Solving -2.4 X (-9.1) A Step-by-Step Guide

Hey guys! Let's dive into a topic that might seem a bit tricky at first glance: multiplying negative decimals. Don't worry, it's not as daunting as it sounds. In this article, we're going to break down the process step by step, ensuring you grasp the concept fully. We'll tackle the problem $-2.4 imes (-9.1)$, but more importantly, we'll equip you with the skills to solve similar problems with confidence. So, grab your calculators (or your mental math hats!), and let's get started!

Before we jump into negative decimals, let’s quickly recap multiplying regular decimals. Think of decimals as fractions in disguise. For instance, 2.4 is the same as 2 and 4/10, or 24/10. When we multiply decimals, we essentially multiply these fractions.

The key is to ignore the decimal points initially and treat the numbers as whole numbers. After you've multiplied the whole numbers, you'll need to figure out where to place the decimal point in your answer. This is where the counting game begins! Count the total number of digits to the right of the decimal points in the original numbers. This count tells you how many digits should be to the right of the decimal point in your final answer. For example, if you're multiplying 2.5 (one digit to the right of the decimal) by 3.25 (two digits to the right of the decimal), your answer should have three digits to the right of the decimal point.

This foundational understanding is crucial because the same principles apply when we introduce negative signs into the mix. We're just adding one extra layer: the rules of sign multiplication.

Now, let’s talk about the golden rules of signs. These are super important because they dictate whether your final answer will be positive or negative. Here's the lowdown:

• A positive number multiplied by a positive number results in a positive number. (+ × + = +)
• A negative number multiplied by a negative number results in a positive number. (- × - = +)
• A positive number multiplied by a negative number (or vice versa) results in a negative number. (+ × - = - or - × + = -)

Notice a pattern? When the signs are the same (both positive or both negative), the result is positive. When the signs are different, the result is negative. Keep these rules in your back pocket; they're your best friends in decimal multiplication and beyond!

Alright, let's put our knowledge to the test and tackle the problem $-2.4 imes (-9.1)$. We'll break it down into manageable steps so you can follow along easily.

Step 1: Ignore the Signs and Multiply

First things first, let’s pretend the negative signs aren't there for a moment. We'll focus on multiplying 2.4 by 9.1. This makes the calculation less cluttered and helps us concentrate on the numbers themselves.

So, we multiply 24 by 91 as if they were whole numbers:

  24
× 91
-----
  24
216
-----
2184

We get 2184. Great! We've handled the multiplication part. Now, let's bring the decimals back into the picture.

Step 2: Place the Decimal Point

Remember the decimal counting game? We need to figure out how many digits should be to the right of the decimal point in our answer. In 2.4, there's one digit to the right of the decimal point, and in 9.1, there's also one digit. That means our answer should have a total of 1 + 1 = 2 digits to the right of the decimal point.

So, we take our result, 2184, and count two digits from the right: 21.84.

Step 3: Apply the Sign Rule

Here comes the crucial part: figuring out the sign. We're multiplying $-2.4$ by $-9.1$. According to our rules, a negative number multiplied by a negative number results in a positive number.

Therefore, our final answer is positive 21.84. We don't need to write a plus sign; it's understood that a number without a sign is positive.

To really solidify your understanding, let's walk through a couple more examples and then give you some practice problems to try on your own.

Example 1: $-3.5 imes 1.2$

1. Multiply without signs: $35 imes 12 = 420$
2. Place the decimal: 3. 5 (one digit) and 1.2 (one digit) mean our answer should have two digits to the right of the decimal point: 4.20
3. Apply the sign rule: Negative times positive is negative. So, the answer is -4.20 (or -4.2).

Example 2: $4.7 imes (-2.5)$

1. Multiply without signs: $47 imes 25 = 1175$
2. Place the decimal: 4. 7 (one digit) and 2.5 (one digit) mean our answer should have two digits to the right of the decimal point: 11.75
3. Apply the sign rule: Positive times negative is negative. So, the answer is -11.75.

Practice Problems:

1. $-1.8 imes (-6.3)$
2. $5.2 imes (-3.1)$
3. $-7.5 imes 2.4$
4. $-4.6 imes (-8.2)$

Take your time, work through the steps, and remember those sign rules! The more you practice, the more confident you'll become.

Now, let's talk about some common pitfalls people encounter when multiplying negative decimals. Being aware of these mistakes can save you a lot of headaches.

• Forgetting the Sign: This is the most frequent error. Always remember to apply the sign rules after you've done the multiplication. Circle the signs before you start multiplying, or write down the sign of the answer before you even begin the calculation. Whatever helps you remember!
• Misplacing the Decimal Point: This often happens when people rush or lose track of the digits. Double-check your counting and make sure you've placed the decimal point correctly. A good strategy is to estimate the answer beforehand. For example, when multiplying 2.4 by 9.1, you know the answer should be around 2 × 9 = 18. This helps you spot if you've misplaced the decimal by a large margin.
• Calculator Dependence: While calculators are handy, relying on them too much can hinder your understanding of the underlying concepts. Try doing the calculations manually from time to time to strengthen your mental math skills. This not only helps you in exams but also builds your number sense.
• Mixing Up Addition and Multiplication Rules: Remember, the rules for multiplying signed numbers are different from those for adding or subtracting them. Don't confuse them! A negative plus a negative is still negative, but a negative times a negative is positive.

You might be wondering,