Solving Proportions: Cross Products Property Explained

Welcome! In this article, we're diving deep into the cross products property, a handy tool for solving proportions. Specifically, we'll tackle the proportion $\frac{36}{42}=\frac{24}{r}$ step by step. If you've ever felt a little lost when dealing with fractions and equal ratios, you're in the right place. We'll break down each step in a clear, friendly way so you can confidently solve similar problems on your own. Let’s get started and make math a little less mysterious!

Understanding Proportions

Before we jump into solving our specific proportion, let's make sure we're all on the same page about what a proportion actually is. Proportions are, at their core, statements of equality between two ratios or fractions. A ratio, simply put, compares two quantities. Think of it as a way of saying how much of one thing there is compared to another. For example, if you have 3 apples and 5 oranges, the ratio of apples to oranges is 3 to 5, which can be written as 3:5 or as a fraction, ${\frac{3}{5}}$.

Now, when we say two ratios are in proportion, we mean they are equivalent. This means that the relationship between the two quantities in one ratio is the same as the relationship in the other ratio. Mathematically, a proportion looks like this: ${\frac{a}{b} = \frac{c}{d}}$, where ${a}$, ${b}$, ${c}$, and ${d}$ are numbers. This equation is telling us that the ratio of ${a}$ to ${b}$ is equal to the ratio of ${c}$ to ${d}$. This concept is the foundation for many real-world applications, from scaling recipes to calculating distances on a map.

To truly grasp the concept, consider a practical example. Imagine you're baking a cake and the recipe calls for 2 cups of flour for every 1 cup of sugar. If you want to make a larger cake and use 4 cups of flour, you'll need to maintain the same ratio to ensure the cake turns out right. This means you'll need 2 cups of sugar. The proportion here is ${\frac{2 \text{ cups of flour}}{1 \text{ cup of sugar}} = \frac{4 \text{ cups of flour}}{2 \text{ cups of sugar}}}$. See how the relationship between flour and sugar remains consistent? That’s the essence of a proportion. Understanding this foundational concept is crucial because it sets the stage for using tools like the cross products property, which we'll explore in detail next. So, keep in mind that proportions are all about maintaining equivalent ratios, and they're incredibly useful in solving problems where relationships between quantities need to stay consistent.

The Cross Products Property: A Powerful Tool

The cross products property is a neat trick that simplifies solving proportions. It's a fundamental concept in mathematics, particularly useful when you're faced with equations where two fractions are set equal to each other. This property offers a straightforward method to eliminate fractions and transform the proportion into a more manageable equation. So, what exactly is this property? In simple terms, the cross products property states that if you have a proportion like ${\frac{a}{b} = \frac{c}{d}}$, then the product of ${a}$ and ${d}$ is equal to the product of ${b}$ and ${c}$. Mathematically, this looks like: ${a imes d = b imes c}$.

Why does this work? Think of it as a shortcut to clearing fractions. If you were to solve the proportion ${\frac{a}{b} = \frac{c}{d}}$ using traditional algebraic methods, you might multiply both sides of the equation by ${b}$ and then by ${d}$ to eliminate the denominators. This is precisely what the cross products property achieves in a single step. By multiplying diagonally, you're essentially doing the same thing, but in a more streamlined way. This makes it incredibly efficient for solving for unknown variables in proportions.

Let's illustrate this with a simple example. Suppose we have the proportion ${\frac{2}{3} = \frac{4}{6}}$. Applying the cross products property, we multiply 2 by 6 and 3 by 4. This gives us ${2 imes 6 = 12}$ and ${3 imes 4 = 12}$. Since both products are equal, this confirms that the proportion is indeed valid. Now, let's take it a step further and see how this property helps us solve for an unknown. Imagine we have the proportion ${\frac{1}{2} = \frac{x}{4}}$, where ${x}$ is the unknown variable we want to find. Using the cross products property, we get ${1 imes 4 = 2 imes x}$, which simplifies to ${4 = 2x}$. To solve for ${x}$, we divide both sides by 2, resulting in ${x = 2}$. This illustrates how the cross products property not only verifies proportions but also provides a clear path to finding unknown values.

The power of the cross products property lies in its simplicity and effectiveness. It transforms a potentially complex fraction equation into a basic algebraic one, making it much easier to solve. As we move on to tackling our main problem, ${\frac{36}{42} = \frac{24}{r}}$, you'll see just how valuable this tool can be. Keep this concept in mind, as it’s the key to unlocking solutions for many proportion-related problems.

Applying the Cross Products Property to Our Proportion

Now that we have a solid understanding of the cross products property, let's put it to work on our original problem: solving the proportion ${\frac{36}{42} = \frac{24}{r}}$. This proportion presents us with an unknown variable, ${r}$, in the denominator of one fraction, and our goal is to find its value. The cross products property is perfectly suited for this task, as it will help us eliminate the fractions and isolate ${r}$.

To begin, we apply the cross products property, which means we multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. In our case, this means multiplying 36 by ${r}$ and 42 by 24. This gives us the equation ${36 imes r = 42 imes 24}$. Notice how the fractions have disappeared, and we're left with a straightforward algebraic equation. This is the beauty of the cross products property – it transforms a proportional relationship into a simple multiplication problem.

