Rounding & Estimation | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Rounding & Estimation

 How do I round numbers to a given place value? 

• Identify the digit in the required place value and circle the number to the right to determine whether to round up or round off.
  • For example, to round 1294 to the nearest 100, locate the 2 digit and use the 9 to decide how to round .
• Find the two options that the number could round to. For example, the two nearest 100's to 1294 are 1200 and 1300.
  • Be cautious when the digit is 9, as rounding up may affect higher place values. For instance, the nearest 2 decimal places to 3.497 are 3.49 and 3.50.
• Rounding Rules: If the digit following the specified decimal place is 5 or greater, round up to the next higher number. If it is less than 5, round down to the lower number.
  • Adding Zeroes: When rounding, ensure that you include any necessary zero placeholders after the decimal point. For example, rounding 1297 to the nearest 100 gives 1300.
  • Rounding to Nearest 100th: Consider rounding 3.497 to two decimal places. Since the digit after the hundredth place is 5, we round up to get 3.50, maintaining exactly two decimal places in the answer.

How do I round to significant figures?

• Rounding to significant figures is akin to rounding to place value but with a focus on significant digits.
• To round to significant figures, identify the first significant figure and the relevant place value.
• For instance, the first significant figure of 3097 is 3, and for 0.006207, it is 6.
• Start with this number and count along to the right
  • You do count the following zeros, for example:
    • 0 is the second significant figure of 3.097
    • 9 is the third significant figure of 30.97
  • Use the normal rules for rounding:
    • Circle the number to the right
    • Use this to determine whether the given significant figure rounds up or rounds off

Why do we use estimation?

• We estimate to find approximations for difficult sums
• Or to check if our answers are about the correct size (right order of magnitude)

How do I estimate?

• We usually round numbers to a sensible value before performing calculations.
• General Rule
  • Round numbers to 1 significant figure: 
    • 7.8 → 8, 
    • 18 → 20, 
    • 3.65 × 10^-4 → 4 × 10^-4, 
    • 1080 → 1000
  • Exceptions
  • It can be more practical to round to a convenient value like 
    • 16.2 → 15, 
    • 9.1 → 10, 
    • 1180 → 1200

How do I know if I have underestimated or overestimated?

• Rounding for addition:
  • Rounding both numbers up leads to an overestimation
  • Rounding both numbers down leads to an underestimation
• Rounding for multiplication:
  • Rounding both numbers up results in an overestimation
  • Rounding both numbers down results in an underestimation
• Subtraction and division complexities:
  • For subtraction (a - b):
    • Increasing 'a' and/or decreasing 'b' leads to an overestimated result.
    • Decreasing 'a' and/or increasing 'b' leads to an underestimated result.
    • If both numbers are increased or decreased, it's challenging to determine if it's an overestimate or underestimate.
  • For division (a ÷ b):
    • Increasing 'a' and/or decreasing 'b' results in an overestimated answer.
    • Decreasing 'a' and/or increasing 'b' results in an underestimated answer.
    • If both numbers are increased or decreased, determining overestimation or underestimation becomes complex.