Understanding Fraction Simplification

To write a fraction in its simplest form, we need to divide both the numerator (the top number) and the denominator (the bottom number) by their Greatest Common Divisor (GCD). The simplest form is also called the reduced form of the fraction, where the only common factor between the numerator and the denominator is 1.

The given fraction is $\frac{182}{130}$.

Finding the Greatest Common Divisor (GCD)

Let's find the GCD of 182 and 130. We can do this by listing the factors of each number or by using prime factorization.

Using prime factorization:

• Prime factors of 182:
• $182 = 2 \times 91$
• $91 = 7 \times 13$
• So, the prime factorization of 182 is $2 \times 7 \times 13$.

• Prime factors of 130:
• $130 = 2 \times 65$
• $65 = 5 \times 13$
• So, the prime factorization of 130 is $2 \times 5 \times 13$.

The common prime factors are 2 and 13. To find the GCD, we multiply these common factors:

GCD(182, 130) = $2 \times 13 = 26$.

Simplifying the Fraction 182/130

Now we divide both the numerator and the denominator of the fraction $\frac{182}{130}$ by their GCD, which is 26.

$$ \frac{182}{130} = \frac{182 \div 26}{130 \div 26} $$

Performing the division:

• $182 \div 26 = 7$
• $130 \div 26 = 5$

So, the simplified fraction is $\frac{7}{5}$.

Comparing with Options

Let's look at the given options:

• Option 1: 28/10. This can be simplified by dividing both by 2: 14/5. Not 7/5.
• Option 2: 91/65. Let's check if these numbers have common factors. 91 = 7 × 13, 65 = 5 × 13. Both are divisible by 13. $\frac{91 \div 13}{65 \div 13} = \frac{7}{5}$. This simplifies to 7/5.
• Option 3: 14/12. This can be simplified by dividing both by 2: 7/6. Not 7/5.
• Option 4: 7/5. This fraction already has 7 and 5 as numerator and denominator, which have no common factors other than 1. It is already in simplest form.

Both options 2 (91/65) and 4 (7/5) simplify to 7/5. However, the question asks for 182/130 in its simplest form. While 91/65 is an intermediate step in reduction (dividing by 2), the complete simplest form is 7/5.

Therefore, the simplest form of $\frac{182}{130}$ is $\frac{7}{5}$.

Revision Table: Fraction Simplification

Additional Information: Why Simplify Fractions?

Simplifying fractions makes them easier to understand and compare. For example, it's clearer to understand what 7/5 represents than 182/130. Simplification is a fundamental skill in mathematics used in various contexts, such as solving equations, comparing quantities, and working with ratios.

To check if a fraction is in simplest form, you can try dividing the numerator and denominator by small prime numbers (2, 3, 5, 7, 11, etc.). If no prime number can divide both evenly, the fraction is in its simplest form.