Next, we perform the multiplication on the right side of the equation. Multiplying 42 by 24 gives us 1008. So, our equation now looks like this: ${36r = 1008}$. Our objective is to isolate ${r}$ on one side of the equation. Currently, ${r}$ is being multiplied by 36. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by 36. This step is crucial because it maintains the equality of the equation while moving us closer to solving for ${r}$.

Dividing both sides by 36, we get ${r = \frac{1008}{36}}$. Now, we simply need to perform the division. Dividing 1008 by 36 gives us 28. Therefore, the solution to our proportion is ${r = 28}$. This means that the value of ${r}$ that makes the proportion ${\frac{36}{42} = \frac{24}{r}}$ true is 28. We've successfully used the cross products property to solve for the unknown variable. This step-by-step process demonstrates how powerful and efficient this property can be. By turning a proportion into a simple algebraic equation, we can easily find the value of any unknown variable.

Step-by-Step Solution

Let's recap the steps we took to solve the proportion ${\frac{36}{42} = \frac{24}{r}}$ using the cross products property. This step-by-step breakdown will not only reinforce the method we used but also provide a clear guide for tackling similar problems in the future. Each step is crucial, and understanding why we take each action is just as important as the action itself. So, let’s walk through it methodically.

Step 1: Apply the Cross Products Property. The first action we take is to apply the cross products property. This involves multiplying the numerator of the first fraction by the denominator of the second fraction and setting it equal to the product of the denominator of the first fraction and the numerator of the second fraction. For our proportion, ${\frac{36}{42} = \frac{24}{r}}$, this translates to multiplying 36 by ${r}$ and 42 by 24, giving us the equation ${36 imes r = 42 imes 24}$. This step is the cornerstone of our approach, as it transforms the proportion into a manageable equation without fractions.

Step 2: Perform the Multiplication. Next, we carry out the multiplication on both sides of the equation. On the left side, we have ${36 imes r}$, which we write as ${36r}$. On the right side, we multiply 42 by 24, which equals 1008. So, our equation now looks like ${36r = 1008}$. This step simplifies the equation, making it easier to isolate the variable we're trying to solve for.

Step 3: Isolate the Variable. Our goal is to get ${r}$ by itself on one side of the equation. Currently, ${r}$ is being multiplied by 36. To undo this multiplication, we perform the inverse operation: division. We divide both sides of the equation by 36. This keeps the equation balanced and moves us closer to our solution. Dividing ${36r}$ by 36 gives us ${r}$, and dividing 1008 by 36 gives us 28. This leads us to our final step.

Step 4: Solve for r. After dividing both sides by 36, we find that ${r = 28}$. This is the solution to our proportion. It means that if we substitute 28 for ${r}$ in the original proportion, the two ratios will be equal. We've successfully used the cross products property to find the value of the unknown variable. These four steps provide a clear, repeatable process for solving proportions. By understanding each step and why it's necessary, you can confidently tackle a wide range of proportion problems.

Verification and Conclusion

After solving for ${r}$, it's always a good practice to verify our solution. This step ensures that our answer is correct and that we haven't made any mistakes along the way. Verification involves substituting the value we found for ${r}$ back into the original proportion and checking if the two ratios are indeed equal. This not only confirms our solution but also deepens our understanding of proportions.

Our original proportion was ${\frac{36}{42} = \frac{24}{r}}$, and we found that ${r = 28}$. To verify, we substitute 28 for ${r}$ in the proportion, giving us ${\frac{36}{42} = \frac{24}{28}}$. Now, we need to check if these two fractions are equivalent. One way to do this is to simplify both fractions to their lowest terms and see if they are the same.

Let's simplify ${\frac{36}{42}}$. Both 36 and 42 are divisible by 6. Dividing both the numerator and the denominator by 6, we get ${\frac{36 ext{ ÷ } 6}{42 ext{ ÷ } 6} = \frac{6}{7}}$. Next, let's simplify ${\frac{24}{28}}$. Both 24 and 28 are divisible by 4. Dividing both the numerator and the denominator by 4, we get ${\frac{24 ext{ ÷ } 4}{28 ext{ ÷ } 4} = \frac{6}{7}}$. Now we can clearly see that both fractions, when simplified, are equal to ${\frac{6}{7}}$. This confirms that our solution, ${r = 28}$, is correct. The two ratios are indeed proportional, and our verification step has given us confidence in our answer.

In conclusion, we've successfully solved the proportion ${\frac{36}{42} = \frac{24}{r}}$ using the cross products property. We walked through each step, from applying the property to isolating the variable and finally verifying our solution. The cross products property is a powerful tool for solving proportions, and by mastering this technique, you can tackle a wide range of mathematical problems involving ratios and proportions. Remember, the key is to understand the underlying principles and practice applying them. With each problem you solve, your confidence and skills will grow. Keep practicing, and you'll become a proportion-solving pro! For further learning on proportions and related mathematical concepts, check out resources like Khan Academy's Proportions Section. Happy solving!