Approximation: Methods, Tricks and Solved Questions
Approximation is the process of replacing exact numbers with nearby convenient values to simplify calculations. This topic covers rounding rules, compatible numbers, approximate multiplication and division, estimation of percentages, and error calculation. The symbol ≈ means “approximately equal to” and is used when the exact and estimated values are different.
What Is Approximation?
A number may be approximated to the nearest integer, tenth, hundredth, or another place value. For rounding, inspect the digit immediately after the required place: a digit from 0 to 4 leaves the retained digit unchanged, while a digit from 5 to 9 increases it by 1. For example, 47.63 ≈ 47.6 to one decimal place and 47.63 ≈ 48 to the nearest integer.
Approximation Formula & Tricks
Important Formulas
Absolute error gives the positive difference between the exact and estimated values.
This expresses the approximation error as a percentage of the exact value.
Use the first digit after the required place to decide how to round.
Replace the numbers with nearby values that make multiplication or division simple, while preserving the approximate scale.
Quick Tricks
Round numbers to values that work easily together, such as multiples of 10, 100, 25, or 50. Choose values close enough to the originals so that the estimate remains reliable.
In an expression containing brackets, multiplication, division, addition and subtraction, first approximate the relevant numbers and then follow BODMAS. Do not add or subtract before completing a required multiplication or division.
For square roots, replace the number by a nearby perfect square when a quick estimate is required. The result should be close to the corresponding integer.
Before accepting an approximation, estimate the number of digits and the rough size of the result. This detects errors such as placing a decimal point incorrectly.
Approximation Concepts
Rounding to a Place Value
If the next digit is 0, 1, 2, 3 or 4, the retained digit remains unchanged. If it is 5, 6, 7, 8 or 9, increase the retained digit by 1 and remove the remaining digits. For example, 8.746 rounded to two decimal places is 8.75, while 8.743 rounded to two decimal places is 8.74.
Approximation in Addition and Subtraction
The same place value is not compulsory for every term, but the rounded values should retain the required accuracy. Approximation can slightly change the result, so use values that are close to the original numbers. In subtraction, preserving the relative size of the minuend and subtrahend helps avoid a misleading estimate.
Approximation in Multiplication and Division
Round factors to simple multiples such as 10, 20, 50 or 100. For division, choose a nearby divisor that divides the rounded dividend conveniently. The answer is an estimate, not the exact value, unless the replacements happen to preserve the exact calculation.
Approximation of Fractions, Percentages and Roots
For a fraction, use nearby numerator and denominator values when the ratio remains close. Common percentages include 10% = 1/10, 25% = 1/4 and 50% = 1/2. For roots, use nearby perfect squares or cubes. For example, 24.8% of 398 is approximately 25% of 400, which equals 100.
Error in an Approximation
Absolute error is always non-negative because it uses the modulus of the difference. Percentage error compares this difference with the exact value. If the exact value is 50 and the approximate value is 49, the absolute error is 1 and the percentage error is (1 ÷ 50) × 100 = 2%.
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Approximation and Estimation Techniques
Learn how to simplify numerical calculations using approximation and estimation techniques, including rounding values and evaluating results efficiently while maintaining reasonable accuracy.
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Approximation Revision Points
Use these rules for quick estimation and checking.
- The symbol ≈ means approximately equal to.
- For rounding, the next digit 0–4 keeps the retained digit unchanged; 5–9 increases it by 1.
- Use compatible nearby numbers for multiplication and division.
- Follow BODMAS after replacing numbers with approximate values.
- For square roots, use nearby perfect squares such as 64, 81 or 100.
- Absolute error = |Exact value − Approximate value|.
- Percentage error = (Absolute error ÷ Exact value) × 100.
- Always check the approximate answer’s sign, decimal position and overall size.
Approximation FAQs
What is 48.76 rounded to the nearest tenth?
The hundredths digit is 6, so increase the tenths digit. Therefore, 48.76 ≈ 48.8.
How is 0.00486 rounded to three decimal places?
The fourth decimal digit is 8, so the third decimal digit increases from 4 to 5. Thus, 0.00486 ≈ 0.005.
What is the approximate value of 398 × 51?
Use 400 × 50. Therefore, 398 × 51 ≈ 20,000.
How can 997 ÷ 49.8 be approximated?
Use 1,000 ÷ 50, giving 997 ÷ 49.8 ≈ 20.
What is the approximate value of 19.8% of 502?
Use 20% of 500. Since 20% of 500 = 100, the approximation is 100.
What is the difference between approximation and exact calculation?
Exact calculation gives the precise value, while approximation uses a nearby convenient value. For example, 49.8 ≈ 50, but 49.8 is the exact given number.