Since in all Complex Fractions we will have a fraction in the both numerator and the denominator or in any one of them, we should simplify the complex fraction first before we perform any operation like addition, subtraction, multiplication or division on it.

### Problems on Simplifying Complex fractions

**1) Find the sum of $\frac{2}{1/4} + \frac{3}{5/7}$ ?****Solution:**

** Step 1:** Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{2/1}{1/4} \rightarrow \frac{2}{1} \times \frac{4}{1} \rightarrow \frac{8}{1}$

$\frac{3/5}{7/1} \rightarrow \frac{3}{5} \times \frac{1}{7} \rightarrow \frac{3}{35}$

**Step 2:** LCD of the fractions = 35

**Step 3:** Express the fractions as equivalent fractions with the common denominator 35

$\frac{8}{1} = \frac{(8 \times 35)}{(1 \times 35)} = \frac{280}{35}$

$\frac{3}{35} = \frac{(3 \times 1)}{(35 \times 1)} = \frac{3}{35}$

**Step 4:** Add the numerators:

$\frac{280}{35} + \frac{3}{35} \rightarrow \frac{(280 + 3)}{35} \rightarrow \frac{283}{35}$

**Step 5:** $\frac{283}{35}$

**2) Find the sum of $\frac{8/7}{9/7}$+ $\frac{1/5}{9/8}$ ?**

**Solution: **

**Step 1:** Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{2/1}{1/4} \rightarrow \frac{2}{1} \times \frac{4}{1} \rightarrow \frac{8}{1}$

$\frac{3/5}{7/1} \rightarrow \frac{3}{5} \times \frac{1}{7} \rightarrow \frac{3}{35}$

**Step 2:** LCD of the fractions = 35

**Step 3:** Express the fractions as equivalent fractions with the common denominator 35

$\frac{8}{1} = \frac{8}{35} = \frac{280}{35}$

$\frac{3}{35} = \frac{3 \times 1}{35 \times 1} = \frac{3}{35}$

**Step 4:** Add the numerators:

$\frac{280}{35} + \frac{3}{35} \rightarrow \frac{(280 + 3)}{35} \rightarrow \frac{283}{35}$

**Step 5:** $\frac{283}{35}$

**3) Simplify $\frac{7/3}{8/7}$ X $\frac{5/2}{5/7}$**

**Solution:**

**Step 1:** Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{7/3}{8/7} \rightarrow \frac{7}{3} \times \frac{7}{8} \rightarrow \frac{49}{24}$

$\frac{5/2}{5/7} \rightarrow \frac{5}{2} \times \frac{7}{5} \rightarrow \frac{7}{2}$

**Step 2:** Multiply the numerator (top number) and denominator of the new fraction.

$\frac{49}{24} \times \frac{7}{2} = \frac{(49 x 7)}{(24 x 2)} = \frac{343}{48}$

**Step 3:** $\frac{343}{48}$

**4) Divide $\frac{8/3}{2/3}$ with $\frac{9/5}{4/3}$**

**Solution :**

**Step 1:** Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{8/3}{2/3} \rightarrow \frac{8}{3} \times \frac{3}{2} \rightarrow \frac{4}{1}$

$\frac{9/5}{4/3} \rightarrow \frac{9}{5} \times \frac{3}{4} \rightarrow \frac{27}{20}$

**Step 2:** Find the reciprocal of the second fraction (the one you want to divide by) and change the division sign to multiplication sign.

$\frac{4}{1} \times \frac{20}{27}$

**Step 3:** Multiply the numerator and denominator of both the new fractions:

$\frac{4}{1} \times \frac{20}{27}$ = $\frac{(4 \times 20)}{(1 \times 27)}$ = $\frac{80}{27}$

**Step 4:**$\frac{80}{27}$