Solving $3 \frac{2}{3} \times \frac{2}{5}$ Find The Missing Numerator

Hey guys! Let's dive into this math problem where we need to find the product of mixed numbers and then figure out a missing numerator. It might sound a bit tricky at first, but trust me, we'll break it down step by step and make it super easy to understand. We're dealing with the equation $3 \frac{2}{3} \times \frac{2}{5} = 1 \frac{[?]}{4}$, and our mission is to find the number that goes in that bracket to make the equation true. So, grab your thinking caps, and let's get started!

Before we jump into solving, let's make sure we understand what the problem is asking. We have a mixed number, $3 \frac{2}{3}$, which we're multiplying by a fraction, $\frac{2}{5}$. The result is another mixed number, $1 \frac{[?]}{4}$, but with a missing numerator. Our goal is to find that missing number. This involves a few key steps: first, we'll need to multiply the mixed number and the fraction. Then, we'll compare our result with the mixed number on the right side of the equation to figure out the missing piece. It's like solving a puzzle, and each step gets us closer to the final answer.

To make the multiplication easier, the first thing we need to do is convert the mixed number, $3 \frac{2}{3}$, into an improper fraction. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). Here’s how we do it:

1. Multiply the whole number part (3) by the denominator (3): 3 * 3 = 9
2. Add the numerator (2) to the result: 9 + 2 = 11
3. Put this new number (11) over the original denominator (3). So, $3 \frac{2}{3}$ becomes $\frac{11}{3}$.

Now, instead of multiplying $3 \frac{2}{3} \times \frac{2}{5}$, we'll be multiplying $\frac{11}{3} \times \frac{2}{5}$. See how much simpler that looks?

Now that we have two fractions, $\frac{11}{3}$ and $\frac{2}{5}$, we can multiply them together. Multiplying fractions is pretty straightforward. You just multiply the numerators together and the denominators together.

So, here’s how it looks:

• Multiply the numerators: 11 * 2 = 22
• Multiply the denominators: 3 * 5 = 15

That gives us $\frac{22}{15}$. Great! We've found the product of the two fractions, but we're not quite done yet. Remember, we need to express our answer as a mixed number in the simplest form.

We've got the improper fraction $\frac{22}{15}$, but we need to turn it back into a mixed number so we can compare it with the $1 \frac{[?]}{4}$ form in the original equation. To do this, we'll divide the numerator (22) by the denominator (15).

• How many times does 15 go into 22? It goes in 1 time.
• What's the remainder? 22 - 15 = 7

So, we have a whole number of 1 and a remainder of 7. This means our mixed number will be 1 and $\frac{7}{15}$. We write this as $1 \frac{7}{15}$. Now we’re getting closer to solving for that missing numerator!

Okay, we've found that $3 \frac{2}{3} \times \frac{2}{5} = 1 \frac{7}{15}$. But remember, the original problem gives us the answer in a slightly different form: $1 \frac{[?]}{4}$. This means we need to figure out how the fraction $\frac{7}{15}$ relates to a fraction with a denominator of 4. Hmm, this might need a little bit of extra work.

Let's analyze this carefully. We have $1 \frac{7}{15}$ and we want to express it in the form of $1 \frac{[?]}{4}$. The whole number part is already the same (1), so we just need to focus on the fractional parts. We need to find an equivalent fraction to $\frac{7}{15}$ but with a denominator of 4. This is where we hit a snag! You see, 15 and 4 don't have a common factor that we can easily use to convert the fraction. This suggests there might be a slight issue with the problem as it's presented.

It looks like there might be a small error in the problem statement. The fraction $\frac{7}{15}$ cannot be directly converted into a fraction with a denominator of 4 while keeping the numerator as a whole number. This is because 15 and 4 do not share any common factors other than 1, meaning we can't simplify or multiply the fraction $\frac{7}{15}$ to have a denominator of 4.

However, let's think about what we've done so far. We correctly converted the mixed number to an improper fraction, multiplied the fractions, and converted the result back to a mixed number. We know that $3 \frac{2}{3} \times \frac{2}{5}$ is indeed equal to $1 \frac{7}{15}$.

Given the problem's format, it's likely that the intended answer should have a denominator that is compatible with 15. Perhaps there was a typo, and the denominator should have been something else.

Based on our calculations, the correct answer in the simplest form is $1 \frac{7}{15}$. If we were to express the answer as a mixed number with a different denominator, we would need to find a common denominator between 15 and the desired denominator, and then adjust the numerator accordingly. However, since we cannot directly convert $\frac{7}{15}$ to a fraction with a denominator of 4, we'll stick with our correct result.

If the question intended to ask for the numerator when the mixed number is in the simplest form, then the answer would be 7. So, the numerator is 7.

So, guys, we've tackled this problem step by step! We converted a mixed number to an improper fraction, multiplied fractions, and converted back to a mixed number. We even encountered a little hiccup with the final comparison, but we worked through it logically. Remember, math is all about understanding the process and being careful with each step. Keep practicing, and you'll become math pros in no time! If you have more questions or problems you'd like to solve, just let me know. Keep up the great work!