A fraction is considered to be improper if the numerator is bigger in value than the denominator and this means that the number on top would be smaller in value compared to the number at the bottom.
It is a fraction that is larger than one.

The number $\frac{9}{5}$ could be considered as an improper fraction, because the numerator is bigger than the denominator.

In the above scenario the numerator 9 is greater than the denominator 5.
In some cases, we have to convert improper fractions into mixed numbers and this can be done by estimating the approximate value of the mixed fraction.

When the value of the Numerator is bigger than the denominator, we call the fraction as an Improper fraction. Let's look more deep into each term in the definition of an Improper fraction.

Words to remember:
Numerator: The top number of a fraction.
Denominator: The bottom number of a fraction, a number that names a part of a whole or part of a set.
Improper fraction: A fraction in which the numerator is larger than the denominator.
Mixed fraction: When a numeral has a whole number and a fraction number, it is known as a mixed fraction. Example: 4$\frac{2}{3}$

Adding and subtracting improper fractions takes a little extra effort.
We can add quarts and gallons if we change these to the same units.
The addition of improper fractions is the same as that.
We can add fifths and sixths if we find the common denominator first.

Steps for adding improper fractions:
Step 1: Convert the improper fractions so that they have the same number in the denominators.
Step 2: Add the numerators but leave the denominators alone.
Step 3: Reduce the answer if required.

Example for adding improper fractions

Add $\frac{13}{8}$ and $\frac{11}{3}$

Solution:

Step 1: Find the LCD of both 8 and 3 (8 x 3 = 24)
Step 2: Multiply both the numerator and the denominator with the factor of the LCD with each fractions $\frac{13 \times 3}{8 \times 3}$+$\frac{11 \times 8}{3 \times 8}$
Step 3: Add the numerators $\frac{39+38}{24}$ = $\frac{127}{24}$
Step 4: Reduce the answer if required 5 $\frac{7}{24}$

Adding Improper Fractions with Different Denominators:

Step 1: Find the lowest or least common multiple of the denominators (LCD) of the given fractions.
Step 2: Express the fractions as equivalent fractions with the denominator as the LCD.
Step 3: Add the numerators of the given fractions.
Step 4: Simplify the fraction if necessary.

Example: Find the sum $\frac{11}{6}$ + $\frac{17}{8}$?

Solution:

Step 1: The LCD of the given fractions = least common multiple (LCM) of 6 and 8 = 24
Step 2: Express the fractions as equivalent fractions with the common denominator 24
$\frac{11}{6}$ = $\frac{11 \times 4}{6 \times 4}$ = $\frac{44}{24}$
$\frac{17}{8}$ = $\frac{17 \times 3}{8 \times 3}$ = $\frac{51}{24}$
Step 3: Add the numerators: $\frac{44}{24}$ + $\frac{51}{24}$ $\rightarrow$ $\frac{44 + 51}{24}$ = $\frac{95}{24}$
Step 4: $\frac{95}{24}$

While subtracting improper fractions, follow the same methodology as any other operation in fractions.

Type A: same denominators
Step 1: Simply subtract the numerators.
Step 2: Leave alone the denominators.
Type B: different denominators
Step 1: Convert the improper fractions into equivalent fractions (having a common or same denominator).
Step 2: Choose the lowest common multiple of the denominators.
Step 3: Subtract the numerators.
Step 4: Leave alone denominators.
Step 5: Reduce the answer if required.

Example for different denominators

Subtract $\frac{5}{3}$ and $\frac{7}{5}$
Steps for subtraction:
Step 1: Choose the LCM for 3 and 5
Step 2: Convert these into equivalent fraction
Step 3: Subtract the numerators 25 and 21
$\frac{5}{3}$ - $\frac{7}{5}$ = $\frac{5 \times 5}{3 \times 5}$ - $\frac{7 \times 3}{5 \times 3}$

Example for same denominator
Subtract $\frac{5}{2}$ and $\frac{11}{2}$
Step 1: Denominators are same so leave alone the denominators
Step 2: Subtract 5 from 11
Step 3: Write the answer keeping in mind the integer rule and reduce it
$\frac{5}{2}$ - $\frac{11}{2}$ = $\frac{5 - 11}{2}$ = $\frac{-6}{2}$ = -3

When multiplying an improper fraction by a whole number, turn the whole number into a fraction with a denominator of 1.

Step 1: Write the whole number as an improper fraction with 1 as denominator.
Step 2: Then multiply across.
Step 3: Change the improper fraction to a whole or mixed number if necessary.

