Finding The Missing Fraction: Product Is 24, One Fraction 4 1/5

by TextBrain Team 64 views

Hey guys! Ever found yourself scratching your head over a math problem that seems to mix fractions and whole numbers? Don't worry, we've all been there! Today, we're going to tackle a classic problem: finding a missing fraction when you know the product of two fractions and one of the fractions. Specifically, we're diving into a problem where the product is 24, and one of the fractions is 4 1/5. Sounds intriguing, right? Let’s break it down step by step, making sure it’s super clear and easy to follow. By the end of this, you'll be a pro at solving these types of problems! So, grab your thinking caps, and let’s get started on this mathematical adventure together. We're going to explore the ins and outs of fractions, multiplication, and how they all connect to help us solve this puzzle. Remember, math is like a puzzle – each piece fits perfectly, and finding that fit is what makes it so satisfying!

Understanding the Problem

Before we jump into calculations, let's really understand what the problem is asking. The core question is: If we multiply two fractions together and get 24, and we know one of those fractions is 4 1/5, how do we find the other fraction? This type of problem is all about understanding the relationship between multiplication and division. Think of it like this: if you know that 2 x 3 = 6, and you know the answer is 6 and one of the numbers is 2, you can find the other number by dividing 6 by 2. We’re going to use the same logic here, but with fractions! The key is to recognize that finding a missing factor in a multiplication problem is essentially a division problem in disguise. We need to divide the product (24) by the known fraction (4 1/5) to find the missing fraction. This might sound a bit daunting at first, especially with the mixed number involved, but don't worry! We're going to take it slow and make sure we understand each step. We'll also touch on why this method works, reinforcing your understanding of the fundamentals of fraction multiplication and division. So, let's get ready to translate this word problem into a mathematical equation and then solve it.

Converting the Mixed Number to an Improper Fraction

The first step in solving our problem is to deal with the mixed number, 4 1/5. A mixed number is a combination of a whole number and a fraction, and it can be a bit tricky to work with directly in multiplication and division. So, we need to convert it into an improper fraction. An improper fraction is one where the numerator (the top number) is larger than or equal to the denominator (the bottom number). To convert 4 1/5 into an improper fraction, we follow a simple process:

  1. Multiply the whole number (4) by the denominator of the fraction (5): 4 * 5 = 20
  2. Add the result to the numerator of the fraction (1): 20 + 1 = 21
  3. Keep the same denominator (5). So, 4 1/5 becomes 21/5.

Why do we do this? Converting to an improper fraction makes the multiplication and division processes much smoother. When we have a single fraction, it's easier to apply the rules of fraction arithmetic. Think of it as changing the form of the number without changing its value. We're just expressing the same quantity in a way that's more convenient for our calculations. This is a crucial step in many fraction problems, so mastering this conversion is a valuable skill. Now that we've transformed our mixed number into an improper fraction, we're one step closer to solving the problem. Let's move on to the next phase: setting up the division problem.

Setting Up the Division Problem

Now that we've converted 4 1/5 to its improper fraction form, 21/5, we can set up the division problem. Remember, we're trying to find the missing fraction that, when multiplied by 21/5, gives us 24. So, we need to divide the product (24) by the known fraction (21/5). This can be written as:

24 ÷ (21/5)

But wait, how do we divide a whole number by a fraction? This is where the concept of division as the inverse of multiplication comes into play. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is simply the fraction flipped upside down. So, the reciprocal of 21/5 is 5/21. This is a key concept to grasp because it transforms our division problem into a multiplication problem, which is often easier to handle. So, we rewrite our division problem as a multiplication problem:

24 * (5/21)

This transformation is super important because it allows us to use the rules of fraction multiplication that we're probably already familiar with. It's like finding a secret code that unlocks the solution! By changing the division into multiplication by the reciprocal, we've made the problem much more manageable. Now, we're ready to actually perform the multiplication and find our missing fraction. Let's dive into the next step and see how it's done.

