Multiplying $20 \times -1 \frac{7}{8}$: A Step-by-Step Guide

Let's dive into how to multiply $20$ by $-1 \frac{7}{8}$. This might seem a bit tricky at first, especially with that mixed number, but don't worry! We'll break it down step by step to make it super clear. Understanding how to work with mixed numbers and negative signs is a crucial skill in mathematics, and by the end of this article, you'll be a pro at solving similar problems.

Understanding the Problem

Before we jump into the calculation, let's make sure we understand what the problem is asking. We need to find the product of $20$ and $-1 \frac{7}{8}$. Remember, the product is the result of multiplication. The presence of a negative sign and a mixed number adds a bit of complexity, but nothing we can't handle. The key to solving this effectively is to convert the mixed number into an improper fraction. This simplifies the multiplication process and makes it less prone to errors. So, let's get started with converting that mixed number!

Step 1: Convert the Mixed Number to an Improper Fraction

The first hurdle is dealing with the mixed number $-1 \frac{7}{8}$. To make our multiplication easier, we need to convert this to an improper fraction. An improper fraction is one where the numerator (the top number) is larger than the denominator (the bottom number).

Here's how we do it:

1. Multiply the whole number part (1) by the denominator (8): $1 \times 8 = 8$.
2. Add the result to the numerator (7): $8 + 7 = 15$.
3. Place this new number (15) over the original denominator (8). So, we get $\frac{15}{8}$.
4. Since our original mixed number was negative, we keep the negative sign. So, $-1 \frac{7}{8}$ becomes $-\frac{15}{8}$.

Now, our problem looks simpler: $20 \times -\frac{15}{8}$. We've successfully transformed the mixed number into a more manageable form. Next, we'll rewrite $20$ as a fraction to set up the multiplication clearly. This step is crucial because it allows us to multiply fractions directly, making the process straightforward and less confusing. Remember, every whole number can be written as a fraction with a denominator of 1.

Step 2: Rewrite 20 as a Fraction

To multiply a whole number by a fraction, it helps to write the whole number as a fraction as well. Any whole number can be written as a fraction by simply putting it over 1. So, $20$ can be written as $\frac{20}{1}$.

Now our problem looks even clearer: $\frac{20}{1} \times -\frac{15}{8}$. We're all set to multiply these two fractions together! This step is important because it unifies the terms into a single format, making the multiplication process much simpler and visually clearer. Without this step, multiplying a whole number by a fraction can seem abstract, but by rewriting the whole number, we've made the process concrete and easy to follow.

Step 3: Multiply the Fractions

Now that we have two fractions, $\frac{20}{1}$ and $-\frac{15}{8}$, we can multiply them together. To multiply fractions, we multiply the numerators (the top numbers) and then multiply the denominators (the bottom numbers).

So, here's how it works:

• Multiply the numerators: $20 \times -15 = -300$.
• Multiply the denominators: $1 \times 8 = 8$.

This gives us the fraction $-\frac{300}{8}$. We've now performed the multiplication, but our answer is in the form of an improper fraction. To make it more readable and useful, we need to simplify this fraction. This involves reducing the fraction to its simplest form and possibly converting it back into a mixed number. Simplifying fractions is an essential skill in mathematics, as it allows us to express results in a clear and concise manner.

Step 4: Simplify the Fraction

We now have the fraction $-\frac{300}{8}$. This is an improper fraction, and it's not in its simplest form. To simplify it, we need to find the greatest common divisor (GCD) of the numerator (300) and the denominator (8) and divide both by it.

First, let's find the GCD of 300 and 8. The factors of 8 are 1, 2, 4, and 8. The factors of 300 include 1, 2, 3, 4, 5, 6, 10, and so on. The greatest common factor is 4.

Now, divide both the numerator and the denominator by 4:

• $-300 \div 4 = -75$
• $8 \div 4 = 2$

So, our simplified fraction is $-\frac{75}{2}$. This is still an improper fraction, but it's in its simplest form. For many applications, it's helpful to convert this back into a mixed number. Converting back to a mixed number helps us to better understand the magnitude of the number and makes it easier to visualize. It's a practical skill for real-world applications, such as measuring ingredients in a recipe or calculating lengths and distances.

Step 5: Convert the Improper Fraction to a Mixed Number (Optional)

While $-\frac{75}{2}$ is a perfectly valid answer, it's often helpful to convert an improper fraction back to a mixed number. This gives us a better sense of the value.

To convert $-\frac{75}{2}$ to a mixed number, we divide 75 by 2:

• $75 \div 2 = 37$ with a remainder of 1.

This means that $-\frac{75}{2}$ is equal to $-37 \frac{1}{2}$. So, our final answer is $-37 \frac{1}{2}$. We've successfully converted the improper fraction into a mixed number, providing a more intuitive representation of the result. This final step helps solidify our understanding of the problem and ensures we can express our answer in the most appropriate form.

Final Answer

So, $20 \times -1 \frac{7}{8} = -37 \frac{1}{2}$.

Conclusion

We've successfully navigated this multiplication problem by breaking it down into manageable steps. First, we converted the mixed number to an improper fraction. Then, we rewrote the whole number as a fraction. Next, we multiplied the fractions and simplified the result. Finally, we converted the improper fraction back to a mixed number for clarity. Each of these steps is a valuable tool in your mathematical toolkit. By mastering these skills, you'll be well-equipped to tackle a wide range of mathematical problems. Remember, practice makes perfect, so keep working at it, and you'll see your skills improve over time.

If you're looking to further enhance your understanding of fractions and multiplication, consider exploring additional resources. Websites like Khan Academy offer comprehensive lessons and practice exercises on various math topics, including fractions and mixed numbers. Engaging with these resources can help solidify your knowledge and build confidence in your mathematical abilities.