Solving $4 1/2 \times 3/4$ Find The Missing Numerator

Hey guys! Today, we're going to dive into a fun math problem that involves multiplying mixed numbers and fractions. The problem we're tackling is:

$4 \frac{1}{2} \times \frac{3}{4} = 3 \frac{[?]}{4}$

Our mission is to find the missing numerator that makes this equation true. Don't worry, it's not as scary as it looks! We'll break it down step by step so you can see exactly how to solve it. So, grab your pencils and let's get started!

Understanding Mixed Numbers and Fractions

Before we jump into solving the problem, let's quickly recap what mixed numbers and fractions are. This will make the whole process much clearer. A fraction represents a part of a whole. It's written with a numerator (the top number) and a denominator (the bottom number). For example, in the fraction $\frac{3}{4}$, 3 is the numerator and 4 is the denominator. It means we have 3 parts out of a total of 4 parts. A mixed number, on the other hand, is a combination of a whole number and a fraction. Our problem includes the mixed number $4 \frac{1}{2}$. This means we have 4 whole units and an additional half unit. To effectively multiply them, we'll need to convert mixed numbers into improper fractions. An improper fraction is one where the numerator is greater than or equal to the denominator, like $\frac{9}{2}$. Converting mixed numbers to improper fractions is crucial because it simplifies the multiplication process, allowing us to work with fractions in a more straightforward manner. This conversion is a foundational step in solving many mathematical problems involving fractions and mixed numbers, so mastering it is super helpful.

Converting Mixed Numbers to Improper Fractions

The first thing we need to do is convert the mixed number $4 \frac{1}{2}$ into an improper fraction. Here's how we do it:

1. Multiply the whole number by the denominator: 4 (whole number) × 2 (denominator) = 8
2. Add the numerator to the result: 8 + 1 (numerator) = 9
3. Keep the same denominator: The denominator stays as 2.

So, $4 \frac{1}{2}$ is equal to $\frac{9}{2}$ as an improper fraction. This conversion is a game-changer because it turns our mixed number into a format that's much easier to work with when multiplying. Think of it like translating from one language to another – we're just changing the way the number looks without changing its actual value. This step is essential for making the multiplication process smooth and accurate. Once you get the hang of converting mixed numbers to improper fractions, you'll find that many fraction-related problems become much more manageable. It's a fundamental skill that will boost your confidence in handling fractions and mixed numbers.

Multiplying Fractions

Now that we've converted the mixed number to an improper fraction, our problem looks like this:

$\frac{9}{2} \times \frac{3}{4} = 3 \frac{[?]}{4}$

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

• Numerator: 9 × 3 = 27
• Denominator: 2 × 4 = 8

So, $\frac{9}{2} \times \frac{3}{4} = \frac{27}{8}$. Multiplying fractions might seem like a basic operation, but it's a building block for more complex math. When you multiply fractions, you're essentially finding a fraction of a fraction. In this case, we're finding $\frac{3}{4}$ of $\frac{9}{2}$. This concept is super useful in real-life situations, like when you're calculating portions of a recipe or figuring out discounts at a store. The key to mastering fraction multiplication is to remember the simple rule: multiply the tops (numerators) and multiply the bottoms (denominators). This straightforward process makes it easy to handle even more complex calculations down the road. Practice makes perfect, so keep at it, and you'll become a fraction multiplication pro in no time!

Converting Improper Fractions to Mixed Numbers

We have our answer as an improper fraction, $\frac{27}{8}$, but the original problem asks for the answer as a mixed number. So, let's convert $\frac{27}{8}$ back into a mixed number. Here’s how:

1. Divide the numerator by the denominator: 27 ÷ 8 = 3 with a remainder of 3.
2. The quotient (3) becomes the whole number part.
3. The remainder (3) becomes the new numerator.
4. Keep the same denominator (8).

So, $\frac{27}{8}$ is equal to $3 \frac{3}{8}$. Converting improper fractions to mixed numbers is like putting the pieces of a puzzle back together. We started with a mixed number, turned it into an improper fraction to make multiplication easier, and now we're converting it back to a mixed number to match the format of the original problem. This process highlights the flexibility of fractions and how we can move between different forms to solve problems effectively. Understanding this conversion is crucial because it allows you to express your answers in the most appropriate way, whether it's a mixed number for everyday understanding or an improper fraction for calculations. It’s a valuable skill that makes working with fractions much more intuitive and practical.

Finding the Missing Numerator

Now we can see that $\frac{9}{2} \times \frac{3}{4} = \frac{27}{8} = 3 \frac{3}{8}$.

Comparing this to the original equation, $4 \frac{1}{2} \times \frac{3}{4} = 3 \frac{[?]}{4}$, we can see that the missing numerator is 3, but the denominator needs to be transformed from 8 to 4.

$\frac{3}{8}$ to $\frac{[?]}{4}$ by dividing both numerator and denominator by 2.

$\frac{3}{8} = \frac{3 \div 2}{8 \div 2} = \frac{1.5}{4}$

Oops! We have an unpredicted result. How about checking the calculation again?

$\frac{9}{2} \times \frac{3}{4} = \frac{27}{8}$ is correct. $\frac{27}{8} = 3 \frac{3}{8}$ is correct.

Let's go back to the original equation:

$4 \frac{1}{2} \times \frac{3}{4} = 3 \frac{[?]}{4}$

Something is wrong here. Let's verify whether the left part is equal to the right part, but, instead of [?], we have $\frac{3}{4}$ in right part.

$4 \frac{1}{2} \times \frac{3}{4} = 3 \frac{3}{4}$ $\frac{9}{2} \times \frac{3}{4} = 3 + \frac{3}{4}$ $\frac{27}{8} = \frac{12}{4} + \frac{3}{4}$ $\frac{27}{8} = \frac{15}{4}$ $\frac{27}{8} = \frac{30}{8}$

The equation is not correct. It should be:

$4 \frac{1}{2} \times \frac{3}{4} = 3 \frac{[3]}{8}$

So, the missing numerator is 3.

Answer

The missing numerator is 3. Great job, guys! You've successfully navigated through multiplying mixed numbers and fractions. Remember, the key is to break down the problem into smaller, manageable steps. Convert mixed numbers to improper fractions, multiply the fractions, and then convert back to a mixed number if needed. Keep practicing, and you'll become a master of fractions in no time!