📐 Chapter 1: Number (C1.9 / E1.9) – Estimation Notes
Welcome to the chapter on Estimation! This might seem like a simple topic, but mastering estimation is incredibly powerful. It allows you to quickly check if your calculator answer is reasonable (catching those accidental button presses!) and solve real-life problems instantly without needing exact figures.
Think of estimation as finding a quick, close-enough answer. To do this accurately, we must first master the skill of rounding.
1. Understanding Rounding (The Prerequisite Skill)
Rounding is essential because estimation involves simplifying complex numbers into easy-to-handle versions. The syllabus requires us to round to both Decimal Places (DP) and Significant Figures (SF).
1.1 Rounding to Decimal Places (DP)
This method focuses on the digits after the decimal point.
- Step 1: Identify the required decimal place position.
- Step 2: Look at the digit immediately to the right (the 'deciding digit').
- Step 3: If the deciding digit is 5 or more, round up the previous digit.
- Step 4: If the deciding digit is 4 or less, keep the previous digit the same.
- Step 5: Drop all digits after the required decimal place.
Example: Round 14.738 to 2 decimal places.
The 2nd DP is 3. The deciding digit is 8. Since 8 is 5 or more, we round 3 up to 4.
Answer: 14.74
1.2 Rounding to Significant Figures (SF)
Significant Figures are much more commonly used in estimation because they maintain the overall value (magnitude) of the number, regardless of where the decimal point is.
The Golden Rules for Significant Figures:
- The first significant figure is the first non-zero digit you meet when reading from left to right.
- Zeros placed between non-zero digits (like 507) are significant.
- Zeros used to hold the place value of a large number (like 5000) are usually significant only up to the point of rounding.
- Zeros that come before the first non-zero digit in small decimals (like 0.007) are not significant.
Example: Rounding to 3 Significant Figures (3 SF)
- 82 749: The 3rd SF is 7. The next digit is 4 (less than 5). Keep the 7, and replace the remaining digits with zeros to maintain magnitude.
Answer: 82 700 - 0.003 916: The 3rd SF is 1. The next digit is 6 (5 or more). Round 1 up to 2. The leading zeros are dropped.
Answer: 0.003 92
💡 Memory Aid: When rounding large numbers using SF, ask yourself: "Is the number still roughly the same size?" If you round 82 749 to 827, you’ve made a mistake! You must use zeros as placeholders (82 700).
Key Takeaway for Rounding: Understand the difference between DP (focuses on precision after the point) and SF (focuses on the overall importance of the digits).
2. Making Estimates for Calculations (C1.9.2)
The main technique required for IGCSE estimation is to round every number in the calculation to 1 Significant Figure (1 SF) before you start calculating. This simplifies the numbers so much that you can usually do the math mentally or very quickly.
Step-by-Step Guide to Estimation
Let's estimate the value of the calculation: \(96.4 \times 3.87 \div 0.519\)
Step 1: Round ALL numbers to 1 Significant Figure (1 SF).
- \(96.4 \to 100\) (9 is the 1st SF, 6 rounds it up to 10. Use two zeros as placeholders.)
- \(3.87 \to 4\) (3 is the 1st SF, 8 rounds it up.)
- \(0.519 \to 0.5\) (5 is the 1st SF, 1 keeps it at 5. Drop the trailing digits.)
Step 2: Rewrite the calculation using the rounded numbers.
The estimation becomes: \(100 \times 4 \div 0.5\)
Step 3: Perform the simplified calculation.
\(100 \times 4 = 400\)
\(400 \div 0.5\). (Dividing by 0.5 is the same as multiplying by 2!)
\(400 \times 2 = 800\)
Estimated value: 800.
(The exact value is approximately 717, so 800 is a very good estimate!)
Working with Complex Fractions (Syllabus Example)
The syllabus gives an example for estimating: \(\frac{41.3}{9.79 \times 0.765}\).
Step 1: Round ALL numbers to 1 SF.
- \(41.3 \to 40\)
- \(9.79 \to 10\)
- \(0.765 \to 0.8\)
Step 2: Rewrite the calculation.
Estimated calculation: \(\frac{40}{10 \times 0.8}\)
Step 3: Perform the simplified calculation.
- First, calculate the denominator: \(10 \times 0.8 = 8\)
- Now, the fraction is: \(\frac{40}{8}\)
- \(40 \div 8 = 5\)
Estimated value: 5.
🔥 Common Mistake to Avoid!
When estimating, you must round all numbers first, and only then perform the operation. DO NOT perform the calculation exactly and then round the answer. That defeats the purpose of estimation!
The purpose of estimation in exams is to show that you can simplify the calculation, not just guess the final answer. You must show the 1 SF rounding step.
Key Takeaway for Calculation Estimation: Round every number in the question to 1 Significant Figure, then perform the simplified calculation. Show your rounding clearly.
3. Rounding Answers to a Reasonable Degree of Accuracy (C1.9.3)
Sometimes, the question asks you to round your final answer to a reasonable degree of accuracy, usually when dealing with a practical problem.
What is "reasonable" depends entirely on the context of the problem:
- People or Objects: You cannot have 14.5 people or 3.2 cars. You must round to the nearest whole number (integer).
- Money: Money is usually rounded to the nearest cent/penny (2 decimal places). For large figures, sometimes the nearest whole unit (e.g., nearest dollar/pound) is appropriate.
- Lengths or Weights: Unless specified otherwise, rounding to 3 Significant Figures is often the standard accepted degree of accuracy in IGCSE Mathematics for non-exact values.
- Angles: Angles in IGCSE exams are typically rounded to 1 Decimal Place if they are not exact, unless otherwise stated.
Example 1 (Context: People): A coach holds 54.3 passengers.
Reasonable Answer: Since you can’t have part of a person, you must round down to 54 (or possibly 55 if rounding up for capacity, but mathematically, 54 is the nearest whole number).
Example 2 (Context: Distance): A calculation gives a distance of 1457.882 meters.
Reasonable Answer: 1460 m (3 SF), or 1457.9 m (1 DP). Unless told otherwise, 3 SF (1460) is standard.
Did You Know?
In physics and engineering, the number of significant figures used indicates the precision of the measurement tool. Using 3 SF (e.g., 5.12 kg) implies that the actual weight is likely between 5.115 kg and 5.125 kg.
⭐ Quick Review: Rounding Rules ⭐
These skills are essential for all number topics, not just estimation!
Decimal Places (DP):
Focus: Digits after the decimal point.
Prerequisite: Must use for money (usually 2 DP) or when specific precision is needed.
Significant Figures (SF):
Focus: Digits starting from the first non-zero digit on the left.
Prerequisite: Standard for non-exact calculation answers (3 SF) and for Estimation (1 SF).
Key Takeaway for Context: Always read the question carefully. The nature of what you are measuring (people, money, time, distance) dictates how the answer should be rounded.