Simplifying Fractions: A Step-by-Step Guide

Nov 10, 2025 by Admin 44 views

Fractions, fractions, everywhere! Sometimes they look a little scary, but don't worry, guys! This article will walk you through simplifying fractions and evaluating expressions with fractions. We'll tackle a couple of examples, breaking down each step to make it super easy. So, let's dive in and become fraction masters!

Evaluating Expressions with Fractions

Let's kick things off with evaluating the expression: $\frac{7}{8} \div \frac{14}{15} \cdot \frac{8}{25}$ . This might seem intimidating at first, but we can handle it by following the order of operations and understanding how to divide fractions. Remember guys, the key to solving these problems is to take it one step at a time.

Step 1: Division of Fractions

Our expression includes both division and multiplication. Following the order of operations (PEMDAS/BODMAS), we perform division before multiplication, working from left to right. So, the first thing we need to do is divide $\frac{7}{8}$ by $\frac{14}{15}$ .

Dividing fractions can seem tricky, but it's actually quite simple. To divide one fraction by another, we flip the second fraction (the divisor) and then multiply. Flipping a fraction means swapping its numerator (the top number) and its denominator (the bottom number). This flipped fraction is called the reciprocal. So, the reciprocal of $\frac{14}{15}$ is $\frac{15}{14}$ .

Now, we can rewrite the division as multiplication:

$\frac{7}{8} \div \frac{14}{15} = \frac{7}{8} \cdot \frac{15}{14}$

To multiply fractions, we multiply the numerators together and the denominators together:

$\frac{7}{8} \cdot \frac{15}{14} = \frac{7 \cdot 15}{8 \cdot 14} = \frac{105}{112}$

Before we move on, let's see if we can simplify this fraction. Both 105 and 112 are divisible by 7. Dividing both by 7, we get:

$\frac{105 \div 7}{112 \div 7} = \frac{15}{16}$

So, $\frac{7}{8} \div \frac{14}{15}$ simplifies to $\frac{15}{16}$ . We're one step closer to solving the whole expression!

Step 2: Multiplication of Fractions

Now we need to multiply our result, $\frac{15}{16}$ , by the remaining fraction in the expression, which is $\frac{8}{25}$ . So, we have:

$\frac{15}{16} \cdot \frac{8}{25}$

Again, we multiply the numerators and the denominators:

$\frac{15}{16} \cdot \frac{8}{25} = \frac{15 \cdot 8}{16 \cdot 25} = \frac{120}{400}$

This fraction looks pretty big, so let's simplify it. We can start by finding the greatest common divisor (GCD) of 120 and 400. Both numbers are divisible by 40, so let's divide both numerator and denominator by 40:

$\frac{120 \div 40}{400 \div 40} = \frac{3}{10}$

Alternatively, we could have simplified before multiplying. Notice that 15 and 25 share a common factor of 5, and 8 and 16 share a common factor of 8. We can simplify diagonally:

$\frac{15}{16} \cdot \frac{8}{25} = \frac{3 \cdot \cancel{5}}{2 \cdot \cancel{8}} \cdot \frac{\cancel{8}}{5 \cdot \cancel{5}} = \frac{3}{2} \cdot \frac{1}{5} = \frac{3}{10}$

Either way, we arrive at the same simplified answer: $\frac{3}{10}$ .

Step 3: The Final Answer

Therefore, the expression $\frac{7}{8} \div \frac{14}{15} \cdot \frac{8}{25}$ evaluates to $\frac{3}{10}$ . See, guys? We did it! By breaking down the problem into smaller steps, we made it much easier to manage.

Simplifying Fractions: How to Do It

Now, let's move on to simplifying a single fraction. The goal here is to express the fraction in its simplest form, where the numerator and denominator have no common factors other than 1. Let's look at the fraction $\frac{7}{72}$ .

Step 1: Finding the Greatest Common Divisor (GCD)

The most efficient way to simplify a fraction is to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

In our case, we need to find the GCD of 7 and 72.

The factors of 7 are 1 and 7 (since 7 is a prime number).
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

Looking at these lists, we see that the only common factor of 7 and 72 is 1. Therefore, the GCD(7, 72) = 1.

Step 2: Simplifying the Fraction

Since the greatest common divisor of 7 and 72 is 1, the fraction $\frac{7}{72}$ is already in its simplest form. This means we can't reduce it any further. Sometimes, guys, the fraction is already as simple as it gets!

Understanding Why This Works

The key to simplifying fractions is understanding that dividing both the numerator and the denominator by the same number doesn't change the value of the fraction. It's like cutting a pizza: whether you cut it into 8 slices or 16 slices, the total amount of pizza remains the same. The slices are just smaller when you cut it into 16.

For example, consider the fraction $\frac{2}{4}$ . Both 2 and 4 are divisible by 2. If we divide both by 2, we get $\frac{1}{2}$ . $\frac{2}{4}$ and $\frac{1}{2}$ represent the same amount, but $\frac{1}{2}$ is in its simplest form.

Tips and Tricks for Simplifying Fractions

Here are a few tips and tricks to help you simplify fractions like a pro:

Look for Common Factors: Start by looking for small common factors, like 2, 3, 5, and 10. If both the numerator and denominator are even, they're both divisible by 2. If they end in 0 or 5, they're both divisible by 5.
Divide by the GCD: If you can find the greatest common divisor, dividing both the numerator and denominator by the GCD will simplify the fraction in one step.
Simplify Incrementally: If you can't immediately find the GCD, simplify in smaller steps. For example, if you see that both numbers are divisible by 4, divide by 4. Then, see if you can simplify further.
Prime Factorization: If you're struggling to find common factors, try finding the prime factorization of both the numerator and denominator. Then, cancel out any common prime factors.

Conclusion

So, there you have it, guys! We've covered evaluating expressions with fractions and simplifying fractions. Remember, the key is to take it one step at a time, look for common factors, and don't be afraid to simplify incrementally. With a little practice, you'll be simplifying fractions like a math whiz in no time!

Fractions are a fundamental concept in mathematics, and mastering them will help you in various areas of math and everyday life. Keep practicing, and you'll become more confident and skilled in working with fractions. And remember, guys, math can be fun!