Example: $\frac{12}{7}$ $\times$ 5
Now let us convert the whole number into an improper fraction
Step 1: Convert 5 into an improper fraction.
Step 2: Multiply across (12 x 5).
Step 3: Convert into a mixed number if necessary.

The main idea to dividing with improper fractions is to understand reciprocals.
A reciprocal is two numbers that have the product of 1.
Knowing what a reciprocal is would be the key to understanding how to divide improper fractions.

Reciprocal samples:
The fractions $\frac{4}{3}$ and $\frac{3}{4}$ are reciprocals because $\frac{4}{3}$ * $\frac{3}{4}$ = 1
Improper Fraction
The product of the fraction and its reciprocal is always equal to 1.

Dividing an improper fraction with another improper fraction:

To divide by a fraction, we have to invert the fraction we are dividing by and then multiply.

Steps for dividing an improper fraction with another:

Step 1: Invert the improper fraction we are dividing by
Step 2: Change the division symbol into multiplication.
Step 3: Multiply across numerators and denominators.
Step 4: Reduce or change an improper fraction to a mixed or whole number if necessary or possible.

Example: Let us divide $\frac{11}{4}$ with $\frac{12}{7}$

Step 1: Invert the improper fraction $\frac{12}{7}$
Step 2: Multiply across numerators and denominators $\frac{11}{4}$ x $\frac{7}{12}$ = $\frac{77}{48}$
Step 3: Change the answer to a mixed number 1 $\frac{29}{48}$

Tips:
The fraction to invert is always the fraction to the right of the division sign.

Cancelling in dividing fractions:

Step 1: Invert and change to multiplication.
Step 2: Cancel and divide the numerator and the denominator wherever necessary.
Step 3: Multiply across the numerator and the denominator.

Example:
Let us divide $\frac{3}{2}$ with $\frac{5}{3}$
Step 1: Invert $\frac{5}{3}$
Step 2: Invert and multiply $\frac{3}{2}$ x $\frac{3}{5}$
Step 3: Write the answer $\frac{9}{10}$

Dividing improper fraction with a whole number:

When dividing a fraction by a whole number, first change the whole number into a fraction by putting a denominator of 1

Example:
A cake recipe needs half pound of butter divided into 3 equal parts.
Hint: We have to divide $\frac{1}{2}$ with 3
Step 1: Change the whole number 3 into a fraction 3 = $\frac{1}{3}$
Step 2: Invert the divisor 3 into $\frac{1}{3}$
Step 3: Multiply across the numerator and the denominator $\frac{1}{2}$ x $\frac{3}{1}$ = $\frac{3}{2}$
Step 4: Reduce the answer into a mixed number $\frac{3}{2}$ = 1 $\frac{1}{2}$

If we need to convert an improper fraction into a mixed fraction we have to follow the following steps.

Step 1: The numerator is divided by the denominator.
Step 2: Place the remainder over the original denominator to help form the fraction.
Step 3: State the answer as a whole number quotient plus the fraction that has the remainder as the numerator.
To change an improper fraction to a mixed fraction
1. Divide the numerator by the denominator
2. The quotient is the whole number part of the mixed number.
3. The remainder is the numerator of the fraction part
4. The original denominator is the denominator of the fraction part.
Begin with $\frac{9}{4}$
Divide 9 with 4 = 2 remainder 1
Rewrite : 2$\frac{1}{4}$

Mixed fraction to improper fraction:

For writing a mixed fraction as an improper fraction, we need to multiply the whole number with the denominator and then add the numerator to it.

The sum obtained is called the numerator and the fraction so formed is an improper fraction.

3$\frac{5}{8}$ = $\frac{8 \times 3+5}{8}$ = $\frac{24+5}{8}$ = $\frac{29}{8}$

Changing mixed fractions into improper fractions:
How could I change 7$\frac{3}{5}$ into an improper fraction?
Step 1: Multiply the denominator by the whole number.
Step 2: Add the answer (35) to the numerator (3), making the fraction improper.

7$\frac{3}{5}$ = $\frac{5*7+3}{5}$ = $\frac{38}{5}$

Changing improper fraction into mixed fractions:

How could I change $\frac{12}{7}$ into a mixed fraction?

When the numerator and denominator of a fraction are multiplied by the same digit, the value of the fraction remains unchanged.

This form of modification is called reducing of a fraction to higher terms.

In case of an improper fraction when we reduce it to a whole number or mixed number we just need to divide the numerator by the denominator.

