Understanding Decimal Multiplication and Scaling

The question provides a known multiplication fact: $17 \times 29 = 493$. We are then asked to find the result of a similar multiplication: $170 \times 0.029$. This problem tests our understanding of how multiplying or dividing numbers by powers of 10 affects the product in multiplication.

Let's break down the relationship between the two multiplication problems:

• The first number in the second problem is $170$. This is related to $17$ from the first problem. Specifically, $170 = 17 \times 10$.
• The second number in the second problem is $0.029$. This is related to $29$ from the first problem. To get $0.029$ from $29$, we need to move the decimal point three places to the left. This is equivalent to dividing by $1000$, or multiplying by $0.001$. So, $0.029 = 29 \times 0.001$.

Calculating the New Product

Now, let's substitute these relationships into the second multiplication problem:

We want to calculate $170 \times 0.029$.

Substitute the relationships we found:

$(17 \times 10) \times (29 \times 0.001)$

Using the associative and commutative properties of multiplication, we can rearrange the terms:

$(17 \times 29) \times (10 \times 0.001)$

We already know that $17 \times 29 = 493$. So, we can substitute this value:

$493 \times (10 \times 0.001)$

Now, let's calculate the value of $10 \times 0.001$. When you multiply by 10, you move the decimal point one place to the right.

$10 \times 0.001 = 0.010 = 0.01$

So, the problem simplifies to:

$493 \times 0.01$

Multiplying a number by $0.01$ is the same as dividing it by $100$. This means we need to move the decimal point in $493$ two places to the left.

The number $493$ can be thought of as $493.0$. Moving the decimal point two places to the left gives us $4.93$.

Therefore, $170 \times 0.029 = 4.93$.

Step-by-Step Solution for Decimal Multiplication

Here is a summary of the steps:

1. Start with the known fact: $17 \times 29 = 493$.
2. Identify the numbers in the problem to be solved: $170$ and $0.029$.
3. Compare these numbers to the original numbers ($17$ and $29$) in terms of powers of 10.
   • $170 = 17 \times 10$
   • $0.029 = 29 \times 0.001$ (or $29 \div 1000$)
4. Rewrite the new multiplication problem using these relationships: $(17 \times 10) \times (29 \times 0.001)$.
5. Rearrange the terms: $(17 \times 29) \times (10 \times 0.001)$.
6. Substitute the known product: $493 \times (10 \times 0.001)$.
7. Calculate the product of the powers of 10: $10 \times 0.001 = 0.01$.
8. The problem becomes: $493 \times 0.01$.
9. Perform the final multiplication: $493 \times 0.01 = 4.93$.

This result matches one of the given options.

Conclusion

Based on the calculations, $170 \times 0.029 = 4.93$. This demonstrates how understanding the relationship between multiplication and division by powers of 10 simplifies complex-looking problems when a related base calculation is provided.

Revision Table: Decimal Operations

Here's a quick review of decimal point movement with powers of 10:

Additional Information: Properties of Multiplication

The solution used properties of multiplication to simplify the calculation. It's helpful to remember these properties:

• Commutative Property: The order of factors does not change the product. $a \times b = b \times a$.
• Associative Property: The grouping of factors does not change the product. $(a \times b) \times c = a \times (b \times c)$.
• Identity Property: Any number multiplied by 1 is the number itself. $a \times 1 = a$.
• Zero Property: Any number multiplied by 0 is 0. $a \times 0 = 0$.

In our solution, we used the commutative and associative properties to regroup the factors $(17 \times 10) \times (29 \times 0.001)$ into $(17 \times 29) \times (10 \times 0.001)$. This allowed us to use the given information ($17 \times 29 = 493$) directly.