International Mathematics (0607) | Estimation

Welcome to Chapter 1.9: Estimation!

Hello! This chapter is super practical. Have you ever tried to quickly check if a restaurant bill is correct, or if the calculation you just did on your calculator looks reasonable? That’s estimation in action!
Estimation is all about getting a quick, sensible answer that is close to the exact value. It’s essentially making a mathematical "smart guess." In your exams, knowing how to estimate accurately proves you have a great feel for number—a key goal of the syllabus.

Section 1: The Essential Prerequisite—Rounding

You cannot estimate calculations without mastering rounding first. The syllabus requires you to round to both decimal places (DP) and significant figures (SF).

1.1 Rounding to Decimal Places (DP)

Rounding to decimal places is usually the easier skill because the position you need to round to is fixed relative to the decimal point.

Step-by-Step Guide: Decimal Places

1. Identify the decimal place you are rounding to (this is your Target Digit).
2. Look at the digit immediately to the right of the Target Digit (this is the Deciding Digit).
3. If the Deciding Digit is 5 or more (5, 6, 7, 8, 9), you round up the Target Digit.
4. If the Deciding Digit is less than 5 (0, 1, 2, 3, 4), the Target Digit stays the same.
5. After rounding, drop all the digits that come after the Target Digit.

Example: Round 14.7381 to 2 decimal places.

Target Digit (2nd DP): 3
Deciding Digit: 8
Since 8 is 5 or more, we round up the 3 to 4.
Result: 14.74

1.2 Rounding to Significant Figures (SF)

Rounding to significant figures is crucial for estimation because we often use 1 Significant Figure (1 SF) for calculations.

The biggest difference here is deciding which digits are "significant" (important) and which are just placeholders.

Key Rules for Significant Figures (SF)

To find the 1st significant figure, start counting from the leftmost digit that is NOT ZERO.

• Rule 1: Non-zero digits are always significant.

  In 45.9, all three digits (4, 5, 9) are significant.

• Rule 2: Zeros between non-zero digits (trapped zeros) are always significant.

  In 1007, the 1st SF is 1, and the two zeros are trapped, so the 4th SF is 7.

• Rule 3: Leading zeros (zeros at the very front of small decimals) are never significant. They are just placeholders.

  In 0.0025, the 1st SF is 2. The three leading zeros are ignored.

• Rule 4: Trailing zeros (zeros at the end of a number):
  • If the number is a whole number (e.g., 4000), these are placeholders and usually not significant unless specified.
  • If the number is a decimal (e.g., 4.50), the trailing zero is significant (it shows precision).

Step-by-Step Guide: Significant Figures

1. Locate the Target Digit (the 1st, 2nd, or 3rd SF). Remember, start counting from the first non-zero digit.
2. Look at the digit immediately to the right (the Deciding Digit).
3. Apply the standard rounding rule (5 or more: round up; less than 5: stay the same).
4. Crucial Step: Any digits *after* the Target Digit must be replaced by zeros if they are before the decimal point, to maintain the correct place value. If they are after the decimal point, drop them.

Example 1: Round 5764 correct to the nearest thousand (which is 1 significant figure).

1st SF is 5.
Deciding Digit is 7 (round up).
The 5 becomes 6. We must replace 7, 6, and 4 with zeros to keep the number in the thousands.
Result: 6000

Example 2: Round 0.04509 to 3 significant figures.

1st SF is 4. 2nd SF is 5. Target Digit (3rd SF) is 0.
Deciding Digit is 9 (round up).
The Target Digit 0 becomes 1.
Result: 0.0451 (The leading zeros are dropped, but the significant figures remain.)

Key Takeaway for Section 1: Rounding is about location and applying the 5-or-more rule. For whole numbers rounded to SF, remember to use zeros as placeholders so you don't accidentally change 5764 into 6!

Section 2: Estimation for Calculations

This is the core skill of C1.9.2: making estimates for calculations involving numbers, quantities, and measurements.

The standard, most reliable method for estimation in IGCSE is to round every single number in the calculation to 1 significant figure (1 SF) first. Then, perform the calculation using these simplified numbers.

Why 1 Significant Figure?

Using 1 SF makes multiplication and division incredibly simple because you are usually only multiplying single digits or powers of ten.

