Multiplying Fractions A Simple Guide With Examples

Hey guys! Let's dive into multiplying fractions. It might seem tricky at first, but trust me, it's super straightforward once you get the hang of it. We're going to break down the problem ${\frac{7}{2} \cdot \frac{1}{5} \cdot \frac{7}{3}}$ step by step, so you'll be multiplying fractions like a pro in no time!

Understanding Fractions

Before we jump into the multiplication, let's quickly recap what a fraction actually is. A fraction represents a part of a whole. It's written as two numbers separated by a line: the numerator (the top number) and the denominator (the bottom number). The numerator tells you how many parts you have, and the denominator tells you how many parts make up the whole. For example, in the fraction ${\frac{1}{2}}$, the numerator (1) means you have one part, and the denominator (2) means the whole is made up of two parts. So, you have one out of two parts, or half.

Key terms:

• Numerator: The top number in a fraction, indicating the number of parts you have.
• Denominator: The bottom number in a fraction, indicating the total number of parts in the whole.

The Simple Rule for Multiplying Fractions

Here's the golden rule: To multiply fractions, you simply multiply the numerators together and the denominators together. That's it! Seriously, it's that simple.

Multiply the Numerators:

You multiply the top numbers (numerators) of the fractions.

Multiply the Denominators:

You multiply the bottom numbers (denominators) of the fractions.

Write the New Fraction:

The result of multiplying the numerators becomes the new numerator, and the result of multiplying the denominators becomes the new denominator.

Let's write it out in a general formula:

${\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}}$

Where:

• a and c are the numerators.
• b and d are the denominators.

Solving ${\frac{7}{2} \cdot \frac{1}{5} \cdot \frac{7}{3}}$ Step-by-Step

Now, let's apply this rule to our problem: ${\frac{7}{2} \cdot \frac{1}{5} \cdot \frac{7}{3}}$. We have three fractions here, but don't worry, the rule still applies. We just multiply all the numerators together and all the denominators together.

Step 1: Multiply the Numerators

We have the numerators 7, 1, and 7. Let's multiply them:

${7 \cdot 1 \cdot 7 = 49}$

So, our new numerator is 49.

Step 2: Multiply the Denominators

Next, we multiply the denominators 2, 5, and 3:

${2 \cdot 5 \cdot 3 = 30}$

Our new denominator is 30.

Step 3: Write the New Fraction

Now, we combine our new numerator and denominator to form the resulting fraction:

${\frac{49}{30}}$

So, ${\frac{7}{2} \cdot \frac{1}{5} \cdot \frac{7}{3} = \frac{49}{30}}$.

This fraction, ${\frac{49}{30}}$, is our answer! It's an improper fraction, meaning the numerator is larger than the denominator. While this is a perfectly valid answer, we can also convert it to a mixed number if we want to.

Simplifying Fractions (If Possible)

Before we declare our final answer, it's always a good idea to check if the fraction can be simplified. Simplifying a fraction means reducing it to its lowest terms. To do this, we look for the greatest common factor (GCF) of the numerator and the denominator and divide both by it.

In our case, we have ${\frac{49}{30}}$. The factors of 49 are 1, 7, and 49. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The only common factor is 1, which means ${\frac{49}{30}}$ is already in its simplest form. Great!

When to Simplify?

Always check if your final fraction can be simplified. It's like putting the finishing touches on your masterpiece! Simplifying makes the fraction easier to understand and work with in future calculations.

Converting Improper Fractions to Mixed Numbers (Optional)

As we mentioned earlier, ${\frac{49}{30}}$ is an improper fraction. Sometimes, it's helpful to convert an improper fraction to a mixed number. A mixed number has a whole number part and a fractional part. To convert ${\frac{49}{30}}$ to a mixed number, we divide the numerator (49) by the denominator (30).

Step 1: Divide

Divide 49 by 30:

${49 \div 30 = 1}$ with a remainder of 19.

Step 2: Write the Mixed Number

The quotient (1) becomes the whole number part, the remainder (19) becomes the new numerator, and the denominator (30) stays the same.

So, ${\frac{49}{30}}$ as a mixed number is ${1\frac{19}{30}}$.

Why Convert to Mixed Numbers?

Mixed numbers can give you a better sense of the quantity you're dealing with. For instance, ${1\frac{19}{30}}$ tells you that you have one whole and a little more. It's often easier to visualize and compare mixed numbers in real-world scenarios.

