1.2 Approximations and Limits
Rounding with Decimal Places (d.p)
When rounding using decimal places, you count from the decimal point. Look at the digit to the right of your target position to decide whether to round up or down.
When rounding using significant figures, you count from the first non-zero digit. This is especially important when dealing with very small or very large numbers.
When rounding using decimal places, you count from the decimal point. Look at the digit to the right of your target position to decide whether to round up or down.
Example 1: Round 7.839 to 2 decimal places
Step 1: Identify the second decimal place → 7.839
Step 2: Look at the digit to the right → 9
Step 3: Since 9 ≥ 5, round up → 7.84
Step 2: Look at the digit to the right → 9
Step 3: Since 9 ≥ 5, round up → 7.84
Example 2: Round 54.83629 to 1 d.p
Rounding with Significant Figures (s.f)
Step 1: Identify the first decimal place → 54.83629
Step 2: Look at the digit to the right → 3
Step 3: Since 3 < 5, round down → 54.8
Step 2: Look at the digit to the right → 3
Step 3: Since 3 < 5, round down → 54.8
When rounding using significant figures, you count from the first non-zero digit. This is especially important when dealing with very small or very large numbers.
Example 3: Round 0.000043213 to 3 s.f
Step 1: Find the first non-zero digit → 0.000043213
Step 2: Count 3 significant figures → 4, 3, 2
Step 3: Look at the next digit → 1
Step 4: Since 1 < 5, round down → 0.0000432
Step 2: Count 3 significant figures → 4, 3, 2
Step 3: Look at the next digit → 1
Step 4: Since 1 < 5, round down → 0.0000432
Example 4: Round 3456.789 to 2 s.f
Step 1: Count from the first digit → 3456.789
Step 2: Look at the next digit → 5
Step 3: Since 5 ≥ 5, round up → 3500
(Note: The zeros are needed to maintain the correct magnitude)
Step 2: Look at the next digit → 5
Step 3: Since 5 ≥ 5, round up → 3500
(Note: The zeros are needed to maintain the correct magnitude)
🧮 Number Rounder
Number:
Round to:
Number:
Round to:
Real Life Uses:
• Finance: Bank balances are rounded to 2 decimal places (pence/cents)
• Science: Measurements in experiments are rounded to appropriate significant figures based on instrument precision
• Engineering: Dimensions are rounded to practical measurements (e.g., cutting materials)
• Statistics: Population figures and percentages are often rounded to make them easier to communicate
• Finance: Bank balances are rounded to 2 decimal places (pence/cents)
• Science: Measurements in experiments are rounded to appropriate significant figures based on instrument precision
• Engineering: Dimensions are rounded to practical measurements (e.g., cutting materials)
• Statistics: Population figures and percentages are often rounded to make them easier to communicate
Estimating with Rounding
Estimating is a useful skill to have when you want to quickly find an approximate answer without having to do the full calculation. One of the most common methods is rounding numbers to make calculations easier.
Since you should know the squares of numbers from 1 to 12, you should be able to estimate square roots from 1 to 144 with reasonable accuracy.
Estimating is a useful skill to have when you want to quickly find an approximate answer without having to do the full calculation. One of the most common methods is rounding numbers to make calculations easier.
Example 1: Estimate 47.8 + 32.6
Estimating Square Roots
Step 1: Round 47.8 to the nearest ten → 50
Step 2: Round 32.6 to the nearest ten → 30
Step 3: Add the rounded numbers → 50 + 30 = 80
(Actual answer: 80.4 - very close!)
Step 2: Round 32.6 to the nearest ten → 30
Step 3: Add the rounded numbers → 50 + 30 = 80
(Actual answer: 80.4 - very close!)
Since you should know the squares of numbers from 1 to 12, you should be able to estimate square roots from 1 to 144 with reasonable accuracy.
Example 2: Estimate √63
Step 1: Find the perfect squares either side → √49 = 7 and √64 = 8
Step 2: Therefore √63 is between 7 and 8
Step 3: Since 63 is very close to 64, √63 is closer to 8
Step 4: Estimate → 7.9
(Actual value: 7.937... - Veeery close)
Step 2: Therefore √63 is between 7 and 8
Step 3: Since 63 is very close to 64, √63 is closer to 8
Step 4: Estimate → 7.9
(Actual value: 7.937... - Veeery close)
Example 3: Estimate 298 × 19
Step 1: Round 298 to the nearest hundred → 300
Step 2: Round 19 to the nearest ten → 20
Step 3: Multiply → 300 × 20 = 6000
(Actual answer: 5662 - Good enough)
Step 2: Round 19 to the nearest ten → 20
Step 3: Multiply → 300 × 20 = 6000
(Actual answer: 5662 - Good enough)
🧮 Square root Finder: Square root any number from 1-144 using estimates, use this to check if you're correct.
