Dylan and Lucas know the digits and in the product 6 x 3.01 are 1806. Dylan says the decimal should be placed between the 8 and 0. Lucas says the decimal should be placed between the 1 and 8.

Question

Decide which student is correct. Explain your thinking. Use estimation and place value in your responds

Answer 1

To determine where the decimal should be placed in the product of \(6 \times 3.01\), we first need to calculate the product.

1. Calculate the Exact Product: \[ 6 \times 3.01 = 6 \times (3 + 0.01) = 6 \times 3 + 6 \times 0.01 = 18 + 0.06 = 18.06 \]

The result of the multiplication is \(18.06\).

2. Analyzing the Decimal Placement: Now, we can evaluate the positions Dylan and Lucas propose:

   • Dylan suggests placing the decimal between the 8 and the 0, which would give \(180.6\).
   • Lucas suggests placing the decimal between the 1 and the 8, which would give \(18.06\).

3. Using Estimation: To validate our result, we can use estimation:

   • We can round \(3.01\) to \(3\). Thus, estimating: \[ 6 \times 3 = 18 \] This estimate suggests our result should be somewhere around \(18\).

4. Understanding Place Value:

   • In \(18.06\), the number before the decimal (18) indicates that the product is a little over 18 (since we calculated \(18.06\)).
   • Placing the decimal after the 8 to form \(180.6\) would imply the product is much larger than \(18\), which is incorrect based on our calculations and estimations.

5. Conclusion: Dylan's placement of the decimal resulting in \(180.6\) is incorrect, as it would indicate a product significantly larger than \(18\). Lucas’s placement resulting in \(18.06\) is correct because it properly reflects the calculated product.

Final answer: Lucas is correct; the decimal should be placed between the 1 and the 8, resulting in \(18.06\).