Question

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.

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 \(18.06\).