Number and Calculation | Year 9 Mathematics (Cambridge) PDF Download

Fractions, Decimals, and Percentages

Converting Fractions to Decimals

• To convert a fraction to a decimal, divide the numerator (top number) by the denominator (bottom number).
• Example: Convert 3/4  to a decimal: 3 ÷ 4 = 0.75

Converting Decimals to Percentages

• To convert a decimal to a percentage, multiply by 100 and add the percentage symbol (%).
• Example: Convert 0.75 to a percentage: 0.75 × 100 = 75%

Converting Percentages to Decimals

• To convert a percentage to a decimal, divide by 100.
• Example: Convert 75% to a decimal: 75 ÷ 100 = 0.75

Converting Fractions to Percentages

• To convert a fraction to a percentage, first convert the fraction to a decimal, then convert the decimal to a percentage.
• Example: Convert 3/5 to a percentage: 3/5 = 0.6
  0.6 × 100 = 60%

Converting Percentages to Fractions

• To convert a percentage to a fraction, write the percentage as a fraction with 100 as the denominator and simplify if possible.
• Example: Convert 50% to a fraction: 50/100 = 1/2

Standard Form and Rounding

Standard Form (Scientific Notation)

• Standard form is a way of writing very large or very small numbers using powers of 10.
• Example: 
  • Write 4500 in standard form: 4.5 × 10³
  •  Write 0.007 in standard form: 7 × 10⁻³

Rounding to a Given Number of Decimal Places

To round a number to a specific number of decimal places, look at the digit immediately after the last required decimal place:

• If it's 5 or more, round up.
• If it's less than 5, round down.

Example: 

• Round 3.456 to 2 decimal places: 3.456 ≈ 3.46
• Round 8.742 to 1 decimal place: 8.742 ≈ 8.7

Rounding to Significant Figures

Significant figures are used to express the precision of a number. The first non-zero digit is the first significant figure.
Example:

• Round 0.004567 to 2 significant figures: 0.004567 ≈ 0.0046
• Round 12345 to 3 significant figures: 12345 ≈ 12300

Mental Strategies and Estimation

Estimation Techniques

Estimation involves making a rough calculation or judgement of a value. This is useful for checking the reasonableness of answers.

Front-end Estimation

• Focus on the most significant digits.
• Example: Estimate the sum of 47.3 and 62.8: 47 + 63 ≈ 110

Rounding for Estimation

• Round numbers to the nearest ten, hundred, etc., and then perform the operation.
• Example: Estimate the product of 23 and 57: 20 × 60 = 1200

Mental Arithmetic Strategies

Developing mental arithmetic skills can help solve problems quickly without a calculator.

Breaking Down Numbers

Break numbers into parts to simplify calculations.
Example:

• Calculate 56 + 47 mentally:
  (50 + 6) + (40 + 7)
  50 + 40 = 90
  6 + 7 = 13
  90 + 13 = 103