Let's analyze and estimate each of the given calculations to determine which results are definitely incorrect. Estimation can help us quickly determine if an answer is reasonable.
### Calculation a: [tex]\(95 \times 212 = 20140\)[/tex]
- Estimation: Round 95 to 100, and 212 to 200.
[tex]\[
100 \times 200 = 20000
\][/tex]
- The given answer is 20140, which is very close to our estimate. This answer is reasonable and likely correct.
### Calculation b: [tex]\(44 \times 17 = 748\)[/tex]
- Estimation: Round 44 to 50, and 17 to 20.
[tex]\[
50 \times 20 = 1000
\][/tex]
- The given answer is 748, which is significantly less than 1000. This answer is not reasonable and likely incorrect.
### Calculation c: [tex]\(689 \times 413 = 28457\)[/tex]
- Estimation: Round 689 to 700, and 413 to 400.
[tex]\[
700 \times 400 = 280000
\][/tex]
- The given answer is 28457, which is significantly lower than our estimate. This answer is not reasonable and definitely incorrect. The transition to 284557 shows a closer reflection.
### Calculation d: [tex]\(142656 \div 8 = 17832\)[/tex]
- Estimation: Round 142656 to 140000.
[tex]\[
140000 \div 8 = 17500
\][/tex]
- The given answer is 17832, which is quite close to our estimate of 17500. This answer is reasonable and likely correct.
### Calculation e: [tex]\(77.9 \times 22.6 = 2512.54\)[/tex]
- Estimation: Round 77.9 to 80, and 22.6 to 20.
[tex]\[
80 \times 20 = 1600
\][/tex]
- The given answer is 2512.54, which is significantly higher than our estimate of 1600. This answer is not reasonable and likely incorrect.
### Calculation f: [tex]\(\frac{8.42 \times 46}{0.2} = 19366\)[/tex]
- Estimation: Round 8.42 to 10, and 46 to 50. Also, simplifying the division by 0.2.
[tex]\[
\frac{10 \times 50}{0.2} = \frac{500}{0.2} = 2500
\][/tex]
- The given answer is 19366, which is significantly higher than our estimate. This answer is not reasonable and definitely incorrect.
### Summary
The incorrect answers based on our estimation are:
- [tex]\(44 \times 17 = 748\)[/tex]
- [tex]\(689 \times 413 = 28457\)[/tex] (with a correction leading to 284557)
- [tex]\(77.9 \times 22.6 = 2512.54\)[/tex]
- [tex]\(\frac{8.42 \times 46}{0.2} = 19366\)[/tex]
So, the calculations that are definitely incorrect are:
[tex]\[ \text{b, c, e, and f} \][/tex]
