Solving Decimal Division Problems Using Known Facts

This problem asks us to find the value of a decimal division using a related integer division fact. We are given that $\text{493} \div \text{29} = \text{17}$ and we need to find the value of $\text{4.93} \div \text{0.0017}$.

Analyzing the Given Information

We are given the division relationship:

\(\frac{493}{29} = 17\)

From this relationship, we can also deduce other related facts. If $493$ divided by $29$ is $17$, it means that $29$ multiplied by $17$ is $493$. Also, $493$ divided by $17$ must be $29$. This last relationship is particularly useful for solving the problem:

\(\frac{493}{17} = 29\)

Calculating the Decimal Division

We need to calculate the value of $\text{4.93} \div \text{0.0017}$. We can write this as a fraction:

\(\frac{4.93}{0.0017}\)

To make this calculation easier, we can express the decimal numbers using powers of 10, relating them back to the integer $493$ and $17$ from the given fact.

• $4.93$ can be written as $493 \times 0.01$, which is $493 \times 10^{-2}$.
• $0.0017$ can be written as $17 \times 0.0001$, which is $17 \times 10^{-4}$.

Now substitute these expressions into the fraction:

\(\frac{493 \times 10^{-2}}{17 \times 10^{-4}}\)

We can separate this fraction into two parts: the division of the integer parts and the division of the powers of 10:

\(\left(\frac{493}{17}\right) \times \left(\frac{10^{-2}}{10^{-4}}\right)\)

Step 1: Evaluate the integer division

Using the fact we derived from the given information, $\frac{493}{17} = 29$.

Step 2: Evaluate the division of powers of 10

Using the rule for dividing exponents with the same base (\(\frac{a^m}{a^n} = a^{m-n}\)), we have:

\(\frac{10^{-2}}{10^{-4}} = 10^{-2 - (-4)} = 10^{-2 + 4} = 10^2\)

And \(10^2\) is equal to $100$.

Step 3: Combine the results

Now multiply the results from Step 1 and Step 2:

\(29 \times 10^2 = 29 \times 100 = 2900\)

So, $\text{4.93} \div \text{0.0017} = 2900$.

Verification using an alternative method

Alternatively, we can remove the decimals by multiplying both the numerator and the denominator by a power of 10 such that the number with the most decimal places becomes an integer. In $\frac{4.93}{0.0017}$, the denominator $0.0017$ has four decimal places. So we multiply both by $10^4$ (or $10000$).

\(\frac{4.93 \times 10000}{0.0017 \times 10000} = \frac{49300}{17}\)

Now, we perform the integer division $\text{49300} \div \text{17}$. We know that $\text{493} \div \text{17} = \text{29}$.

\(\frac{49300}{17} = \frac{493 \times 100}{17} = \left(\frac{493}{17}\right) \times 100\)

\(\left(\frac{493}{17}\right) \times 100 = 29 \times 100 = 2900\)

Both methods give the same result, $2900$.

Final Answer

The value of $\text{4.93} \div \text{0.0017}$ is $2900$. This matches option 4.

Revision Table: Key Concepts for Decimal Division

Additional Information: Understanding Decimal Place Values

Understanding place values and how they relate to powers of 10 is crucial for working with decimals. Each position to the right of the decimal point represents a decreasing power of 10.

• 0.1 is $10^{-1}$ (one tenth)
• 0.01 is $10^{-2}$ (one hundredth)
• 0.001 is $10^{-3}$ (one thousandth)
• 0.0001 is $10^{-4}$ (one ten-thousandth)

This means any decimal number can be written as an integer multiplied by a power of 10. For example:

• $4.93 = 493 \times 0.01 = 493 \times 10^{-2}$
• $0.0017 = 17 \times 0.0001 = 17 \times 10^{-4}$

This conversion is very helpful in simplifying calculations involving decimals, especially when a relationship to integers is known, as in this problem.