Approximation and Estimation
This topic covers how to approximate complex calculations to produce a quick, reasonable estimate of the answer. It is a vital skill for a non-calculator exam to quickly check the magnitude of a result or to choose the most plausible option in a multiple-choice question.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Round numbers in a calculation to 1 or 2 significant figures to simplify the arithmetic.
- Choose convenient numbers for rounding. For example, round 98.7 to 100, and 0.49 to 0.5 (or 1/2).
- Use standard approximations for constants: π is often approximated as 3 or 22/7.
- Approximate surds by finding the nearest perfect square. For example, √(63.5) is close to √(64), which is 8.
- When approximating a fraction, be mindful if you are rounding the numerator up and the denominator down (or vice versa), as this will affect whether your estimate is larger or smaller than the true value.
Formulae
π ≈ 3 A quick and simple approximation for general calculations.
π ≈ 22/7 A more accurate approximation, especially useful when dealing with numbers that are multiples of 7.
Definitions
- Approximation
- The process of finding a value that is close enough to the true value for a specific purpose, such as estimation.
- Significant Figures
- The digits in a number that are reliable and necessary to indicate the quantity of something. Counting starts from the first non-zero digit.
Worked example
Without a calculator, estimate the value of: (√(80.5) × 10.22) / (π × 4.95)
- 1
Step 1:
Approximate each term in the expression to a simpler number.
- 2
√(80.5) is very close to √(81), so we approximate it as 9.
- 3
10.22 is very close to 102, so we approximate it as 100.
- 4
π can be approximated as 3 for simplicity.
- 5
4.95 is very close to 5.
- 6
Step 2:
Substitute these approximated values back into the expression.
- 7
The expression becomes approximately (9 × 100) / (3 × 5).
- 8
Step 3:
Perform the simplified calculation.
- 9
(9 × 100) / (3 × 5) = 900 / 15.
- 10
To calculate 900 / 15, we can simplify the fraction.
Both are divisible by 3:
300 / 5.
This is equal to 60.
Answer: 60
Common mistakes
- ×Incorrectly rounding significant figures, especially for decimals (e.g., rounding 0.0289 to 0.02 instead of 0.03 for 1 s.f.).
- ×Losing track of the decimal point or order of magnitude, particularly when dividing by a number less than 1.
- ×Choosing approximations that are not strategic. For example, rounding 24.8 to 25 is more useful for division by 5 than rounding it to 20 or 30.
- ×Forgetting to apply powers or roots to the approximated numbers before performing other operations.
No-calculator tips
- ✓Convert tricky decimals to fractions before calculating. For example, dividing by 0.25 is the same as multiplying by 4.
- ✓When dealing with very large or small numbers, approximate them using scientific notation (e.g., 0.0042 ≈ 4 x 10-3) to make multiplication and division simpler.
- ✓If you round some numbers up and others down, try to balance them to keep the estimate accurate. Or, be aware that if you round all numbers up, your estimate will likely be an overestimate.
- ✓Simplify fractions before you multiply. In the expression (A × B) / (C × D), look to see if you can cancel any common factors between the numerator and denominator.