Expressions Equivalent To 0.05 * 0.6: Find The Matches!

Hey guys! Let's dive into a fun math problem where we need to figure out which expressions have the same value as $0.05 \cdot 0.6$. It's like a little puzzle, and we're going to break it down step by step. Get ready to put on your math hats and explore some cool tricks with decimals and fractions!

Understanding the Base Expression: $0.05 \cdot 0.6$

Okay, so our starting point is the expression $0.05 \cdot 0.6$. To really understand what's going on, let's calculate this first. We need to multiply 0.05 by 0.6. Think of it like this: 0.05 is the same as 5 hundredths, and 0.6 is the same as 6 tenths. When we multiply them, we're finding a fraction of a fraction, which can sometimes feel a bit tricky, but don't worry, we've got this!

To multiply decimals, a super helpful trick is to ignore the decimal points for a moment and just multiply the numbers as if they were whole numbers. So, we multiply 5 by 6, which gives us 30. Now, we need to figure out where to put the decimal point in our answer. To do this, we count the total number of decimal places in our original numbers. 0.05 has two decimal places, and 0.6 has one decimal place, giving us a total of three decimal places. This means our answer needs to have three digits after the decimal point. So, 30 becomes 0.030. And guess what? We can drop that trailing zero, so our final result for $0.05 \cdot 0.6$ is 0.03. Awesome! Now we know what we're aiming for – any other expression that equals 0.03 is a match!

Analyzing the Given Expressions

Now, let's roll up our sleeves and dive into each of the expressions provided. We need to carefully evaluate each one and see if it equals our target value of 0.03. It's like being a math detective, and each expression is a clue! We'll use our knowledge of fractions, decimals, and multiplication to crack the code. Let's get started!

Expression 1: $5 \cdot \frac{1}{100} \cdot 6 \cdot \frac{1}{10}$

Alright, let's break down this first expression: $5 \cdot \frac{1}{100} \cdot 6 \cdot \frac{1}{10}$. We've got a mix of whole numbers and fractions here, which is totally manageable. Remember, multiplying fractions is all about multiplying the numerators (the top numbers) and the denominators (the bottom numbers). But before we jump into that, let's rewrite this expression in a way that might make it a bit clearer. We can rearrange the terms because multiplication is commutative, meaning the order doesn't change the result. So, let's group the whole numbers together and the fractions together: $(5 \cdot 6) \cdot (\frac{1}{100} \cdot \frac{1}{10})$.

Now, let's simplify. 5 multiplied by 6 is 30. Next, we multiply the fractions: $\frac{1}{100} \cdot \frac{1}{10}$. To do this, we multiply the numerators (1 times 1, which is 1) and the denominators (100 times 10, which is 1000). So, we have $\frac{1}{1000}$. Now our expression looks like this: $30 \cdot \frac{1}{1000}$. This means we're multiplying 30 by one thousandth. Another way to think about this is dividing 30 by 1000. When we do that, we get 0.030, which is the same as 0.03! Woohoo! This expression matches our target value.

Expression 2: $5 \cdot 6 \cdot \frac{1}{1000}$

Next up, we have the expression $5 \cdot 6 \cdot \frac{1}{1000}$. This one looks pretty straightforward. Let's start by multiplying the whole numbers: 5 times 6 is 30. So now we have $30 \cdot \frac{1}{1000}$. Just like in the previous expression, this means we're multiplying 30 by one thousandth, or dividing 30 by 1000. And as we already figured out, 30 divided by 1000 is 0.03. Awesome! This expression is another match.

Expression 3: $5 \cdot 0.001 \cdot 6 \cdot 0.01$

Okay, let's tackle the expression $5 \cdot 0.001 \cdot 6 \cdot 0.01$. This one involves multiplying decimals, so let's use our trick of ignoring the decimal points for a moment and just multiplying the numbers as whole numbers. We have 5, 0.001, 6, and 0.01. Let's rearrange them to group the whole numbers together: $(5 \cdot 6) \cdot (0.001 \cdot 0.01)$. 5 times 6 is 30. Now we need to multiply 0.001 by 0.01. Think of 0.001 as one thousandth and 0.01 as one hundredth.

To multiply 0.001 by 0.01, we can again ignore the decimal points and multiply 1 by 1, which gives us 1. Now, we count the decimal places. 0.001 has three decimal places, and 0.01 has two decimal places, giving us a total of five decimal places. So, our result needs to have five digits after the decimal point. That means 1 becomes 0.00001. Now we have $30 \cdot 0.00001$. This means we're multiplying 30 by one hundred-thousandth. When we do this, we get 0.00030, which is the same as 0.0003. Hmm, 0.0003 is not equal to 0.03. Bummer! This expression is not a match.

Expression 4: 0.03

Well, this one's pretty straightforward! The expression is simply 0.03. And guess what? That's our target value! Hooray! This expression is a match.

Expression 5: 0.003

Last but not least, we have the expression 0.003. This is three thousandths. We're looking for 0.03, which is three hundredths. These are different values. 0. 003 is smaller than 0.03. So, this expression is not a match. Better luck next time!

Conclusion: Identifying the Equivalent Expressions

Alright, math detectives, we've cracked the code! We started with the expression $0.05 \cdot 0.6$, calculated its value to be 0.03, and then carefully analyzed each of the given expressions. We used our knowledge of decimals, fractions, and multiplication to determine which ones matched our target value.

So, drumroll please… The expressions that have the same value as $0.05 \cdot 0.6$ (which is 0.03) are:

• $5 \cdot \frac{1}{100} \cdot 6 \cdot \frac{1}{10}$
• $5 \cdot 6 \cdot \frac{1}{1000}$
• 0.03

Great job, everyone! You've successfully navigated through this math puzzle and identified the equivalent expressions. Keep up the awesome work, and remember, math can be fun when we break it down step by step!