Find √
Find √
Real Life Uses:
• Shopping: Quickly estimating the total cost of groceries to stay within budget
• Construction: Estimating materials needed (e.g., paint, tiles) before precise calculations
• Time Management: Estimating journey times or how long tasks will take
• Mental Arithmetic: Checking if calculator answers are reasonable (catching input errors)
• Business: Quick profit/loss estimates before detailed financial analysis
• Shopping: Quickly estimating the total cost of groceries to stay within budget
• Construction: Estimating materials needed (e.g., paint, tiles) before precise calculations
• Time Management: Estimating journey times or how long tasks will take
• Mental Arithmetic: Checking if calculator answers are reasonable (catching input errors)
• Business: Quick profit/loss estimates before detailed financial analysis
Understanding Upper and Lower Bounds
Upper and lower bounds are used to give an idea of the range within which a value lies. They are particularly useful when dealing with measurements that have been rounded or estimated.
• Lower Bound: The smallest possible value that a measurement could be
• Upper Bound: The largest possible value that a measurement could be
When performing calculations with measurements, it's important to consider the bounds of the values involved to find the maximum and minimum possible results.
Upper and lower bounds are used to give an idea of the range within which a value lies. They are particularly useful when dealing with measurements that have been rounded or estimated.
• Lower Bound: The smallest possible value that a measurement could be
• Upper Bound: The largest possible value that a measurement could be
Example 1: Find the bounds of 5.4 cm (to 1 d.p)
Calculations with Bounds
Step 1: The number is given to 1 decimal place, so the range is ±0.05
Step 2: Lower bound = 5.4 - 0.05 = 5.35 cm
Step 3: Upper bound = 5.4 + 0.05 = 5.45 cm
Therefore, the actual length could be anywhere in the range: 5.35 cm ≤ length < 5.45 cm
Step 2: Lower bound = 5.4 - 0.05 = 5.35 cm
Step 3: Upper bound = 5.4 + 0.05 = 5.45 cm
Therefore, the actual length could be anywhere in the range: 5.35 cm ≤ length < 5.45 cm
When performing calculations with measurements, it's important to consider the bounds of the values involved to find the maximum and minimum possible results.
Example 2: Calculate the area bounds of a rectangle
Length = 5.4 cm (to 1 d.p), Width = 3.2 cm (to 1 d.p)
Length = 5.4 cm (to 1 d.p), Width = 3.2 cm (to 1 d.p)
For Minimum Area (use lower bounds):
Lower bound of length = 5.35 cm
Lower bound of width = 3.15 cm
Minimum area = 5.35 × 3.15 = 16.8525 cm²
For Maximum Area (use upper bounds):
Upper bound of length = 5.45 cm
Upper bound of width = 3.25 cm
Maximum area = 5.45 × 3.25 = 17.7125 cm²
Therefore, the area could be anywhere in the range: 16.8525 cm² ≤ area < 17.7125 cm²
Lower bound of length = 5.35 cm
Lower bound of width = 3.15 cm
Minimum area = 5.35 × 3.15 = 16.8525 cm²
For Maximum Area (use upper bounds):
Upper bound of length = 5.45 cm
Upper bound of width = 3.25 cm
Maximum area = 5.45 × 3.25 = 17.7125 cm²
Therefore, the area could be anywhere in the range: 16.8525 cm² ≤ area < 17.7125 cm²
Quick Reference: Finding Bounds
• Rounded to nearest 10: ±5
• Rounded to nearest 1: ±0.5
• Rounded to 1 d.p: ±0.05
• Rounded to 2 d.p: ±0.005
• Rounded to 1 s.f: depends on magnitude (e.g., 400 → ±50)
• Rounded to nearest 10: ±5
• Rounded to nearest 1: ±0.5
• Rounded to 1 d.p: ±0.05
• Rounded to 2 d.p: ±0.005
• Rounded to 1 s.f: depends on magnitude (e.g., 400 → ±50)
🧮 Bound Calculator
Rounded value:
Rounded to: decimal places
Rounded value:
Rounded to: decimal places
Real Life Uses:
• Manufacturing: Tolerance levels in parts (e.g., bolts must be 10mm ± 0.1mm)
• Medicine: Drug dosages and concentration ranges for safe administration
• Quality Control: Ensuring products meet specifications (weight, size, temperature)
• Science Experiments: Recording the precision of measurements and calculating error margins
• Construction: Ensuring materials fit together despite measurement uncertainties
• GPS & Navigation: Accuracy ranges for location data (±5 meters)
• Manufacturing: Tolerance levels in parts (e.g., bolts must be 10mm ± 0.1mm)
• Medicine: Drug dosages and concentration ranges for safe administration
• Quality Control: Ensuring products meet specifications (weight, size, temperature)
• Science Experiments: Recording the precision of measurements and calculating error margins
• Construction: Ensuring materials fit together despite measurement uncertainties
• GPS & Navigation: Accuracy ranges for location data (±5 meters)