Example: Let us reduce $\frac{43}{8}$

Step 1: Divide 43 by 8.
Step 2: Express it in a mixed form (Whole number +$\frac{Remainder}{Divisor}$)
Rule for reducing improper fraction.
Divide the numerator by the denominator.
$\frac{43}{8}$ = 5$\frac{3}{8}$

Objectives of simplifying improper fractions are as follows:
  • Reduce higher term fractions into its lowest terms
  • Confirm whether the improper fraction is in its simplest form
  • Simplifying the improper fraction by changing it into mixed numerals
  • In simplifying improper fraction the main objective is to convert these into their mixed form.

1) Reduce to lowest terms: $\frac{36}{30}$?
Solution:
Step 1: Factors of the numerator $\rightarrow$ 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of the denominator $\rightarrow$ 30 = 1, 2, 3, 5, 6, 10, 15, 30
Step 2: Greatest common factor (GCD) = 6
Step 3: Divide numerator and denominator by 6 $\rightarrow$ $\frac{36 ÷ 6}{30 ÷ 6}$ $\rightarrow$ $\frac{6}{5}$

2) Reduce to lowest terms: $\frac{36}{24}$?
Solution:
Step 1: Factors of the numerator $\rightarrow$ 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of the denominator $\rightarrow$ 24 = 1, 2, 3, 4, 6, 8, 12, 24
Step 2: Greatest common factor (GCD) = 12
Step 3: Divide numerator and denominator by 12 $\rightarrow$ $\frac{36 ÷ 12}{24 ÷ 12}$ $\rightarrow$ $\frac{3}{2}$

3) Reduce to lowest terms: $\frac{64}{56}$?
Solution:
Step 1: Factors of the numerator $\rightarrow$ 64 = 1, 2, 4, 8, 16, 32, 64
Factors of the denominator $\rightarrow$ 56 = 1, 2, 4, 7, 8, 14, 28, 56
Step 2: Greatest common factor (GCD) = 8
Step 3: Divide numerator and denominator by 8 $\rightarrow$ $\frac{64 ÷ 8}{56 ÷ 8}$ $\rightarrow$ $\frac{8}{7}$

4) Reduce to lowest terms: $\frac{69}{24}$?
Solution:
Step 1: Factors of the numerator $\rightarrow$ 69 = 1, 3, 23, 69
Factors of the denominator $\rightarrow$ 24 = 1, 2, 3, 4, 6, 8, 12, 24
Step 2: Greatest common factor (GCD) = 3.
Step 3: Divide numerator and denominator by 3 $\rightarrow$ $\frac{69 ÷ 3}{24 ÷ 3}$ $\rightarrow$ $\frac{23}{8}$

5) Find the sum $\frac{10}{7}$ + $\frac{8}{7}$?
Solution:
Step 1: Add the numerators = (10 + 8) = 18
Step 2: Retain the common denominator = 7
Step 3: Sum of the fractions = $\frac{Sum\ of \ Numerators}{Common\ Denominator}$ $\rightarrow$ $\frac{18}{7}$
Step 4: $\frac{18}{7}$

6) Find the sum $\frac{14}{9}$ + $\frac{17}{9}$?
Solution:
Step 1: Add the numerators = (14 + 17) = 31
Step 2: Retain the common denominator = 9
Step 3: Sum of the fractions = $\frac{Sum\ of\ Numerators}{Common\ Denominator}$ $\rightarrow$ $\frac{31}{9}$
Step 4:$\frac{31}{9}$

7) Find $\frac{10}{7}$ - $\frac{9}{7}$?
Solution:
Step 1: Difference between the numerators = 10-9 = 1
Step 2: Retain the common denominator = 7
Step 3: Difference between the improper fractions = $\frac{Difference\ between\ the\ Numerators}{Common\ Denominator}$ = $\frac{1}{7}$
Step 4: $\frac{1}{7}$

8) Find $\frac{11}{3}$ - $\frac{5}{3}$?
Solution:
Step 1: Difference between the numerators = 11 - 5 = 6.
Step 2: Retain the common denominator = 3.
Step 3: Difference between the improper fractions = $\frac{Difference\ between\ the\ Numerators}{Common\ Denominator}$ = $\frac{6}{3}$
Step 4: Divide the numerator and denominator by 3 $\rightarrow$ $\frac{(6 ÷ 3)}{(3 ÷ 3)}$ = $\frac{2}{1}$ $\rightarrow$ 2

9) Find the product of $\frac{8}{7}$ and $\frac{9}{7}$?
Solution:
Step 1: Multiply the numerators = 8 x 9 = 72
Step 2: Multiply the denominators = 7 x 7 = 49
Step 3: The product of the improper fractions = $\frac{(Product\ of\ the\ Numerators)}{(Product\ of\ the\ Denominators)}$ = $\frac{72}{49}$
Step 4: $\frac{72}{49}$.

