Choose an easy benchmark
An estimate uses nearby numbers to make a calculation easier. The sign ≈ means approximately equal; it does not mean exactly equal.
For 49 × 21, try 50 × 20 = 1000. One number goes up and the other goes down, but the changes do not necessarily cancel.
Nearby dimensions, not identical areas.
Check the original value
The original product is 49 × 21 = 49 × 20 + 49 = 980 + 49 = 1029.
Among 800, 1000, 1200 and 1400, the closest option is 1000. Its distance from 1029 is 29. The benchmark helped locate the answer without pretending that 1000 was exact.
Upper and lower bounds
Both endpoints are included.
10 ≤ a ≤ 12; 4 ≤ b ≤ 5
For positive factors, use both lower endpoints for the lower bound and both upper endpoints for the upper bound.
10 × 4 ≤ a × b ≤ 12 × 5; 40 ≤ a × b ≤ 60
For a positive quotient, the smallest result uses the smallest numerator and largest denominator. Reverse these for the largest result.
10 ÷ 5 ≤ a ÷ b ≤ 12 ÷ 4; 2 ≤ a ÷ b ≤ 3
Accuracy limits
Exact result: 19 × 21 = 399.
The estimate is 400. Is its error at most 1?
Absolute error: |400 − 399| = 1.
Allowed error: 1. Equality is within the limit.
Answer: Within the limit.
The estimate is 400. Is its error at most 1% of the exact result?
399 × 1 ÷ 100 = 3.99; 1 ≤ 3.99
Answer: Within the limit.
Above or below a threshold
How does the exact result compare with 400?
19 × 21 = 399 < 400
Answer: Below.
Try a nearby square
Which option is closest to √2510: 50 or 60? Use nearby perfect squares.
2500 = 50² and 2601 = 51². Locate 2510 between those squares, then compare the possible square roots.
50² = 2500
2500 < 2510 < 2601
51² = 2601
Try it
Which option is closest to √2510?
- 50
- 60
Show the answer
50
50. Since 2500 < 2510 < 2601, the root lies between 50 and 51. It is less than 1 from 50, but more than 9 from 60.
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