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 $\rightarrow$ $\frac{(154 ÷ 11)}{(165 ÷ 11)}$ $\rightarrow$ $\frac{14}{15}$