Understanding the Decimal Division Problem

The question asks us to calculate the value of $0.0713 \div 3.1$ using the given multiplication fact $23 \times 31 = 713$. This type of problem tests our understanding of how decimal places affect the results of multiplication and division.

Relating Division to Multiplication

We are given the fact that $23 \times 31 = 713$. This implies two related division facts:

• $713 \div 31 = 23$
• $713 \div 23 = 31$

The division problem we need to solve is $0.0713 \div 3.1$. Notice that the digits in $0.0713$ are the same as in $713$, and the digits in $3.1$ are the same as in $31$. This suggests that the result of the division $0.0713 \div 3.1$ will involve the digits $23$, but the position of the decimal point will be different.

Analyzing the Decimal Places

Let's compare the numbers in the given multiplication fact and the division problem:

• From $23 \times 31 = 713$, we have $713 \div 31 = 23$.
• We want to calculate $0.0713 \div 3.1$.

We can write the numbers with decimals as fractions or by considering the shift in the decimal point from the original integers (23, 31, 713).

• $0.0713$ is $713$ with the decimal point moved 4 places to the left ($713 \times 10^{-4}$).
• $3.1$ is $31$ with the decimal point moved 1 place to the left ($31 \times 10^{-1}$).

So, the problem $0.0713 \div 3.1$ is equivalent to:

$\dfrac{0.0713}{3.1} = \dfrac{713 \times 10^{-4}}{31 \times 10^{-1}}$

Performing the Decimal Division Calculation

Using the fact $713 \div 31 = 23$, we can simplify the expression:

$\dfrac{713 \times 10^{-4}}{31 \times 10^{-1}} = \left(\dfrac{713}{31}\right) \times \left(\dfrac{10^{-4}}{10^{-1}}\right)$

We know $\dfrac{713}{31} = 23$.

For the powers of 10, when dividing exponents, we subtract the powers:

$\dfrac{10^{-4}}{10^{-1}} = 10^{-4 - (-1)} = 10^{-4 + 1} = 10^{-3}$

So, the calculation becomes:

$23 \times 10^{-3}$

$10^{-3}$ means moving the decimal point 3 places to the left from the number $23$.

$23.0 \times 10^{-3} = 0.023$

Alternatively, consider the total number of decimal places in the division:

• Dividend ($0.0713$): 4 decimal places
• Divisor ($3.1$): 1 decimal place

When dividing, the number of decimal places in the quotient is the number of decimal places in the dividend minus the number of decimal places in the divisor.

Number of decimal places in quotient = (Decimal places in 0.0713) - (Decimal places in 3.1)

Number of decimal places in quotient = $4 - 1 = 3$

Since $713 \div 31 = 23$, the digits in $0.0713 \div 3.1$ will be $23$. We need the result to have 3 decimal places. Starting with $23$, we move the decimal 3 places to the left: $23.0 \to 2.30 \to 0.230 \to 0.0230$. So, the result is $0.023$.

Conclusion on the Decimal Division Result

Based on the calculation and analysis of decimal places, the value of $0.0713 \div 3.1$ is $0.023$. This matches one of the provided options.

The final answer is $\boxed{0.023}$.

Revision Table: Key Concepts in Decimal Division

Additional Information on Decimal Arithmetic

Working with decimals is fundamental in mathematics. Here are a few related points:

• Multiplication: The number of decimal places in the product is the sum of the decimal places in the numbers being multiplied. For example, $0.12 \times 0.3$ (2 dp + 1 dp = 3 dp). $12 \times 3 = 36$. Result is $0.036$.
• Addition/Subtraction: Align the decimal points vertically before adding or subtracting. The result will have the decimal point in the same position.
• Converting to Fractions: Decimals can be written as fractions with denominators that are powers of 10. For instance, $0.0713 = \dfrac{713}{10000}$, $3.1 = \dfrac{31}{10}$. This can sometimes help in understanding the division, as $\dfrac{713/10000}{31/10} = \dfrac{713}{10000} \times \dfrac{10}{31} = \dfrac{713}{31} \times \dfrac{10}{10000} = 23 \times \dfrac{1}{1000} = \dfrac{23}{1000} = 0.023$.
• Estimating: Before performing exact calculations, estimating the answer can help catch errors. For $0.0713 \div 3.1$, $0.0713$ is a small number, and $3.1$ is around 3. Dividing a small number by about 3 will give an even smaller number. $0.023$ is a small number, fitting the estimate.