10) Find the product of $\frac{7}{5}$ and $\frac{10}{9}$?
Solution:
Step 1: Multiply the numerators = 7 x 10 = 70.
Step 2: Multiply the denominators = 5 x 9 = 45.
Step 3: The product of the improper fractions = $\frac{(Product\ of\ the\ Numerators)}{(Product\ of\ the\ Denominators)}$ = $\frac{70}{45}$.
Step 4: Divide numerator and denominator by 5 $\rightarrow$ $\frac{(70 ÷ 5)}{(45 ÷ 5)}$ $\rightarrow$ $\frac{14}{9}$

11) Find the product of $\frac{17}{15}$ and $\frac{23}{15}$?
Solution:
Step 1: Multiply the numerators = 17 x 23 = 391.
Step 2: Multiply the denominators = 15 x 15 = 225.
Step 3: The product of the improper fractions = $\frac{(Product\ of\ the\ Numerators)}{(Product\ of\ the\ Denominators)}$ = $\frac{391}{225}$
Step 4: $\frac{391}{225}$

12) Find the product of $\frac{13}{12}$ and $\frac{5}{4}$?
Solution:
Step 1: Multiply the numerators = 13 x 5 = 65.
Step 2: Multiply the denominators = 12 x 4 = 48.
Step 3: The product of the improper fractions = $\frac{(Product\ of\ the\ Numerators)}{(Product\ of\ the\ Denominators)}$ = $\frac{65}{48}$
Step 4: $\frac{65}{48}$

13) Divide $\frac{8}{7}$ with $\frac{4}{3}$?
Solution:
Step 1: Reciprocal of the second fraction = $\frac{4}{3}$ $\rightarrow$ $\frac{3}{4}$
Step 2: Multiply the numerators and denominators of both the fractions
$\rightarrow$ $\frac{8}{7}$ x $\frac{3}{4}$ $\rightarrow$ $\frac{(8 \times 3)}{(7 \times 4)}$ $\rightarrow$ $\frac{24}{28}$

Step 3: Divide the numerator and denominator by 4 $\rightarrow$ $\frac{(24 ÷ 4)}{(28 ÷ 4)}$ = $\frac{6}{7}$

14) Divide $\frac{20}{13}$ with $\frac{22}{9}$?
Solution:
Step 1: Reciprocal of the second fraction = $\frac{22}{9}$ $\rightarrow$ $\frac{9}{22}$.
Step 2: Multiply the numerators and denominators of both the fractions
$\rightarrow$ $\frac{20}{13}$ x $\frac{9}{22}$ $\rightarrow$ $\frac{(20 \times 9)}{13 \times 22}$ $\rightarrow$ $\frac{180}{286}$
Step 3: Divide the numerator and denominator by 2 $\rightarrow$ $\frac{(180 ÷ 2)}{(286 ÷ 2)}$ = $\frac{90}{143}$

15) Divide $\frac{16}{15}$ with $\frac{17}{15}$?
Solution:
Step 1: Reciprocal of the second fraction = $\frac{17}{15}$ $\rightarrow$ $\frac{15}{17}$
Step 2: Multiply the numerators and denominators of both the fractions
$\rightarrow$ $\frac{16}{15}$ x $\frac{15}{17}$ $\rightarrow$ $\frac{(16 \times 15)}{(15 \times 17)}$ $\rightarrow$ $\frac{240}{255}$
Step 3: Divide numerator and denominator by 15 $\rightarrow$ $\frac{(240 ÷ 15)}{(255 ÷ 15)}$ $\rightarrow$ $\frac{16}{17}$

16) Divide $\frac{14}{11}$ with $\frac{15}{11}$?
Solution:
Step 1: Reciprocal of the second fraction = $\frac{15}{11}$ $\rightarrow$ $\frac{11}{15}$
Step 2: Multiply the numerators and denominators of both the fractions
$\rightarrow$ $\frac{14}{11}$ x $\frac{11}{15}$ $\rightarrow$ $\frac{(14 \times 11)}{(11 \times 15)}$ $\rightarrow$ $\frac{154}{165}$
Step 3: Divide numerator and denominator by 11 $\rightarrow$ $\frac{(154 ÷ 11)}{(165 ÷ 11)}$ $\rightarrow$ $\frac{14}{15}$