A fraction is a rational number. It can also be represented as a decimal number. The general form of a fraction is $\frac{a}{b}$ where a is the numerator and b is the denominator. At times the number in the numerator and denominator has common factors which can be cancelled out to have a simpler fraction. This process is known as reducing fractions or simplifying fraction. Reducing fractions makes it easier to perform calculations on two or more fractions.

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The following steps are followed to reduce fractions,

**Step 1: **First, write the prime factors of the numerator quantity and the denominator quantity.

**Step 2:** Find the common factors of both the numerator quantity and the denominator quantity.

**Step 3: **Divide both numerator or denominator by common factors or in other word, cancel out all the common factors.

### Solved Examples

**Question 1: **Reduce $\frac{30}{105}$, using prime factorization.

** Solution: **

**Step 1:** Prime factors of 30 = 2 $\times$ 3 $\times$ 5 and

105 = 3 $\times$ 5 $\times$ 7

**Step 2:**

$\frac{30}{105}$ = $\frac{2\times3\times5}{3\times5\times7}$

**Step 3: **

$\frac{30}{105}$ = $\frac{2}{7}$

**Question 2: **Reduce $\frac{42}{33}$, using prime factorization.

** Solution: **

**Step 1:** Prime factors of 42 = 2 $\times$ 3 $\times$ 7 and

33 = 3 x 11

**Step 2:**

$\frac{42}{33}$ = $\frac{2\times3\times7}{3\times11}$

**Step 3:**

$\frac{42}{33}$ = $\frac{14}{11}$

Examples on reducing fractions with variables:

### Solved Examples

**Question 1: **Reduce $\frac{6xy}{42x^2}$ to lowest terms.

** Solution: **

**Step 1: **Factor of 6xy = 2 x 3 x $x$ x $y$ and

42 $x^2$ = 2 $\times$ 3 $\times$ 7 $\times$ $x$ $\times$ $x$

**Step 2: **

$\frac{6xy}{42x^2}$ = $\frac{2 \times 3 \times x \times y}{2 \times 3 \times 7 \times x \times x}$

**Step 3:** ** **

$\frac{6xy}{42x^2}$ =** **$\frac{y}{7x}$

**Question 2: **Reduce $\frac{30xy}{98xz}$ to lowest terms.

** Solution: **

**Step 1: **Factor of 30 $xy$ = 2 $\times$ 3 $\times$ 5 $\times$ $x$ $\times$ $y$ and

98$xz$ = 2 $\times$ 7 $\times$ 7 $\times$ $x$ $\times$ $z$

**Step 2: **

$\frac{30xy}{98xz}$ = $\frac{2 \times 3 \times 5 \times x \times y}{2 \times 7 \times 7 \times x \times z}$

**Step 3: **

$\frac{30xy}{98xz}$ = $\frac{15y}{49z}$

Problem 1: Simplify $\frac{268}{124}$.

Problem 2: Can the term $\frac{8xy}{2x}$ be simplified?

Problem 3: Reduce the fraction $\frac{265}{165}$

Problem 4: Find the simplified form of the fraction $\frac{65}{13}$.

105 = 3 $\times$ 5 $\times$ 7

$\frac{30}{105}$ = $\frac{2\times3\times5}{3\times5\times7}$

33 = 3 x 11

$\frac{42}{33}$ = $\frac{2\times3\times7}{3\times11}$

$\frac{42}{33}$ = $\frac{14}{11}$

42 $x^2$ = 2 $\times$ 3 $\times$ 7 $\times$ $x$ $\times$ $x$

$\frac{6xy}{42x^2}$ = $\frac{2 \times 3 \times x \times y}{2 \times 3 \times 7 \times x \times x}$

$\frac{6xy}{42x^2}$ =

98$xz$ = 2 $\times$ 7 $\times$ 7 $\times$ $x$ $\times$ $z$

$\frac{30xy}{98xz}$ = $\frac{2 \times 3 \times 5 \times x \times y}{2 \times 7 \times 7 \times x \times z}$

$\frac{30xy}{98xz}$ = $\frac{15y}{49z}$

Problem 2: Can the term $\frac{8xy}{2x}$ be simplified?

Problem 3: Reduce the fraction $\frac{265}{165}$

Problem 4: Find the simplified form of the fraction $\frac{65}{13}$.

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