Step-by-Step Process for Estimation

1. Read the calculation (e.g., a fraction, multiplication, or combination).
2. Round every number in the expression to 1 Significant Figure.
3. Rewrite the simplified expression using the rounded numbers.
4. Calculate the result. This final result is your estimate.

Example: By writing each number correct to 1 significant figure, estimate the value of: $$ \frac{41.3}{9.79 \times 0.765} $$

Step 1 & 2: Rounding to 1 SF:
41.3 \(\approx\) 40 (1st SF is 4, 1 rounds down, use 0 as placeholder)
9.79 \(\approx\) 10 (1st SF is 9, 7 rounds up, use 0 as placeholder)
0.765 \(\approx\) 0.8 (1st SF is 7, 6 rounds up, keep 0 as placeholder, drop non-significant decimal trailing digits)

Step 3 & 4: Calculation:
$$ \text{Estimate} = \frac{40}{10 \times 0.8} $$ $$ = \frac{40}{8} $$ $$ = 5 $$

Analogy: The Quick Shopping Trip

Imagine you are at the supermarket and need to buy three items costing \$1.95, \$4.10, and \$7.85. You want a quick estimate of the total before going to the checkout.

• \$1.95 (rounds to 1 SF) \(\approx\) \$2
• \$4.10 (rounds to 1 SF) \(\approx\) \$4
• \$7.85 (rounds to 1 SF) \(\approx\) \$8

Estimated Total: \$2 + \$4 + \$8 = \$14.
(Actual total is \$13.90. Our estimate of \$14 is very close and much faster to calculate mentally!)

Key Takeaway for Section 2: For calculating estimates, the golden rule is: Round every number to 1 Significant Figure. This simplifies the arithmetic drastically.

Section 3: Rounding Answers to a Reasonable Degree of Accuracy

The final part of estimation (C1.9.3) asks you to round your final answers to a reasonable degree of accuracy in the context of a given problem. This is where common sense comes in!

When you calculate a complicated answer (especially using a calculator for non-exact values like area or volume), the display might show something long, like 13.784562.... You need to decide how to present this answer.

What is "Reasonable"?

The standard expectation in IGCSE Mathematics for non-exact numerical answers is to give them correct to 3 significant figures (3 SF), unless:

• The question specifically asks for a different accuracy (e.g., 2 DP, nearest whole number, or 4 SF).
• The context dictates a different accuracy (e.g., money or angles).

Contextual Rounding Rules:

1. If the answer relates to Money: Always give your answer to 2 decimal places (2 DP), which represents cents or pence (e.g., \$15.42).
2. If the answer relates to Angles (in degrees): Always give the answer to 1 decimal place (1 DP), unless otherwise instructed.
3. If the answer is general (length, area, volume, time): Use the standard 3 significant figures (3 SF).
4. If the number is huge: If the answer is 3,456,890, rounding to the nearest hundred thousand (2 SF: 3,500,000) might be more practical than 3 SF (3,460,000).

Example: A calculation yields the volume of a liquid as \(V = 57.3491 \text{ cm}^3\).

If the question does not specify the accuracy, you round to 3 SF.
57.3491... to 3 SF is 57.3 \(\text{cm}^3\). (The 4 rounds down).

Did You Know?

When using a calculator, always use the unrounded value from previous parts of the calculation, even if you wrote down an answer rounded to 3 SF. If you round too early, your final answer might not be accurate enough. The syllabus states: "avoid rounding figures until they have their final answer."

Key Takeaway for Section 3: When providing a final answer, use 3 SF unless the problem involves money (use 2 DP) or angles (use 1 DP), or if the context suggests a simpler rounding is more sensible.

Estimation Quick Review Checklist

Rounding to 1 SF (For Estimation in Calculations)

• Locate the first non-zero digit.
• Round it based on the next digit (5+ rounds up).
• Replace subsequent whole number digits with zeros (placeholders).

Estimation Checklist

1. Round ALL numbers in the calculation to 1 SF first.
2. Calculate using the simple rounded values.
3. If calculating the final answer for a problem, round to 3 SF (unless money or angles).

Keep practicing your rounding skills! Estimation is a powerful tool that helps you spot mistakes and understand the magnitude of numbers in the real world. You got this!