Sure, let's evaluate each of the given estimates step-by-step to determine if they are reasonable.
### Estimate 1: [tex]\(782 \times 11 = 8000\)[/tex]
Actual Calculation:
[tex]\[ 782 \times 11 = 8602 \][/tex]
Reasonableness Check:
To determine if the estimate of 8000 is reasonable, we need to check if 8602 is close enough to 8000. For this, we can consider an acceptable range. If we allow for a deviation of about [tex]\(\pm1000\)[/tex], we check if:
[tex]\[ 8000 - 1000 \leq 8602 \leq 8000 + 1000 \][/tex]
[tex]\[ 7000 \leq 8602 \leq 9000 \][/tex]
Since 8602 falls within the range [7000, 9000], the estimate [tex]\(782 \times 11 = 8000\)[/tex] is reasonable.
### Estimate 2: [tex]\(34582 + 56722 = 80000\)[/tex]
Actual Calculation:
[tex]\[ 34582 + 56722 = 91304 \][/tex]
Reasonableness Check:
To see if the estimate of 80000 is reasonable, we again consider an acceptable range. If we allow for a deviation of about [tex]\(\pm10000\)[/tex], we check if:
[tex]\[ 80000 - 10000 \leq 91304 \leq 80000 + 10000 \][/tex]
[tex]\[ 70000 \leq 91304 \leq 90000 \][/tex]
Since 91304 falls outside the range [70000, 90000], the estimate [tex]\(34582 + 56722 = 80000\)[/tex] is unreasonable.
### Estimate 3: [tex]\(357 \div 8 = 45\)[/tex]
Actual Calculation:
[tex]\[ 357 \div 8 = 44.625 \][/tex]
Reasonableness Check:
To determine if the estimate of 45 is reasonable, we can check if 44.625 is close enough to 45. If we tolerate a deviation of about [tex]\(\pm5\)[/tex], we check if:
[tex]\[ 45 - 5 \leq 44.625 \leq 45 + 5 \][/tex]
[tex]\[ 40 \leq 44.625 \leq 50 \][/tex]
Since 44.625 falls within the range [40, 50], the estimate [tex]\(357 \div 8 = 45\)[/tex] is reasonable.
### Conclusion:
- Estimate 1: [tex]\(782 \times 11 = 8000\)[/tex] is reasonable.
- Estimate 2: [tex]\(34582 + 56722 = 80000\)[/tex] is unreasonable.
- Estimate 3: [tex]\(357 \div 8 = 45\)[/tex] is reasonable.
So, Estimates 1 and 3 are reasonable, while Estimate 2 is not.
