CIE IGCSE Mathematics (0580) Estimation Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Estimation Study Notes
LEARNING OBJECTIVE
- Rounding to a Specified Degree of Accuracy
Key Concepts:
- Rounding
- Estimation
Rounding to a Specified Degree of Accuracy
Rounding to a Specified Degree of Accuracy
Rounding to Decimal Places
When rounding a number to a certain number of decimal places (dp), keep only that many digits after the decimal point.
To round to \( n \) decimal places:
- Identify the digit in the \( n \)th decimal place.
- Look at the digit immediately after it (the next decimal place).
- If the next digit is 5 or more, round up. If it is less than 5, round down.
Rounding to Significant Figures
Significant figures (sf) count from the first non-zero digit.
To round to \( n \) significant figures:
- Identify the first \( n \) significant digits.
- Look at the digit after the \( n \)th significant figure to decide whether to round up or down.
- Adjust the remaining digits to zero if rounding whole numbers, or truncate if decimals.
Important Notes:
- Trailing zeros after the decimal point count in decimal places but not as significant figures.
- Leading zeros do not count as significant figures.
- Use rounding in final answers only when instructed or when interpreting measurements.
Example:
Round \( 5.87629 \) to 3 decimal places.
▶️ Answer/Explanation
Step 1: Look at the 3rd decimal digit: \( 6 \)
Step 2: Look at the next digit (4th dp): \( 2 \)
Since \( 2 < 5 \), we round down.
Final Answer: \( 5.876 \)
Example:
Round \( 0.004738 \) to 2 significant figures.
▶️ Answer/Explanation
Step 1: First two significant digits are \( 4 \) and \( 7 \)
Step 2: Next digit is \( 3 \), so we round down
Final Answer: \( 0.0047 \)
Rounding Answers to a Reasonable Degree of Accuracy
What Does This Mean?
After solving a problem, your final answer should be rounded to an appropriate number of decimal places or significant figures depending on the context.
Key Considerations
- Do not round too early. Round only in the final step.
- Match the level of accuracy to the precision of the data in the question.
- Use the unit of measurement provided in the question (e.g. cm, kg, ₹, seconds).
- In word problems, round to the nearest sensible value (whole person, full item, full day, etc.).
Examples of Context-Based Rounding
- Money: round to 2 decimal places (e.g. ₹12.47)
- Time: round to nearest minute or hour if exact seconds aren’t practical
- People or objects: round to nearest whole number (you can’t have 3.6 people)
Common Mistakes to Avoid
- Rounding too soon, which can lead to inaccurate final answers.
- Giving answers that are more accurate than the original data.
- Using inconsistent rounding throughout the problem.
Example:
By rounding each number to 1 significant figure, estimate the value of:
\( \frac{41.3}{9.79} \times 0.765 \)
▶️ Answer/Explanation
Step 1: Round each number to 1 significant figure:
- \( 41.3 \approx 40 \)
- \( 9.79 \approx 10 \)
- \( 0.765 \approx 0.8 \)
Step 2: Estimate the expression:
\( \frac{40}{10} \times 0.8 = 4 \times 0.8 = 3.2 \)
Estimated Answer: \( 3.2 \)
Example:
Estimate the value of: \( \left( \frac{63.5}{19.2} \times 4.88 \right) – 2.14 \) by rounding each number to 1 significant figure.
▶️ Answer/Explanation
Step 1: Round each number to 1 significant figure:
- \( 63.5 \approx 60 \)
- \( 19.2 \approx 20 \)
- \( 4.88 \approx 5 \)
- \( 2.14 \approx 2 \)
Step 2: Estimate the expression:
\( \left( \frac{60}{20} \times 5 \right) – 2 = (3 \times 5) – 2 = 15 – 2 = 13 \)
Estimated Answer: \( 13 \)
Estimation in Calculations
Estimation in Calculations
What is Estimation?
Estimation means rounding numbers to make a calculation easier and quicker to perform mentally. It gives an approximate answer, not an exact value.
Why Use Estimation?
- To check whether a calculated result is reasonable.
- To simplify mental arithmetic in real-world situations.
- To quickly compare options, prices, or quantities.
How to Estimate Effectively
- Round numbers to 1 significant figure or a convenient whole number.
- Use compatible numbers (e.g., round 48 to 50, 103 to 100).
- Perform the calculation using rounded values.
- State that the answer is an estimate.
Common Contexts for Estimation
- Calculating cost or totals in shopping.
- Estimating time needed to complete a task.
- Approximating quantities in measurements.
- Checking answers in multiplication or division.
Important Notes:
- Estimates should be close enough to help you check the reasonableness of actual results.
- Never use estimates in final answers unless specifically asked to.
Example:
Estimate the value of \( 49.7 \times 301 \)
▶️ Answer/Explanation
Step 1: Round numbers to 1 significant figure or convenient whole numbers:
\( 49.7 \approx 50 \), \( 301 \approx 300 \)
Step 2: Estimate the product:
\( 50 \times 300 = 15{,}000 \)
Estimated Answer: \( 15{,}000 \)
Example:
You buy items costing ₹198.75, ₹52.30, and ₹18.90. Estimate the total amount you’ll pay.
▶️ Answer/Explanation
Step 1: Round each value to a convenient amount:
₹198.75 → ₹200, ₹52.30 → ₹50, ₹18.90 → ₹20
Step 2: Add the rounded values:
₹200 + ₹50 + ₹20 = ₹270
Estimated Total: ₹270
Example:
Write 5764 correct to the nearest thousand.
▶️ Answer/Explanation
Step 1: Identify the digit in the thousands place: In 5764, it is \( 5 \).
Step 2: Look at the digit to the right (hundreds place): It is \( 7 \).
Step 3: Since \( 7 \geq 5 \), round up.
Final Answer: \( 5764 \approx 6000 \) (to the nearest thousand)