Decimals, Percentages, and Rounding | Year 9 Mathematics (Cambridge) PDF Download

Working with Decimals

Decimals are a way to represent fractions and mixed numbers. They use a decimal point to separate the whole number part from the fractional part.

• Reading Decimals: Read the number to the left of the decimal point as a whole number, then read the digits to the right of the decimal point individually.
  • Example: 3.456 is read as "three point four five six."
• Adding and Subtracting Decimals: Line up the decimal points and proceed as with whole numbers. If necessary, add zeros to ensure the numbers have the same number of decimal places.
  • Example: 2.34 + 1.456 = 2.340 + 1.456 = 3.796
• Multiplying Decimals: Multiply the numbers as if they were whole numbers. Count the total number of decimal places in both factors. Place the decimal point in the product so that it has the same number of decimal places.
  • Example: 1.2 × 3.4 = 12 × 34 = 40.8 (one decimal place from 1.2 + one decimal place from 3.4)
• Dividing Decimals: Move the decimal point in the divisor to make it a whole number, and move the decimal point in the dividend the same number of places. Divide as with whole numbers, then place the decimal point in the quotient.
  • Example: 4.5 ÷ 1.5 = 45 ÷ 15 = 3

Percentage Calculations

Percentages represent parts per hundred. They are used to express how large or small one quantity is relative to another quantity.

• Converting Between Percentages and Decimals: To convert a percentage to a decimal, divide by 100. To convert a decimal to a percentage, multiply by 100.
  • Example: 75% = 75 ÷ 100 = 0.75 and 0.5 = 0.5 × 100 = 50%
• Finding a Percentage of a Number: Multiply the number by the percentage (expressed as a decimal).
  • Example: 20% of 50 = 0.20 × 50 = 10
• Increasing or Decreasing by a Percentage: Calculate the percentage amount and add or subtract it from the original number.
  • Example: Increase 200 by 15% = 200 + (0.15 × 200) = 200 + 30 = 230
  • Example: Decrease 200 by 15% = 200 - (0.15 × 200) = 200 - 30 = 170
• Percentage Change: The percentage change between two values is found by dividing the difference between the values by the original value, then multiplying by 100.
  • Example: From 50 to 75: ((75 - 50) ÷ 50) × 100 = (25 ÷ 50) × 100 = 50%

Rounding and Approximation

Rounding numbers makes them simpler and easier to work with, though they become less precise.

• Rounding to a Certain Decimal Place: Look at the digit one place to the right of your desired decimal place. If it is 5 or greater, round up; if it is less than 5, round down.
  • Example: Round 3.456 to two decimal places. Look at the third decimal place (6). Since it is greater than 5, round up: 3.46
• Rounding to the Nearest Whole Number: Look at the first decimal place. If it is 5 or greater, round up; if it is less than 5, round down.
  • Example: Round 4.7 to the nearest whole number. Since 7 is greater than 5, round up: 5
• Estimating Sums and Differences: Round the numbers to the nearest ten, hundred, etc., then perform the addition or subtraction.
  • Example: Estimate 47 + 52. Round to nearest ten: 50 + 50 = 100
• Using Significant Figures: Round to a certain number of significant figures to reflect precision in measurements.
  • Example: Round 0.00456 to two significant figures. The first two significant figures are 4 and 5, so the result is 0.0046.

Examples

• Adding Decimals: 12.345 + 7.89 = 12.345 + 7.890 = 20.235
• Percentage of a Number: 30% of 200 = 0.30 × 200 = 60
• Rounding to the Nearest Whole Number: 9.82 rounded to the nearest whole number is 10.
• Estimating a Difference: Estimate 234 - 198 by rounding to nearest ten: 230 - 200 = 30.