Approximations

 

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🔹 1. What is Approximation?

Approximation means finding a value that is close enough to the actual number for practical purposes, especially when exact values are not needed.

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🔹 2. Rounding Off Numbers

➤ Steps to Round a Number:

1. Identify the digit to round to.

2. Look at the digit next to it (on the right):

   • If it's 5 or more, increase the rounding digit by 1.

   • If it's 4 or less, keep the rounding digit the same.

3. Replace all digits after it with zeros (whole numbers) or remove them (decimals).

➤ Examples:

• Round 76.849 to the nearest whole number:
  Look at 8 → Round up → 77

• Round 76.849 to 1 decimal place:
  Look at 4 → Round down → 76.8

• Round 76.849 to 2 decimal places:
  Look at 9 → Round up → 76.85

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🔹 3. Decimal Places (dp)

A decimal place shows how many digits are written after the decimal point.

Value	Decimal Places
5.6	1 dp
5.67	2 dp
5.678	3 dp

➤ Example:

• Round 3.1469 to 2 decimal places = 3.15

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🔹 4. Significant Figures (s.f.)

Significant figures are digits that carry meaning in a number.

➤ Rules:

1. Start counting from the first non-zero digit.

2. Zeros in between or after decimals may be significant.

➤ Examples:

• 345.76 to 2 s.f. = 350

• 0.04562 to 2 s.f. = 0.046

• 7.899 to 3 s.f. = 7.90

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🔹 5. Approximations in Calculations

Sometimes we estimate numbers to make mental math or rough calculations easier.

➤ Example 1: Estimation in Addition

Estimate 396 + 248
Round:

$$396 \approx 400, \quad 248 \approx 250 \Rightarrow 400 + 250 = 650$$

➤ Example 2: Estimation in Multiplication

Estimate 21.7 × 4.1
Round:

$$21.7 \approx 22, \quad 4.1 \approx 4 \Rightarrow 22 × 4 = 88$$

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✅ Summary Table

Concept	Description	Example
Rounding	Reduce digits based on the next digit	76.849 → 76.8 (1 dp)
Decimal Places	Digits after the decimal point	3.142 → 3.14 (2 dp)
Significant Figures	Count meaningful digits	0.04562 → 0.046 (2 s.f.)
Estimation	Rough answer	396 + 248 ≈ 650

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