4. $\frac{88}{19}$
    5. $\frac{231}{17}$
  3. Convert following mixed fractions to improper fractions
    1. $1\frac{2}{3}$
    2. $7\frac{5}{6}$
    3. $9\frac{11}{19}$
    4. $15\frac{3}{4}$
    5. $20\frac{7}{9}$
  4. Convert the following improper fractions to decimal numbers
    1. $\frac{7}{2}$
    2. $\frac{27}{20}$
    3. $\frac{89}{25}$
    4. $\frac{129}{125}$
    5. $\frac{237}{50}$
  5. Convert the following decimal numbers to improper fractions
    1. $2.36$
    2. $1.05$
    3. $1.1$
    4. $12.564$
    5. $34.35$

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