Performing the Multiplication

Okay, we've transformed our problem into a multiplication: 24 * (5/21). Now, let's get down to the actual multiplication. To multiply a whole number by a fraction, we can think of the whole number as a fraction with a denominator of 1. So, 24 becomes 24/1. Now we have:

(24/1) * (5/21)

To multiply fractions, we simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together:

Numerator: 24 * 5 = 120 Denominator: 1 * 21 = 21

So, we get 120/21. But hold on! This fraction looks a bit…clunky. It’s an improper fraction, and it might not be in its simplest form. This is where simplification comes in handy. Simplifying fractions makes them easier to understand and work with. We're essentially reducing the fraction to its lowest terms without changing its value. Think of it as tidying up our answer to make it look its best. Before we simplify, let's just recap what we've done so far: we've converted a mixed number to an improper fraction, turned a division problem into a multiplication problem, and multiplied the fractions. We're on the home stretch now! The next step is to simplify our result, 120/21, and then we'll have our answer. Let's get to it!

Simplifying the Fraction

We've arrived at the fraction 120/21, which, as we discussed, isn't in its simplest form. To simplify a fraction, we need to find the greatest common factor (GCF) of the numerator and the denominator and then divide both by that number. The GCF is the largest number that divides evenly into both numbers. Let's find the GCF of 120 and 21.

Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 Factors of 21: 1, 3, 7, 21

The greatest common factor of 120 and 21 is 3. Now, we divide both the numerator and the denominator by 3:

120 ÷ 3 = 40 21 ÷ 3 = 7

So, 120/21 simplifies to 40/7. This is an improper fraction, which is perfectly fine, but sometimes it's more helpful to express the answer as a mixed number, especially if the original problem included one. Converting an improper fraction to a mixed number gives us a better sense of the actual value. It's like switching from a complicated code to plain English! Simplifying fractions is a crucial skill in math, not just for this problem, but for many others as well. It makes calculations easier and the answers more understandable. We're now just one step away from the final answer. Let's convert 40/7 to a mixed number and see what we get.

Converting Back to a Mixed Number (Optional)

Our simplified fraction, 40/7, is a perfectly valid answer, but to make it even clearer and perhaps more relatable to the original problem's form, we can convert it back into a mixed number. To do this, we divide the numerator (40) by the denominator (7):

40 ÷ 7 = 5 with a remainder of 5

The whole number part of our mixed number is the quotient (5), the numerator of the fractional part is the remainder (5), and the denominator stays the same (7). So, 40/7 as a mixed number is 5 5/7. This means that the other fraction we were looking for is 5 5/7. We've come full circle, starting with a mixed number in the problem and, if we choose, ending with one in our answer. Converting back to a mixed number can be particularly useful in real-world contexts where we're dealing with quantities that are easier to visualize as whole numbers and fractions. Think about measuring ingredients for a recipe or calculating distances. And there you have it! We've successfully found the missing fraction. Let's take a moment to recap our steps and celebrate our achievement.

Final Answer and Recap

So, after all our calculations, we've found that the other fraction is 5 5/7. Let's quickly recap the steps we took to get there:

  1. Understood the problem: We identified that we needed to find a missing factor in a multiplication problem.
  2. Converted the mixed number to an improper fraction: 4 1/5 became 21/5.
  3. Set up the division problem: We recognized that dividing by a fraction is the same as multiplying by its reciprocal.
  4. Performed the multiplication: We multiplied 24 by the reciprocal of 21/5, which is 5/21.
  5. Simplified the fraction: We reduced 120/21 to 40/7 by dividing both by their greatest common factor, 3.
  6. Converted back to a mixed number (optional): We changed 40/7 to 5 5/7 for clarity.

Throughout this process, we've not only solved the problem but also reinforced several important concepts in fraction arithmetic. We've seen how converting between mixed numbers and improper fractions can simplify calculations, how division can be transformed into multiplication using reciprocals, and how simplifying fractions gives us the clearest answer. Remember, math problems are often like puzzles, and each step is a piece that fits into the larger picture. By breaking down the problem into manageable steps and understanding the underlying concepts, we can tackle even the most challenging questions. Give yourself a pat on the back for making it through this problem! You've added another tool to your math toolkit, and you're ready to take on more fraction challenges. Keep practicing, and you'll be amazed at how much you can achieve!