Let's Recap and Solidify Our Understanding

Okay, let’s recap the steps to make sure we've got this down. Remember, multiplying fractions is all about multiplying straight across – numerators with numerators, denominators with denominators.

1. Multiply the numerators: Multiply all the top numbers together.
2. Multiply the denominators: Multiply all the bottom numbers together.
3. Write the new fraction: Place the product of the numerators over the product of the denominators.
4. Simplify (if possible): Reduce the fraction to its lowest terms by dividing the numerator and denominator by their greatest common factor.
5. Convert to a mixed number (optional): If the fraction is improper, you can convert it to a mixed number.

We've successfully walked through multiplying ${\frac{7}{2} \cdot \frac{1}{5} \cdot \frac{7}{3}}$, and we found the answer to be ${\frac{49}{30}}$ or ${1\frac{19}{30}}$.

Practice Makes Perfect: More Examples

To really master multiplying fractions, practice is key! Let's work through a few more examples together.

Example 1:

${\frac{3}{4} \cdot \frac{2}{5}}$

1. Multiply the numerators: ${3 \cdot 2 = 6}$
2. Multiply the denominators: ${4 \cdot 5 = 20}$
3. Write the new fraction: ${\frac{6}{20}}$
4. Simplify: Both 6 and 20 are divisible by 2. ${\frac{6 \div 2}{20 \div 2} = \frac{3}{10}}$

So, ${\frac{3}{4} \cdot \frac{2}{5} = \frac{3}{10}}$.

Example 2:

${\frac{1}{3} \cdot \frac{5}{7} \cdot \frac{2}{3}}$

1. Multiply the numerators: ${1 \cdot 5 \cdot 2 = 10}$
2. Multiply the denominators: ${3 \cdot 7 \cdot 3 = 63}$
3. Write the new fraction: ${\frac{10}{63}}$
4. Simplify: 10 and 63 have no common factors other than 1, so the fraction is already in its simplest form.

Thus, ${\frac{1}{3} \cdot \frac{5}{7} \cdot \frac{2}{3} = \frac{10}{63}}$.

Example 3:

${\frac{8}{3} \cdot \frac{1}{4}}$

1. Multiply the numerators: ${8 \cdot 1 = 8}$
2. Multiply the denominators: ${3 \cdot 4 = 12}$
3. Write the new fraction: ${\frac{8}{12}}$
4. Simplify: Both 8 and 12 are divisible by 4. ${\frac{8 \div 4}{12 \div 4} = \frac{2}{3}}$

Therefore, ${\frac{8}{3} \cdot \frac{1}{4} = \frac{2}{3}}$.

Common Mistakes to Avoid

Even though multiplying fractions is straightforward, there are a few common mistakes that students sometimes make. Let's go over them so you can steer clear of these pitfalls.

Mistake 1: Adding Numerators and Denominators

One of the most common mistakes is adding the numerators and denominators instead of multiplying them. Remember, we multiply across, not add!

Incorrect:

${\frac{1}{2} \cdot \frac{1}{3} \neq \frac{1+1}{2+3} = \frac{2}{5}}$

Correct:

${\frac{1}{2} \cdot \frac{1}{3} = \frac{1 \cdot 1}{2 \cdot 3} = \frac{1}{6}}$

Mistake 2: Forgetting to Simplify

Always, always check if your final fraction can be simplified. Leaving a fraction unsimplified is like not quite finishing a puzzle. Make sure you find the greatest common factor and divide both the numerator and denominator by it.

Mistake 3: Mixing Up Numerators and Denominators

It's crucial to keep the numerators and denominators in their correct places. The numerator is always on top, and the denominator is always on the bottom. If you mix them up, you'll get the wrong answer!

Mistake 4: Not Multiplying All Numerators or Denominators

When you're multiplying more than two fractions, make sure you multiply all the numerators together and all the denominators together. Don't leave any out!

Mistake 5: Incorrectly Converting to Mixed Numbers

If you choose to convert an improper fraction to a mixed number, make sure you do the division correctly and place the quotient, remainder, and original denominator in the correct spots. A little mistake in the division can lead to a wrong mixed number.

By being aware of these common mistakes, you can avoid them and ensure you're multiplying fractions accurately.

Real-World Applications of Multiplying Fractions

You might be wondering,