Question
Estimate the product of 23 x 19.
23 -> 20
20 x 20 = _
19 -> _
We could also have multiplied 2 x 2 and then added up the number of zeros at the end of the factors, and that would have given us 400. Does the same approach work for the first example?
5 x 1 = _
Add our 2 zeros to get 500.
23 -> 20
20 x 20 = _
19 -> _
We could also have multiplied 2 x 2 and then added up the number of zeros at the end of the factors, and that would have given us 400. Does the same approach work for the first example?
5 x 1 = _
Add our 2 zeros to get 500.
Answers
GPT-4o mini
To estimate the product of 23 x 19 using the method you've suggested, let's break it down:
1. **Round the numbers**:
- We can round 23 to 20.
- We can round 19 to 20 as well.
2. **Estimate the product**:
- Now multiply the rounded numbers:
\[
20 \times 20 = 400
\]
3. **Refine the estimate**:
- Since we were trying to estimate 23 x 19, we note that both original numbers are less than what we multiplied, so our estimate of 400 is a high estimation.
4. **Check the alternative approach**:
- The second approach you mentioned involves a simplified multiplication:
- For 23, if we break it down as \( (20 + 3) \) and for 19 as \( (20 - 1) \), we can see another way of estimating:
- \( 20 - x \)(where x is a small offset in both cases.
- This shows us that we can say \( 400 - \) some adjustments because one factor (19) is slightly less than 20 and the other factor (23) is slightly more but close.
5. **Final note**:
- As for your second example, multiplying \( 5 \times 1 \) gives us 5, and then if you add the two zeros, you would get 500; however, that's not a direct multiplication but a conceptual application of scaling.
Thus, both methods can help guide our estimation, and realizing the above leads to the understanding that the specific approach can vary depending on your goal. The product of 23 x 19 is actually 437, and we are just refining our estimation techniques to get close to that answer using rounding principles.
1. **Round the numbers**:
- We can round 23 to 20.
- We can round 19 to 20 as well.
2. **Estimate the product**:
- Now multiply the rounded numbers:
\[
20 \times 20 = 400
\]
3. **Refine the estimate**:
- Since we were trying to estimate 23 x 19, we note that both original numbers are less than what we multiplied, so our estimate of 400 is a high estimation.
4. **Check the alternative approach**:
- The second approach you mentioned involves a simplified multiplication:
- For 23, if we break it down as \( (20 + 3) \) and for 19 as \( (20 - 1) \), we can see another way of estimating:
- \( 20 - x \)(where x is a small offset in both cases.
- This shows us that we can say \( 400 - \) some adjustments because one factor (19) is slightly less than 20 and the other factor (23) is slightly more but close.
5. **Final note**:
- As for your second example, multiplying \( 5 \times 1 \) gives us 5, and then if you add the two zeros, you would get 500; however, that's not a direct multiplication but a conceptual application of scaling.
Thus, both methods can help guide our estimation, and realizing the above leads to the understanding that the specific approach can vary depending on your goal. The product of 23 x 19 is actually 437, and we are just refining our estimation techniques to get close to that answer using rounding principles.